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6400fdfc5e ... split-medial-pigeonhole-programme

Author SHA1 Message Date
didericis d3fc4bfc4c Split medial pigeonhole programme into its own paper 
Move Section 5 of "Medial Tire Decompositions of Plane Triangulations"
into a new standalone paper, "The Medial Pigeonhole Programme", which
cites the medial tire paper for its terminology and notation. Convert
the three cross-references that pointed into earlier sections (annular
teeth, bite-face-count, boundary medial vertices) into citations.

Remove Section 5 from the medial tire paper and update its abstract to
drop the moved chain-pigeonhole claim, pointing to the follow-up paper.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-14 21:08:06 -04:00
didericis b2439e4bac Walkthrough: show only the medial graph in panels C and D 
Drop the faint base-graph (G') edges and the dotted restored base edge from the
medial panels, leaving just the medial graph (medial vertices at edge midpoints,
medial edges, colours, halos, and the restored-diagonal medial square). Panels A
and B still show the triangulation.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-13 00:20:25 -04:00
didericis 20e2cc94b4 Fix walkthrough figure to use verified planar embeddings 
The walkthrough previously used a concentric layout whose outer-triangle->ring
spokes can cross -- not a valid plane embedding. Rebuild draw_walkthrough.py on
networkx planar_layout with an explicit crossing check: G, G', and the medial
M(G') drawn at edge midpoints are each verified crossing-free before rendering.
G' is embedded once and reused for panels B/C/D; G reuses it when still planar.

The medial-at-midpoints drawing is planar except for the medial triangle of the
geometric outer face (its midpoint-chords would cut across the unbounded region),
so those three edges are detected via the convex hull and omitted; the remainder
is verified crossing-free. Note updated to describe the embedding.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-13 00:16:03 -04:00
didericis e7e8536559 Draw the minimal failing graph with a verified planar embedding 
draw_failing_graph.py renders seed2 #26 (ring [3,6,3]+face leaf, 12 vertices),
the smallest graph the programme fails on after exhausting sites x tread-phases x
root colour-orders. Uses networkx planar_layout for a straight-line embedding and
verifies no two non-incident edges cross before drawing. Panel A: plain embedding;
panel B: BFS levels with the odd level-2 seam (the inner triangle 9-11-10) bold,
the terminal leaf face shaded -- the face-leaf/gadget spot where removal fails.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-13 00:09:42 -04:00
didericis faf9e01139 Enumerate colour/tread phase over the residue graphs 
residue_phase_sweep.py exhaustively enumerates the two colouring control knobs
-- the per-annulus tread phase {0,1}^A and the root-DFS colour order perms(0,1,2)
-- on top of every insertion-site combo, for the graphs the random-phase site
sweep still fails. canonical_coloring_explicit makes this deterministic.

Result (residue_phase_sweep_results.txt): the two hub graphs are RESCUED once
phase is enumerated rather than sampled (so the random-phase fail count overstates
difficulty); the genuine obstructions that survive sites x phases x colour-orders
are exactly the face-leaf graphs (terminal-triangle leaf gadget). Smallest is
seed2 #26 [3,6,3] face (1 combo, 24 settings, all fail at gadget-removal) -- a
minimal obstruction target. Caveat: try_establish is a bounded local Kempe search,
so STILL FAILS means unreachable by the bounded search from canonical-even over
all knob settings, not that no Kempe path exists.

Findings note updated.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-13 00:03:49 -04:00
didericis 9d296eb9c8 Add worked walkthrough; factor explicit phase/colorder colouring 
Refactor canonical_coloring into coloring_skeleton (phase-independent parts) +
canonical_coloring_explicit (explicit phases + DFS colour order) + a random-phase
wrapper for back-compat. This exposes the two control knobs deterministically so
they can be enumerated rather than only sampled.

Add a fully worked example on the smallest clean graph (ring [3,5]+hub, 9
vertices, one odd seam, no gadgets): even_program_walkthrough.md traces all six
stages -- generate G with embedding, pick source + BFS levels, choose the diamond
site that evens the level-5 seam, build M(G'), the canonical colouring (seam
mono-3, hub annulus alternates, root by DFS), and a real {1,2}-Kempe switch that
makes the diamond quad reducible. dump_walkthrough.py reproduces every number;
draw_walkthrough.py renders the 4-panel figure even_program_walkthrough.png.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-12 23:44:29 -04:00
didericis 2ff712b994 Sweep all diamond insertion sites; report first-match vs full sweep 
run_graph no longer takes the first admissible seam edge per odd seam. It now
enumerates every valid diamond site per odd seam (_candidate_sites), sweeps the
full Cartesian product (capped by --max-combos), runs <=4 colour phases per
combination, and counts a graph ok iff SOME placement fully descends. Reports
both the old first-match tally and the swept tally, plus design-space stats and
how many graphs the sweep rescued.

Finding: most "fail:diamond-switch" cases were heuristic, not intrinsic. The
old 39/60 was the first-match heuristic (one point in the design space, and
seed-sensitive 31-39). Sweeping insertion sites rescues ~20 of ~24 first-match
failures:

  seed 1:  first-match 31 ok / 29 fail  ->  sweep 54 ok / 6 fail  (rescued 23)
  seed 2:  first-match 36 ok / 24 fail  ->  sweep 57 ok / 3 fail  (rescued 21)

Only ~3-6 fail:diamond-switch survive the full site sweep -- those are the real
obstruction targets for the joint {1,3}-cycle bipartiteness solver. The colour/
tread phase is still only randomized over 4 attempts, not enumerated.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-12 23:08:17 -04:00
didericis c6e2c3e1a5 Add even-level-cycle colouring program harness 
Constructive route: surger G so every level cycle is even (two-vertex leaf gadget
on terminal triangles -> 4-wheel, no defect; diamond on odd internal seams), take
the canonical even colouring of M(G') (no 4CT used), Kempe-remove the planted
degree-4/3 vertices to reach a proper 3-colouring of M(G).

Pipeline runs end to end on synthetic ring triangulations: surgery, canonical
colouring, and gadget removal all work; the program lands on the CYCLE LAYER
(39/60 ok, rest fail:diamond-switch). Diagnostic: a descendable colouring always
EXISTS (M(G) is 3-colourable), so failures are Kempe-reachability from the
canonical even colouring, not non-existence -- the entire difficulty is localised
there. Greedy per-diamond switching is insufficient because diamonds share vertical
{1,3}-Kempe cycles; the principled solve is joint (bipartiteness of the diamond /
side-cycle constraint graph), which is the identified next step. Includes the leaf
gadget figure and a findings note.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-12 22:36:15 -04:00
didericis d547076cba Verify chain-pigeonhole exhaustively for n<=14 via R_T composition fixpoint 
Add kempe_rt_composition_probe.py: Ext(T) = boundary necklaces realisable on a
subtree's outer seam by a compatible Kempe-balanced selection; monotone maps over
minimal-antichain families decide whether empty Ext is reachable. Modeling facts
established: the seam is exactly the singleton down apexes (bite apexes have parent
faces on both sides, hence parent-internal); necklace states are exact because a
child attaches with free dihedral placement (dihedral-closed sequence sets).

Result over all no-length-3-boundary tiles n<=14 (7750 tiles, 1966 distinct
relations, 149 leaf, 27 branching): empty Ext is NOT reachable — every assemblable
tree admits a compatible selection, verifying the chain-pigeonhole conjecture
exhaustively for tire trees with treads n<=14 and no separating triangles. The
fixpoint saturates in 2 rounds: restriction does not accumulate along chains.
Tightest subtree pins a size-5 seam to the single necklace 00012; every smallest
minimal Ext contains the blocky/regular state. Relations cached (~6MB) for cheap
extension to larger n. Caveat: terminal facial-triangle leaves not yet modeled.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-12 02:34:27 -04:00
didericis 28d3d55b92 Refute the regular uniform seam family at n=15 
Add kempe_regular_family_test.py (fixed family, per-tile early-exit, --branch-only).
Threads 614/614 at n=12 but FAILS at n=15 on two classes of no-separating-triangle
tiles: non-branching large-even-outer + odd inner (UUUUUUDUDUDUDUD, p=10, face 5) and
branching odd-outer + two even inner faces (UDUDUDDUDUDDDDD bite=(5,12), p=5, [4,4];
11/1022 branching fail). This is the R_T coupling (not a product) biting at scale: the
uniform family sets outer/inner states independently per size. The shortcut was
stronger than the chain-pigeonhole conjecture (which allows per-interface freedom), so
its failure costs a constructive route, not the conjecture; pairwise overlap still
holds. Next line: per-interface R_T composition respecting coupling.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-12 01:57:13 -04:00
didericis 8b47af6036 n=14 branching case feasible with one regular uniform seam family 
Full uniform-family CSP at n=14 --no-tri (4403 tiles, 193 branching) is FEASIBLE:
one family threads every tile incl. branching nodes (outer rim + both inner faces
at once). Independent candidate test threads 193/193 branching tiles. Witness is
fully regular: sigma_m = 0^m if m even (monochromatic), 0^(m-2)12 if m odd. So on
the 4CT-relevant class the chained pigeonhole is constructively resolved throughout
the tested range (n=9,12,14, incl. branching).

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-12 00:54:48 -04:00
didericis 2b016bc1ca Find smallest n admitting branching tiles (n=11 unrestricted, n=14 no-tri) 
Add kempe_branching_min_probe.py (structural: >=2 inner faces with singletons).
Unrestricted branching first appears at n=11; no-separating-triangle branching
(>=2 inner faces each >=4 singletons, p>=4) first appears at n=14 (193 tiles).
Smallest example: word=UUUUDDDDDDDDDD bite=(8,13), p=4, faces root{4,5,6,7} and
bite{9,10,11,12}. n=14 is the smallest place to test the uniform family / R_T
composition on a genuine branching no-separating-triangle tile.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-12 00:31:19 -04:00
didericis bacbdaaf26 No-separating-triangle restriction removes the chained-seam obstruction 
Add --no-tri filter (exclude tiles with a length-3 boundary = separating/non-facial
triangle in G: outer rim of 3 up teeth, or an inner face of exactly 3 singleton
downs) to the trend and uniform-family probes.

The n=12 breaker UUUDUDUDUDUD bite=(3,11) has a size-3 inner face (encloses d5,d7,d9)
and is excluded. With the restriction the size-7 universal at n=12 is restored
(|D[7]| 0->2), every |D[m]|>=1 across n=6..13, and the uniform-family CSP becomes
FEASIBLE at n=12 with the simplest family (monochromatic on even sizes, min-cut on
odd). So the only universal failure was an artifact of admitting non-4-connected
configs; on the 4CT-relevant class gluing is constructively trivial in range.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-12 00:21:26 -04:00
didericis 1aa76a5226 Plot the n=12 size-7 universal breaker tile 
Add plot_breaker_tile.py and figure for word=UUUDUDUDUDUD bite=(3,11): structure
(7 up teeth = size-7 outer rim, bite (3,11), singleton downs d5,d7,d9) plus a
Kempe-balanced colouring. Reconfirms the outer rim realises 9/10 admissible size-7
necklaces, never 0001112 -- the lone tile that empties the size-7 universal at n=12.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 23:51:36 -04:00
didericis a724a50344 Track odd-size universal trend; the n=12 failure is sporadic, not a trend 
Add kempe_universal_trend_probe.py (|D[m]| per size across n). Across n=6..13 and
all sizes, the ONLY empty per-size universal is (n=12, m=7): at n=13 size 7 is back
to |D|=2 with more boundaries (579), so the vanishing is sporadic, not monotone.
The lone n=12 breaker is the outer rim of word=UUUDUDUDUDUD bite=(3,11) (most-
alternating 7-up word, antipodal bite), realising 9/10 size-7 necklaces and missing
only 0001112. Correct the earlier "doomed at scale" reading in the findings note:
the uniform shortcut almost always works (near-total coverage) but is fragile to a
single exceptional tile; pairwise gluability still always holds.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 23:40:48 -04:00
didericis b1100b41d9 Add chained-seam findings note (medial pigeonhole) 
Write up the R_T coupling, the uniform-family result (feasible n=9, infeasible
n=12 via empty size-7 universal, 0001112 blocked 210/211), and the open threads.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 23:22:27 -04:00
didericis b656b6aed3 Add transfer-relation & uniform-family probes (chained-seam / pigeonhole) 
Pursue the paper's medial pigeonhole programme (R_T restriction relation,
chain-pigeonhole conjecture) at the data level.

Findings: R_T (outer<->inner boundary necklace, one Kempe-balanced colouring)
is genuinely coupled, not a product of its projections. A uniform per-size
boundary-state family threading every tile EXISTS at n=9 (unique per size, the
balanced-block necklaces 0011/000011/012/00012 -- not monochromatic), but FAILS
at n=12: size-7 seams admit no universal state (|D[7]|=0; near-universal 0001112
realised on 210/211 boundaries, blocked by one tile). So the uniform "same state
everywhere" shortcut breaks once large odd seams appear and universals vanish as
the tile population grows; the per-interface pigeonhole choice is genuinely
needed. Pairwise gluability still holds, so this locates the conjecture's
difficulty rather than obstructing gluing.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 23:16:27 -04:00
didericis aecbc5ed28 Add tile-overlap probe: per-tile interface subsets always glue 
Each tile realises only a subset of the parity-admissible alphabet on its rim,
and tiles genuinely omit interfaces (n=12 m=8: max 273/274, min 43). But any
two tiles always glue: interface subsets always overlap (n=9 m=3-6, n=12 m=3-8)
-- usually via a global universal seam present on every inner+outer rim, and
where none exists (n=12 m=7) the worst pair still shares 14 seams. The universal
seams are the low-complexity ones (<=2 colours, single contiguous block). No
local gluing obstruction; any obstruction must be global across a nested stack.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 22:52:38 -04:00
didericis c56da7bb23 Add interface-admissibility probe; confirm parity characterization at n=12 
For each interface size m, compare the realized census vocabulary (outer
up-tooth apexes and inner singleton-down apexes) against the full
parity-admissible set. At n=12, m=3..8 every parity-admissible sequence is
realized on both faces (counts 1,4,10,31,91,274; none missing), and up==down
throughout -- the n=9 result is n-independent and scales to m=8. Validated
against the known n=9 answer before running n=12.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 22:47:46 -04:00
didericis d094a310d8 Read up/down apex sequences off the un-deduped census 
The anchored single-representative reading interacted with dihedral graph
dedup to record an arbitrary orientation of each necklace, producing a
spurious up-vs-down split at n=9,m=6 (001212 only up, 010122 only down --
the same necklace). Add dihedral_reading_sequences(), which unions the
canonical reading over all 2n dihedral anchors and exactly reproduces the
brute un-deduped census; make it the default for both experiments, with
--anchored to recover the old behaviour. Document the artifact and fix in
kempe_sequence_orientation_note.md.

Regenerate up + down for n=9, m=3..6. Up and down now agree on sequences
and groupings at every m (m=6: identical 31 sequences, 6 groups; the
001212/010122 pair appears on both sides). Groupings coarsen vs anchored
(m=4: 3 groups; m=5: 2 groups) since the orientation-honest vocabulary
merges previously split sequence-sets.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 22:25:44 -04:00
didericis d8b5975f81 Add inner-face down-apex colour-sequence experiment (n=9 sweep) 
Mirror of the up-tooth experiment with the distinguished valid face moved
from the outer face to an inner non-tooth face (root or bite inner-gap).
For each (M(T), inner face) config holding m singleton down-tooth apexes,
record the apex colour sequence (cyclic order, mod colour permutation) over
Kempe-balanced colourings and group configs by their sequence-set. Runs for
m=3,4,5,6 with per-sequence notes, figures, and a config atlas.

Finding: inner faces realise the same parity-admissible sequence vocabulary
and the same distinct-sequence counts (1/4/10/28) as the outer face, i.e.
the Kempe-parity law acts uniformly on every valid face. At m=6 the configs
are the U<->D embedding mirror of the up-m=6 graphs (matching 7 configs,
28 sequences, 127 colourings).

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 21:19:35 -04:00
didericis d93e8d137a Add up-tooth apex colour-sequence experiment over n=9 up-teeth sweep 
Enumerate Kempe-balanced 3-colourings of every M(T) with |A(T)|=9 and a
fixed number m of up teeth, record the up-tooth apex colour sequence
(cyclic order, mod colour permutation only), and group the M(T) by their
set of unique sequences. Runs for m=3,4,5,6 with per-sequence notes and
figures plus a summary atlas.

Finding: realised sequences obey outer-face Kempe parity (all three
colour-counts share m's parity). Distinct sequences grow 1/4/10/28 while
M(T) count falls 23/29/18/7 across m=3..6.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 21:04:59 -04:00
didericis a4b3a6fb50 Draw per-graph Realized/Unrealized/Invalid colouring notes 
Add draw_tire_realization.py: for each full medial tire graph from the seed-1
analysis, draw every proper 3-colouring (mod colour permutation) in a grid,
each panel coloured by its three colour classes and banner-labelled Realized /
Unrealized / Invalid, and write one standalone note per graph (plus a README
index).  Refactor tire_realization_analysis to expose iter_pieces() yielding
per-piece coloured colourings.

Output: tire_realization_seed1/ with 17 piece notes + figures.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 17:58:41 -04:00
didericis dacef25cbb Add Realized/Unrealized/Invalid tire-colouring analysis 
For a random 12-vertex maximal planar graph (sphere convex hull), enumerate
all proper 3-colourings of M(G), take the BFS-level (tire-tree) decomposition
from every source vertex, and build each full medial tire graph M(T) in the
ambient tread-face model (cycle + teeth + bites).  Recognise each M(T) as a
FullMedialTireGraph and label every proper 3-colouring Realized (Kempe-balanced
and a restriction of a global colouring), Unrealized (balanced but not a
restriction), or Invalid (not balanced).

Findings on seed 1 (17 pieces, M(G) with 90 colourings): zero realized-but-
invalid colourings (confirms Remark 5.8 on a real triangulation), and 12 of 17
pieces carry Unrealized colourings -- Kempe-balance is necessary but not
sufficient for realization; it is sufficient only on cap-like all-up/shallow
treads.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 17:36:34 -04:00
didericis cf035243f6 Verify Remark 5.8 on genuine bite treads 
Bites arise when the inner outerplanar graph O has a bridge: the bridge
edge is traversed twice by the outer-face walk, so its medial vertex is
adjacent to four annular vertices.

- check_remark58_bite.py: a minimal bite tread (outer 4-cycle + interior
  bridge u-w) restricts to Kempe-balanced on all colourings (outer face).
- check_remark58_bite_rich.py: O = triangle abc + pendant bridge a-d gives
  one bite plus three singleton down teeth in the bite's inner-gap face;
  every restriction is Kempe-balanced (the three gap singletons are a
  rainbow in every global colouring).

Update Remark 5.8's verification note: the bite case, including singletons
in the bite-gap face, is now confirmed.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 16:46:53 -04:00
didericis 5bed8b4dfb Verify Remark 5.8 mechanism; correct it to level-cycle conservation 
Computational checks of the necessity of Kempe-balance (Remark 5.8):

- check_medial_face_parity.py shows the naive "even P-coloured vertices
  per medial face" claim is false (odd vertex-faces on the octahedron and
  stacked triangulations), so the original face-parity justification was
  wrong.
- check_remark58_bitefree.py builds genuine bite-free tire pieces (capped
  triangulated annuli) and confirms every proper 3-colouring of M(G)
  restricts to a Kempe-balanced colouring (|A(T)|=6,8,10,12, all
  colourings, zero failures).

Rewrite Remark 5.8 to cite the correct mechanism: the up/down apexes lie
on level cycles, and a P-Kempe cycle meets each level cycle in an even
number of P-coloured incidences (Lemma 5.6).  Note the bite case is not
yet checked end to end.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 16:33:00 -04:00
didericis 79cbca8e00 Add Kempe-balanced colouring definition and validity classifier 
Define Kempe-balanced colourings of a full medial tire graph (Def 5.7):
for each valid face (outer face or interior non-tooth face of B(T)) and
each colour pair {a,b}, the number of tooth apexes incident to the face
coloured a or b must be even.  Add Remark 5.8 (necessity: a colouring of
M(T) extends to M(G) only if it is Kempe-balanced) and rename Lemma 5.5
to "Kempe chains are cycles".

Add kempe_valid_colorings.py: enumerate all proper 3-colourings of a full
medial tire graph, label each Kempe-balanced/valid or invalid, and plot
them with the offending face's Kempe chains and odd apex set highlighted
on invalid panels.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 16:00:10 -04:00
didericis 8cc94fb6b9 Add full medial tire graph generator and n=9 atlas 
Name A(T) the "annular cycle" (Thm 3.3, Def 3.4); clarify the bite-face
condition in Remark 3.8 to count down-tooth apexes interior to each face;
add the non-incidence stipulation for bite edges to Def 3.7.

Add an exhaustive generator over |A(T)| enforcing the 3.1-3.9 properties
(tooth word, non-crossing non-incident bites, >=3 up teeth, bite-face
condition), plus a plotting script and the n=9 atlas (81 dihedral classes).

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-11 12:23:57 -04:00
didericis 4062e87c61 Add figures, Kempe-cycle section, and restriction experiments 
Adds two TikZ figures (boundary-state worst cases and annular cycle
counterexample), a new subsection on Kempe-cycle conservation across
medial tires, and the experiment scripts/findings for the medial tire
restriction search and annular cycle condition check.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
 2026-06-11 01:16:05 -04:00
didericis 20fe6c24ca Add medial tire decomposition paper 2026-06-08 15:34:53 -04:00
404 changed files with 22535 additions and 119 deletions
Expand all files Collapse all files
README.md
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@@ -16,6 +16,7 @@ All papers are at `papers/<name>/paper.tex`. The current set:
| `iterated_reduction_in_reduced_dual` | An Iterated Reduction in the Reduced Dual |
| `level_resolutions_of_maximal_planar_graphs` | Level Resolutions of Maximal Planar Graphs |
| `level_switching` | Level Switching |
| `medial_tire_decompositions_of_plane_triangulations` | Medial Tire Decompositions of Plane Triangulations |
| `nested_tire_decompositions_of_plane_triangulations` | Nested Tire Decompositions of Plane Triangulations |
| `plane_depth` | Plane Depth |
| `plane_depth_sequencing` | Plane Depth Sequencing |
papers/medial_pigeonhole_programme/paper.aux
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@@ -0,0 +1,28 @@
\relax
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-nested-tire-decompositions}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A medial pigeonhole programme}}{1}{}\protected@file@percent }
\newlabel{def:medial-boundary-state}{{2.1}{2}}
\newlabel{conj:medial-chain-pigeonhole}{{2.2}{2}}
\newlabel{conj:medial-route-fct}{{2.3}{2}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Kempe-cycle conservation across medial tires}}{2}{}\protected@file@percent }
\newlabel{lem:kempe-cycles}{{3.1}{2}}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\newlabel{lem:kempe-conservation}{{3.2}{3}}
\newlabel{def:kempe-balanced}{{3.3}{3}}
\newlabel{rem:kempe-balance-necessary}{{3.4}{3}}
\bibcite{bauerfeld-medial-tire}{1}
\bibcite{bauerfeld-nested-tire-decompositions}{2}
\bibcite{tait-original}{3}
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%% filename: amsart-template.tex
%% American Mathematical Society
%% AMS-LaTeX v.2 template for use with amsart
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\documentclass{amsart}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
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\begin{document}
\title{The Medial Pigeonhole Programme}
% author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}
\subjclass[2010]{Primary }
\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Kempe chain, Four Colour Theorem}
\date{}
\dedicatory{}
\begin{abstract}
Building on the medial tire decomposition of a plane triangulation, we
formulate a pigeonhole programme for the Four Colour Theorem in medial
terms. Each tire carries a boundary-state restriction relation, and a
proper vertex $3$-colouring of the full medial graph is a compatible
selection of these boundary states across the tire tree. We state a
chain-pigeonhole conjecture asserting that the restriction relations
cannot remain mutually disjoint along every branch, and we refine the
boundary states by recording how two-colour Kempe cycles are routed
through each annular tire region. This yields a Kempe-enhanced
restriction relation and a notion of Kempe-compatible gluing along level
cycles.
\end{abstract}
\maketitle
\section{Introduction}
This paper continues the medial tire programme begun
in~\cite{bauerfeld-medial-tire}. We use freely the terminology and
notation introduced there. For a plane triangulation $G$ with fixed
embedding, $M(G)$ denotes the full medial graph, and the tire-tree
decomposition $\mathcal{T}(G,S)$ at a level source $S$
of~\cite{bauerfeld-nested-tire-decompositions} induces a decomposition
of $M(G)$ into full medial tire graphs $\mathsf{M}(T)$, one for each
tread $T$, glued along their boundary medial vertex sets
$\partial_{\mathrm{out}}\mathsf{M}(T)$ and
$\partial_{\mathrm{in}}\mathsf{M}(T)$. We also use the annular medial
cycle $A(T)$, its up and down teeth and their apexes, the bites and the
auxiliary plane graph $B(T)$, and the medial tire restriction relation
$R_T$ of~\cite{bauerfeld-medial-tire}.
By the Tait--medial correspondence of~\cite{bauerfeld-medial-tire},
proper vertex $3$-colourings of $M(G)$ are in natural bijection with
proper $3$-edge-colourings of the cubic planar dual $G^*$. Thus the
Four Colour Theorem is the assertion that the full medial graph of every
plane triangulation is properly vertex $3$-colourable, and the medial
tire decomposition turns this into a question about how local boundary
colourings compose across the tire tree.
\section{A medial pigeonhole programme}
The restriction relation $R_T$ records exactly the local information
needed to pass a medial $3$-colouring through a tire. In a nested
chain
\[
T_0 \supset T_1 \supset \cdots \supset T_k,
\]
the outer boundary state of $T_{i+1}$ must match an inner boundary
state allowed by $R_{T_i}$. Thus a proof of the Four Colour Theorem in
this framework would follow from a structural reason that these
restriction sets cannot remain mutually disjoint along every branch of
the tire tree.
\begin{definition}[Medial boundary state]
\label{def:medial-boundary-state}
A \emph{medial boundary state} on a boundary set
$\partial\mathsf{M}(T)$ is a proper vertex $3$-colouring of the
subgraph induced by that boundary set, considered up to permutation of
the three colours and the dihedral symmetries of the boundary walk
when that boundary is a cycle.
\end{definition}
\begin{conjecture}[Medial chain-pigeonhole principle]
\label{conj:medial-chain-pigeonhole}
There is a function $N(k)$ such that the following holds. Let
$T_0 \supset T_1 \supset \cdots \supset T_{N(k)}$ be a nested chain of
tire treads whose relevant boundary medial walks have length at most
$k$. Then two adjacent restriction relations in the chain have
compatible medial boundary states after colour permutation and boundary
symmetry. Equivalently, the chain contains a local gluing step that
cannot be obstructed by disjoint proper vertex $3$-colouring
restrictions.
\end{conjecture}
\begin{conjecture}[Medial tire route to the Four Colour Theorem]
\label{conj:medial-route-fct}
For every plane triangulation $G$ and every level source $S$, the
restriction relations $\{R_T : T \in V(\mathcal{T}(G,S))\}$ admit a
compatible selection of boundary states across the tire tree. Hence
$M(G)$ is properly vertex $3$-colourable, $G^*$ is properly
$3$-edge-colourable, and $G$ is properly $4$-vertex-colourable.
\end{conjecture}
\begin{remark}
Conjecture~\ref{conj:medial-route-fct} is equivalent in strength to
the Four Colour Theorem when combined with Tait's correspondence. The
point of the formulation is not to weaken the target theorem, but to
move the obstruction into finite boundary-state restrictions carried by
annular medial tire pieces.
\end{remark}
\section{Kempe-cycle conservation across medial tires}
We now record an additional structure carried by proper
$3$-colourings of medial graphs. This structure will be useful for
describing how colourings glue across level cycles.
Let $G$ be a plane triangulation and let $M=M(G)$ be its medial graph.
Let
\[
\varphi:V(M)\to\{1,2,3\}
\]
be a proper $3$-colouring of $M$. For a two-element colour set
$P=\{a,b\}\subseteq\{1,2,3\}$, let $M_P$ denote the subgraph of $M$
induced by the vertices of colours $a$ and $b$.
Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of
$M_P$ has degree $2$ in $M_P$. Hence every component of $M_P$ is a
cycle. We call these components the $P$-Kempe cycles of $\varphi$.
\begin{lemma}[Kempe chains are cycles]
\label{lem:kempe-cycles}
Let $G$ be a plane triangulation, let $M=M(G)$, and let
$\varphi$ be a proper $3$-colouring of $M$. For each
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle.
\end{lemma}
\begin{proof}
Let $v\in V(M_P)$. In the medial graph $M$, the vertex $v$ has degree
$4$. Since $\varphi$ is a proper $3$-colouring, none of the neighbours
of $v$ has colour $\varphi(v)$. Thus all four neighbours of $v$ have
one of the two colours different from $\varphi(v)$.
In the medial graph of a plane triangulation, the neighbours of a
medial vertex occur in two opposite pairs corresponding to the two
faces incident with the corresponding edge of $G$. Around each such
triangular face, the three medial vertices receive all three colours.
Consequently, at $v$ there are exactly two neighbours of each colour
different from $\varphi(v)$. It follows that, in the subgraph induced
by any two colours $P$, every vertex has degree $2$. Hence each
component of $M_P$ is a cycle.
\end{proof}
Let $T$ be a medial tire region. We regard $T$ as an annular transition
region whose boundary consists of one outer level cycle and finitely
many inner level cycles:
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
Here $C_0$ is the outer level cycle of $T$, and the cycles
$C_1,\ldots,C_m$ are the inner level cycles. Each inner level cycle
$C_i$ is also the outer level cycle of the corresponding child region
in the tire tree.
The following lemma is the basic conservation principle.
\begin{lemma}[Kempe-cycle conservation across level cycles]
\label{lem:kempe-conservation}
Let $C$ be a level cycle of $M$ separating a parent side from a child
side. Let $K$ be a $P$-Kempe cycle for some
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$. Then $K$ cannot enter the child side
of $C$ without also leaving it.
Equivalently, the incidences of $K$ with $C$ are paired by the
components of $K$ lying on the child side of $C$, and also paired by the
components of $K$ lying on the parent side of $C$.
\end{lemma}
\begin{proof}
By the preceding lemma, $K$ is a cycle. The level cycle $C$ separates
the sphere into two closed regions, which we call the parent side and
the child side. Consider the intersection of $K$ with one of these
regions. Since $K$ is a cycle, no component of this intersection can
have exactly one boundary endpoint on $C$. Each component is either
closed within the region, or is a path with two boundary endpoints on
$C$. Thus every entrance through $C$ is paired with an exit through
$C$.
\end{proof}
We now use these Kempe cycles to single out the colourings of a full
medial tire graph that respect the annular tooth structure.
\begin{definition}[Kempe-balanced colouring]
\label{def:kempe-balanced}
Let $\varphi$ be a proper $3$-colouring of the full medial tire graph
$\mathsf{M}(T)$. For a colour pair $P=\{a,b\}$, let $\mathsf{M}(T)_P$ be
the subgraph induced by the vertices of colours $a$ and $b$. Since
$\mathsf{M}(T)$ need not be $4$-regular, the components of
$\mathsf{M}(T)_P$ are paths or cycles; we call them the $P$-\emph{Kempe
chains} of $\varphi$. Every vertex of colour $a$ or $b$ lies on exactly
one $P$-Kempe chain.
A \emph{valid face} is the outer face of $\mathsf{M}(T)$, or an interior
face of $B(T)$ that is not a tooth---namely the root face or a bite
inner-gap face in the sense of~\cite{bauerfeld-medial-tire}. The
\emph{tooth apexes incident to} a valid face $F$ are:
\begin{itemize}
\item the up-tooth apexes (\cite{bauerfeld-medial-tire}), when
$F$ is the outer face;
\item the singleton down-tooth apexes whose annular edge lies on $F$,
when $F$ is interior---the apex on annular edge $m$ being incident to
the innermost bite $(i,j)$ with $i<m<j$, or to the root face if there
is none.
\end{itemize}
Bite apexes are never incident to a valid face in this sense.
For a colour pair $P=\{a,b\}$ write $\nu_P(F)$ for the number of tooth
apexes incident to $F$ that are coloured $a$ or $b$---equivalently, that
lie on a $P$-Kempe chain. The colouring $\varphi$ is
\emph{Kempe-balanced} if $\nu_P(F)$ is even for every valid face $F$ and
every colour pair $P$.
\end{definition}
\begin{remark}[Necessity of Kempe-balance]
\label{rem:kempe-balance-necessary}
A proper $3$-colouring of $\mathsf{M}(T)$ can be part of a proper
$3$-colouring of the whole medial graph $M(G)$ only when it is
Kempe-balanced: if $\varphi$ is the restriction to $\mathsf{M}(T)$ of a
proper $3$-colouring of $M(G)$, then $\varphi$ is Kempe-balanced.
Equivalently, a colouring of $\mathsf{M}(T)$ that fails the parity
condition at some valid face and colour pair cannot extend to a proper
$3$-colouring of $M(G)$. This is an instance of Kempe-cycle
conservation (Lemma~\ref{lem:kempe-conservation}). The tooth apexes
incident to a valid face are boundary medial vertices
(\cite{bauerfeld-medial-tire}) lying on a single level
cycle of the tire decomposition: the up-tooth apexes lie on the outer
level cycle, and the singleton down-tooth apexes incident to an interior
non-tooth face lie on the inner level cycle bounding that face. In the
$4$-regular graph $M(G)$ each $P$-Kempe chain of $\mathsf{M}(T)$ closes
up into a $P$-Kempe cycle, which by Lemma~\ref{lem:kempe-conservation}
meets each level cycle in an even number of $P$-coloured incidences; for
a given valid face these incidences are exactly its incident tooth
apexes coloured $a$ or $b$, whence $\nu_P(F)$ is even.
This argument is verified computationally. For bite-free pieces---capped
triangulated annuli on annular cycles of length $6,8,10,12$---every proper
$3$-colouring of $M(G)$ restricts to a Kempe-balanced colouring. The same
holds for pieces carrying a bite, including the case where singleton down
teeth lie in the bite's inner-gap face: there the inner level cycle splits
into a child level cycle per gap, and conservation across each child cycle
supplies the parity (in the checked example the three singleton down apexes
of a bite gap are a rainbow in every restriction).
\end{remark}
More generally, let $T$ be a medial tire region with boundary
\[
\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
For a $P$-Kempe cycle $K$, every component of $K\cap T$ is either a
cycle contained in $T$, or a path with two endpoints on
$\partial T$. Thus the $P$-Kempe arcs inside $T$ define a pairing of
the $P$-coloured boundary incidences of
\[
C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
This motivates the following refinement of boundary states.
\begin{definition}[Kempe-enhanced boundary state]
Let $T$ be a medial tire region with outer level cycle $C_0$ and inner
level cycles $C_1,\ldots,C_m$. Let
\[
\mathcal C(T)=C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
A \emph{Kempe-enhanced boundary state} on $T$ consists of the following
data:
\begin{enumerate}
\item a boundary colouring
\[
\alpha:V(\mathcal C(T))\to\{1,2,3\};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
a pairing $\pi_P$ of the $P$-coloured boundary incidences of
$\mathcal C(T)$ induced by the $P$-Kempe arcs lying inside $T$.
\end{enumerate}
We write such a state as
\[
\kappa=(\alpha,\pi_{12},\pi_{23},\pi_{31}).
\]
\end{definition}
Given a proper $3$-colouring $\varphi$ of the medial tire graph
$M(T)$, the restriction of $\varphi$ to the boundary level cycles gives
the boundary colouring $\alpha$, while the two-colour Kempe arcs inside
$T$ give the pairings $\pi_{12},\pi_{23},\pi_{31}$. Thus $\varphi$
determines a Kempe-enhanced boundary state, denoted
\[
\kappa_T(\varphi).
\]
\begin{definition}[Kempe-enhanced restriction relation]
The \emph{Kempe-enhanced restriction relation} of $T$ is
\[
\mathcal K_T
=
\left\{
\kappa_T(\varphi):
\varphi \text{ is a proper }3\text{-colouring of } M(T)
\right\}.
\]
This refines the ordinary boundary-colouring relation by recording not
only which boundary colourings extend across $T$, but also how the
two-colour Kempe cycles are routed through the annular tire region.
\end{definition}
The annular structure of a tire is useful in two distinct ways. First,
it gives a bounded transition region between level cycles: the colouring
of the annular medial cycle controls, and in many cases determines, the
colouring of the remaining medial tire vertices. Thus the number of
possible transition states is bounded in terms of the annular structure,
rather than the total size of the subtree below the tire. Second, it
describes how the outer level cycle and the inner level cycles are
related by Kempe arcs. The level cycles are the gluing interfaces, while
the annular tire is the transition operator between them.
\begin{definition}[Kempe-compatible gluing]
Let $T$ be a medial tire region and let $U$ be a child region glued to
$T$ along a common level cycle $C$. Thus $C$ is an inner level cycle of
$T$ and the outer level cycle of $U$.
Let
\[
\kappa_T=(\alpha_T,\pi^T_{12},\pi^T_{23},\pi^T_{31})
\in \mathcal K_T
\]
and
\[
\kappa_U=(\alpha_U,\pi^U_{12},\pi^U_{23},\pi^U_{31})
\in \mathcal K_U.
\]
We say that $\kappa_T$ and $\kappa_U$ are \emph{Kempe-compatible along
$C$} if:
\begin{enumerate}
\item the boundary colourings agree on $C$:
\[
\alpha_T|_{V(C)}=\alpha_U|_{V(C)};
\]
\item for each colour pair
\[
P\in\{\{1,2\},\{2,3\},\{3,1\}\},
\]
the pairings $\pi^T_P$ and $\pi^U_P$ compose along the
$P$-coloured incidences of $C$ without producing an unpaired endpoint.
\end{enumerate}
When these conditions hold, the composed pairings determine a
Kempe-enhanced boundary state on the exposed boundary of
$T\cup_C U$.
\end{definition}
In these terms, gluing local colourings is not merely a matter of
matching boundary colours. The colourings must also route their
two-colour Kempe arcs compatibly across every shared level cycle. The
ordinary restriction relation records whether a boundary colouring can
be extended locally; the Kempe-enhanced relation additionally records
the conservation of Kempe-cycle flow through the annular transition
region.
For a tire with one outer level cycle and several inner level cycles,
\[
\partial T=C_0\sqcup C_1\sqcup\cdots\sqcup C_m,
\]
the parent tire may correlate the boundary states on the different
inner cycles. The Kempe-enhanced relation records this correlation as
a system of pairings among the $P$-coloured incidences of all boundary
level cycles simultaneously. Thus one should view a medial tire as a
multi-output transition operator
\[
\mathcal K_T:
C_0 \leadsto (C_1,\ldots,C_m),
\]
rather than as an independent collection of binary transitions.
The guiding principle is therefore:
\begin{quote}
Level cycles are the interfaces used for gluing, while annular tire
regions are the bounded transition regions that route Kempe cycles
between those interfaces.
\end{quote}
\begin{thebibliography}{9}
\bibitem{bauerfeld-medial-tire}
E.~Bauerfeld,
\emph{Medial Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\bibitem{bauerfeld-nested-tire-decompositions}
E.~Bauerfeld,
\emph{Nested Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\bibitem{tait-original}
P.~G. Tait,
\emph{Remarks on the colourings of maps},
Proceedings of the Royal Society of Edinburgh \textbf{10} (1880),
729--729.
\end{thebibliography}
\end{document}
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# Chained seams: the medial pigeonhole, located
Companion to the interface-alphabet results (`kempe_interface_admissibility_probe.py`,
`kempe_tile_overlap_probe.py`). This note pursues the paper's §6 *medial pigeonhole
programme* — the restriction relation `R_T` between a tire's outer and inner boundary
states, and the open *chain-pigeonhole conjecture* — at the data level.
Scripts: `kempe_transfer_relation_probe.py`, `kempe_uniform_family_probe.py`.
## Background
A nested chain `T_0 ⊃ T_1 ⊃ …` glues into a proper 3-colouring of `M(G)` iff
consecutive boundary states match: `inner-state(T_i) = outer-state(T_{i+1})`
(compatible family, paper Prop "gluing criterion"). Boundary states are necklaces
(colours permuted, boundary walk rotated/reflected). The restriction relation
```
R_T ⊆ outer-states × inner-states
```
records which (outer, inner) boundary-state pairs one Kempe-balanced colouring of `T`
realises jointly. The chain-pigeonhole conjecture asks whether nested `R_T`'s can ever
compose to empty.
Established earlier: the realised boundary alphabet (each side) is exactly the full
parity-admissible set, identical inner vs outer, `n`-independent (n=9, n=12, m=3..8);
and every *pair* of tiles overlaps, so single seams never dead-end.
## Finding 1 — `R_T` is genuinely coupled
`R_T` is **not** a product of its projections: a tile's inner boundary necklace
constrains its outer one. Seen at n=9 in the `(p,q) = (4,5)` and `(5,4)` size classes
(`product? = False`), and broadly at n=12. So "every seam individually realisable"
does **not** trivially pass through a tile.
## Finding 2 — the uniform shortcut works at n=9, breaks at n=12
Question: is there one boundary state `σ_m` per level-cycle *size* that threads every
tile simultaneously (outer face and every inner face)? If so, painting every level
cycle `σ_m` glues any tree with no pigeonhole.
- **n=9: FEASIBLE.** Unique universal per size (`|D[m]| = 1`), and notably *not*
monochromatic — the balanced-block necklaces
```
σ3 = 012 σ4 = 0011 σ5 = 00012 σ6 = 000011
```
threads all 66 tiles. (Caveat: n=9 has **no** branching tiles.)
- **n=12: INFEASIBLE.** 1237 tiles (1029 bite, 175 genuinely branching). The CSP fails
for one sharp reason: **size-7 seams admit no universal state** (`|D[7]| = 0`). The
211 size-7 boundaries realise all 10 admissible necklaces between them, but their
intersection is empty — the near-universal `0001112` lands on **210 of 211**,
blocked by a single tile.
Per-size universal domain sizes at n=12: `3:1 4:1 5:1 6:2 7:0 8:4 9:1`.
## Finding 3 — the failure is SPORADIC, not a monotone trend
`kempe_universal_trend_probe.py` tracks `|D[m]|` (universal count) per size across
`n = 6..13`. Legend `|D[m]| (best_coverage / num_boundaries)`; `*` = odd size:
```
n= 6: m3*=1(6/6) m4=2(2/2) m6=4(1/1)
n= 7: m3*=1(10/10) m4=1(8/8) m5*=1(2/2) m7*=3(1/1)
n= 8: m3*=1(28/28) m4=2(21/21) m5*=1(10/10) m6=2(3/3) m8=8(1/1)
n= 9: m3*=1(49/49) m4=1(52/52) m5*=1(28/28) m6=1(14/14) m7*=2(3/3) m9*=8(1/1)
n=10: m3*=1(118) m4=2(106) m5*=1(86) m6=2(46) m7*=1(16/16) m8=4(4/4) m10=18(1/1)
n=11: m3*=1(310) m4=1(263) m5*=1(188) m6=1(139) m7*=2(60/60) m8=2(20/20) m9*=3(4/4) m11*=21(1/1)
n=12: m3*=1(849) m4=1(691) m5*=1(534) m6=2(353) m7*=0(210/211) m8=3(88) m9*=1(24/24) m10=6(5/5) m12=48(1/1)
n=13: m3*=1(2457) m4=1(1805) m5*=1(1386) m6=1(1070) m7*=2(579/579) m8=1(315) m9*=2(110) m10=1(28) m11*=7(5) m13*=63(1/1)
```
The earlier "universals vanish as the tile population grows" reading is **wrong**.
Across all `n = 6..13` and all sizes, the **only** empty universal anywhere is
`(n=12, m=7)`. Size 7 has *more* boundaries at n=13 (579) than n=12 (211), yet its
universal is back (`|D|=2`). So the failure is **sporadic**, not monotone — a
number-theoretic quirk of `n=12`, not a scaling law. Note also that best_coverage is
always near-total: the most-shared necklace reaches all-but-rarely-one boundary, so
the universal is "almost there" even when it fails.
### The single breaker (n=12, m=7)
The lone size-7 boundary that kills the universal is the **outer rim** of
```
word = UUUDUDUDUDUD bite = (3,11) (7 up teeth)
```
— the most-alternating word with a near-full-span bite. Its outer rim realises 9 of
the 10 admissible size-7 necklaces, missing exactly `0001112` (the balanced two-block
`3+3+1` pattern); every other size-7 boundary realises `0001112`. One exceptional tile
breaks the shortcut.
## Finding 4 — interpretation
The uniform "paint every seam the same" shortcut **almost always works** (universals
non-empty with near-total coverage, even at thousands of boundaries), but is
**fragile**: a single exceptional tile can empty a per-size universal, as at
`(n=12, m=7)`. So the shortcut cannot be relied on in general — yet its failures are
rare, not systematic. The per-interface pigeonhole choice is needed precisely to
absorb these sporadic breakers. **This is not an obstruction to gluing** — pairwise
overlap always holds, so chains still glue by choosing states per interface. The
conjecture's difficulty is thus concentrated in rare exceptional tiles and the
branch-coupled selection around them, not in any single seam or in a scaling trend.
## Finding 5 — restricting to no separating triangles REMOVES the obstruction
The 4CT reduces to triangulations with no separating triangles (internally
4-connected). In the tread model, a separating (non-facial) triangle in `G` shows up
as a **length-3 boundary walk** of a tread: an outer rim of 3 up teeth, or an inner
face holding exactly 3 singleton down teeth (the `012` seam). Restricting to tiles
with **no length-3 boundary** (`kempe_universal_trend_probe.py --no-tri`,
`kempe_uniform_family_probe.py --no-tri`):
- The n=12 breaker `UUUDUDUDUDUD` bite=(3,11) has a size-3 inner face (the bite
encloses exactly `d5,d7,d9`), so it is **excluded** — along with its kin.
- The size-7 universal at n=12 is **restored**: `|D[7]|` goes `0 → 2`, on a reduced
population (211 → 76 size-7 boundaries). Across `n = 6..13` and all sizes, **every
`|D[m]| ≥ 1`** — no empty universal anywhere.
- The uniform-family CSP becomes **FEASIBLE at n=12** (was infeasible). The threading
family is now the simplest one — **monochromatic on even sizes, min-cut on odd**:
```
σ4 = 0000 σ5 = 00012 σ6 = 000000 σ7 = 0000012 σ8 = 00000000
```
(Without the restriction, `σ4` had to be `0011`, not monochromatic, because
separating-triangle tiles blocked the all-0 rim. Removing them lets monochromatic
even seams work.)
So the **only** universal failure we found (n=12, size 7) was an artifact of admitting
tiles that correspond to **non-4-connected** triangulations. On the 4CT-relevant class
(no separating triangles), the uniform seam family exists and gluing is constructively
trivial throughout the tested range — no pigeonhole needed.
Caveat: even at n=12 the restricted population has **0 branching tiles** (multi-inner
faces all require a size-3 face at this n), so the branching case stays untested under
the restriction; and this is `n ≤ 13` only.
## Finding 6 — smallest branching n (`kempe_branching_min_probe.py`)
Branching tile = `≥2` inner faces carrying singleton-down interfaces (a tree node with
`≥2` children).
```
n #tiles branching no-tri branching
9 81 0 0
10 203 0 0
11 503 30 0 <- unrestricted branching first appears
12 1344 175 0
13 3586 789 0
14 9929 3024 193 <- no-separating-triangle branching first appears
15 27481 10538 1022
```
- Unrestricted branching first appears at **n=11**.
- No-separating-triangle branching first appears at **n=14**. Smallest example:
```
word = UUUUDDDDDDDDDD bite = (8,13) p = 4
inner faces: root {4,5,6,7} (size 4) + bite(8,13) {9,10,11,12} (size 4)
```
4 up teeth, two size-4 inner faces — branching, no length-3 boundary. (Reason for the
gap: a branching node needs `≥2` inner faces each `≥4` singletons plus `≥4` up teeth
plus the bite pair = `4 + 4 + 4 + 2 = 14` edges.)
So **n=14** is the smallest place to test the uniform family / `R_T` composition on a
genuine *branching* no-separating-triangle tile — the conjecture's real case.
## Finding 7 — the branching case (n=14) is FEASIBLE with one regular family
Full uniform-family CSP at n=14 `--no-tri` (4403 tiles, 3766 bite, **193 branching**):
**FEASIBLE** — a single uniform family threads every tile, branching nodes included
(each branching tile shows the uniform state on its outer rim AND both inner faces at
once). Domains `|D[m]|`: `4:2 5:1 6:2 7:2 8:3 9:3 10:5`. The witness family is fully
regular:
```
σ_m = 0^m (monochromatic) if m even
σ_m = 0^(m-2) 1 2 (one block + 1 + 2) if m odd
4:0000 5:00012 6:000000 7:0000012 8:00000000 9:000000012 10:0000000000
```
A direct candidate test (just the 193 branching tiles) confirmed it independently:
the monochromatic-even / min-cut-odd family threads **193/193**.
So on the 4CT-relevant class (no separating triangles), the chained-seam pigeonhole is
**constructively resolved throughout the tested range** (n = 9, 12, 14, including the
first branching nodes) by one explicit regular seam family: paint every even level
cycle one colour, and every odd level cycle as a single monochromatic block plus the
two parity-forced off-colour vertices.
## Finding 8 — the regular family is REFUTED at n=15
`kempe_regular_family_test.py` (tests the fixed regular family with per-tile
early-exit). At n=12 it threads 614/614 (matches the CSP). **At n=15 it FAILS**, on two
distinct classes of no-separating-triangle tiles:
- **non-branching, large even outer + odd inner:** e.g. `UUUUUUDUDUDUDUD` (no bite),
p=10, inner face size 5. Monochromatic `σ10 = 0^10` on the 10-up rim cannot coexist
with `σ5 = 00012` on the inner face.
- **branching, odd outer + two even inner faces:** e.g. `UDUDUDDUDUDDDDD` bite=(5,12),
p=5, faces `[4,4]`. `σ5` outer with monochromatic `σ4` on *both* inner faces is not
jointly realisable. Branching tiles: **1011/1022 threaded, 11 fail.**
So the clean regular conjecture is **false**. Crucially this refutation is exactly the
`R_T` **coupling** (Finding 1) asserting itself at scale: the regular family sets outer
and inner states *independently per size*, but `R_T` is not a product, so a large
monochromatic rim over-constrains the annular cycle and forbids the paired inner
necklace. The failure is not about branching per se — it hits large even rims (non-
branching) and small odd rims with two even children (branching) alike.
### What it means for strategy
The uniform "one state per size, everywhere" family was a **too-strong shortcut** —
much stronger than the chain-pigeonhole conjecture, which only needs *some* compatible
selection per chain with freedom to choose a *different* state at each interface. Its
failure costs a cheap constructive route, **not** the conjecture: pairwise overlap
still always holds. The load transfers to the genuine object — **per-interface
selection respecting `R_T` coupling**, i.e. composing `R_T` along chains/trees with
per-seam freedom rather than a global family.
## Finding 9 — `R_T` composition: the pigeonhole verified exhaustively for n ≤ 14
`kempe_rt_composition_probe.py` — the conjecture proper, with full per-interface
freedom. Two modeling facts were established first:
- **The seam is exactly the singleton down apexes** of one inner face. A bite apex's
edge has parent tread faces on *both* sides, so it is internal to the parent's tile
and shares no medial vertex with the child.
- **Necklace states are exact, not approximate**: a child tile attaches with free
dihedral placement, so its realisable outer-sequence set is dihedral-closed and
"necklace match ⟺ some aligned sequence match". (This is the paper's own Def. of
medial boundary state.)
Define `Ext(T)` = necklaces realisable on a subtree's outer seam by a compatible
Kempe-balanced selection: `Ext(T) = {o : ∃(o, i⃗) ∈ R_T^joint, i_j ∈ Ext(C_j)}`,
leaves = all-bite tiles (no singleton interfaces, degenerate inner boundaries). The
maps are monotone, so the **antichain of minimal reachable Ext sets per seam size**
decides whether ∅ is reachable. Relations are cached (`kempe_rt_relations_cache.json`).
Result over all no-length-3-boundary tiles with `n ≤ 14` (7750 tiles, 1966 distinct
relations, 149 leaf, **27 branching**):
- **∅ is NOT reachable.** Every tree assemblable from this universe — branching
included — admits a compatible Kempe-balanced selection. Since every real tire tree
whose treads have `n ≤ 14` and no separating triangles is such a tree, the
chain-pigeonhole conjecture is **verified exhaustively for that class**.
- **Composition saturates in 2 rounds.** The minimal antichains are already closed
under every tile map after one pass: restriction does *not* accumulate along chains
— it bottoms out immediately. This is exactly the structural behaviour the paper's
§6 programme hoped for ("restriction sets cannot remain mutually disjoint").
- **Restriction is real but bounded.** Tightest subtrees per seam size: m=4 forces
2 of 3 necklaces, **m=5 forces a single necklace `{00012}`**, m=7 forces 3 of 10,
m=8 forces 8 of 34, m=14 forces 133 of 7515. Yet no parent relation ever misses a
minimal set entirely.
- **The blocky states are always offered.** Every smallest minimal Ext set contains
the regular necklace of its size (`0^m` or `0^{m-2}12`-type). The regular family
failed (Finding 8) because a single tile sometimes cannot take blocky-in and
blocky-out *jointly* — but per-interface freedom routes around it, and the data
shows subtrees always keep the blocky option available downward.
Caveats: (i) terminal facial triangles (innermost treads ending on a face of `G`) are
not yet modelled — our leaves are only the degenerate-inner-boundary all-bite tiles;
adding 3-faces as terminal children with `Ext = {012}` is a small extension. (ii) The
universe is abstract: it is a *superset* of real trees (good for the positive verdict;
an abstract obstruction, had one appeared, would still have needed a realisability
check).
## Open threads
- **Terminal-3-face leaves.** Extend the fixpoint with facial-triangle termination
(parent tiles with one 3-singleton face allowed as terminal); rerun.
- **Push `max_n`.** n=15 adds ~1 hr of one-time classification to the cache; the
2-round saturation suggests verdicts stabilise quickly, but a deeper universe is
the only way an obstruction could still appear.
- **Why saturation?** The 2-round fixpoint convergence is the empirical shadow of a
provable statement: composing tire restriction relations stabilises after one
level. A proof of that, with the minimal antichains characterised, would *be* the
pigeonhole lemma for this class.
- **Structural lemma for the n=15 uniform failures** (why a monochromatic large rim
forbids the paired odd-inner necklace) — still open, now lower priority.
papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_medial_annular_cycle_condition.py
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"""Check the medial annular-cycle almost-two-colour condition.
For each generated plane triangulation G and each requested level source:
1. Build the full medial graph M(G).
2. Find depth-component tire annular medial subgraphs.
3. Enumerate simple cycles in those annular subgraphs.
4. Search for a proper vertex 3-colouring of M(G) such that every
such cycle uses two colours except at at most one vertex.
Run with Sage, for example:
sage -python papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_medial_annular_cycle_condition.py --n-min 4 --n-max 8
"""
from __future__ import annotations
import argparse
from collections import defaultdict, deque
from itertools import combinations
from typing import Any, Iterable, Iterator, Sequence, cast
from sage.all import Graph, graphs # type: ignore[attr-defined] # pylint: disable=no-name-in-module
from sage.graphs.graph_coloring import all_graph_colorings # type: ignore[attr-defined] # pylint: disable=no-name-in-module
Edge = tuple[Any, Any]
Coloring = dict[Edge, int]
Source = tuple[Any, ...]
def vertex_key(v: Any) -> str:
return repr(v)
def edge_key(u: Any, v: Any) -> Edge:
return (u, v) if vertex_key(u) <= vertex_key(v) else (v, u)
def is_induced_cycle(g: Graph, vertices: Sequence[Any]) -> bool:
if len(vertices) < 3:
return False
h = cast(Graph, g.subgraph(list(vertices)))
return h.is_connected() and h.num_edges() == len(vertices) and all(
h.degree(v) == 2 for v in h.vertices()
)
def induced_cycle_sources(g: Graph, max_size: int | None = None) -> Iterator[Source]:
vertices = sorted(g.vertices(), key=vertex_key)
upper = len(vertices) if max_size is None else min(max_size, len(vertices))
for k in range(3, upper + 1):
for subset in combinations(vertices, k):
if is_induced_cycle(g, subset):
yield tuple(subset)
def level_sources(g: Graph, mode: str, max_cycle_source_size: int | None) -> Iterator[Source]:
if mode in ("vertex", "all"):
for v in sorted(g.vertices(), key=vertex_key):
yield (v,)
if mode in ("cycle", "all"):
yield from induced_cycle_sources(g, max_cycle_source_size)
def distances_from_source(g: Graph, source: Source) -> dict[Any, int]:
if len(source) == 1:
return dict(g.shortest_path_lengths(source[0]))
distances = {v: 0 for v in source}
queue: deque[Any] = deque(source)
while queue:
v = queue.popleft()
for w in g.neighbor_iterator(v):
if w in distances:
continue
distances[w] = distances[v] + 1
queue.append(w)
return distances
def embedded_copy(g: Graph) -> Graph:
emb = cast(Graph, g.copy())
if not emb.is_planar(set_embedding=True):
raise ValueError("graph is not planar")
return emb
def medial_graph(g: Graph) -> Graph:
"""Build the full medial graph from the embedding rotation at vertices."""
emb = embedded_copy(g)
rotation = emb.get_embedding()
m = Graph()
medial_vertices = [edge_key(u, v) for u, v, _ in emb.edge_iterator()]
m.add_vertices(medial_vertices)
for v, neighbors in rotation.items():
if len(neighbors) < 2:
continue
n = len(neighbors)
for i in range(n):
e1 = edge_key(v, neighbors[i])
e2 = edge_key(v, neighbors[(i + 1) % n])
if e1 != e2:
m.add_edge(e1, e2)
return m
def face_vertices(face: Sequence[tuple[Any, Any]]) -> set[Any]:
out: set[Any] = set()
for u, v in face:
out.add(u)
out.add(v)
return out
def face_edges(face: Sequence[tuple[Any, Any]]) -> set[Edge]:
return {edge_key(u, v) for u, v in face}
def dual_components_by_depth(
g: Graph, source: Source
) -> list[tuple[int, list[int], set[Edge]]]:
"""Return (depth, face-indices, annular-edge-set) for each depth component."""
emb = embedded_copy(g)
distances = distances_from_source(emb, source)
faces = emb.faces()
f_vertices = [face_vertices(face) for face in faces]
f_edges = [face_edges(face) for face in faces]
depths = [min(distances[v] for v in verts) for verts in f_vertices]
edge_faces: dict[Edge, list[int]] = defaultdict(list)
for idx, edges in enumerate(f_edges):
for edge in edges:
edge_faces[edge].append(idx)
dual_adj: dict[int, set[int]] = defaultdict(set)
for incident in edge_faces.values():
for a in range(len(incident)):
for b in range(a + 1, len(incident)):
dual_adj[incident[a]].add(incident[b])
dual_adj[incident[b]].add(incident[a])
components = []
seen = [False] * len(faces)
for start in range(len(faces)):
if seen[start]:
continue
depth = depths[start]
comp = [start]
seen[start] = True
stack = [start]
while stack:
f = stack.pop()
for h in dual_adj[f]:
if not seen[h] and depths[h] == depth:
seen[h] = True
comp.append(h)
stack.append(h)
annular_edges: set[Edge] = set()
for f in comp:
for u, v in f_edges[f]:
if {distances[u], distances[v]} == {depth, depth + 1}:
annular_edges.add(edge_key(u, v))
if len(annular_edges) >= 3:
components.append((depth, comp, annular_edges))
return components
def simple_cycle_vertex_sets(g: Graph) -> set[frozenset[Any]]:
vertices = sorted(g.vertices(), key=repr)
index = {v: i for i, v in enumerate(vertices)}
cycles: set[frozenset[Any]] = set()
def dfs(start: Any, current: Any, path: list[Any], seen: set[Any]) -> None:
for nxt in g.neighbor_iterator(current):
if nxt == start:
if len(path) >= 3:
cycles.add(frozenset(path))
continue
if nxt in seen or index[nxt] <= index[start]:
continue
seen.add(nxt)
path.append(nxt)
dfs(start, nxt, path, seen)
path.pop()
seen.remove(nxt)
for start in vertices:
dfs(start, start, [start], {start})
return cycles
def annular_medial_cycles(g: Graph, source: Source) -> list[frozenset[Edge]]:
m = medial_graph(g)
cycles: list[frozenset[Edge]] = []
seen: set[frozenset[Edge]] = set()
for _depth, _faces, annular_edges in dual_components_by_depth(g, source):
sub = cast(Graph, m.subgraph(list(annular_edges)))
for cycle in simple_cycle_vertex_sets(sub):
typed = frozenset(cast(Iterable[Edge], cycle))
if typed not in seen:
seen.add(typed)
cycles.append(typed)
return cycles
def almost_two_coloured(cycle: frozenset[Edge], coloring: Coloring) -> bool:
counts = defaultdict(int)
for vertex in cycle:
counts[coloring[vertex]] += 1
return min(counts.get(c, 0) for c in range(3)) <= 1
def first_cycle_violation(
cycles: Sequence[frozenset[Edge]], coloring: Coloring
) -> frozenset[Edge] | None:
for cycle in cycles:
if not almost_two_coloured(cycle, coloring):
return cycle
return None
def color_counts(cycle: frozenset[Edge], coloring: Coloring) -> dict[int, int]:
counts = {0: 0, 1: 0, 2: 0}
for vertex in cycle:
counts[coloring[vertex]] += 1
return counts
def coloring_witness(
m: Graph,
cycles: Sequence[frozenset[Edge]],
max_colorings: int | None,
) -> tuple[Coloring | None, int, bool, frozenset[Edge] | None, Coloring | None]:
checked = 0
last_violation = None
for raw in all_graph_colorings(m, 3, vertex_color_dict=True):
coloring = cast(Coloring, raw)
checked += 1
violation = first_cycle_violation(cycles, coloring)
if violation is None:
return coloring, checked, True, None, None
last_violation = violation
if max_colorings is not None and checked >= max_colorings:
return None, checked, False, last_violation, coloring
return None, checked, True, last_violation, coloring
def source_label(source: Source) -> str:
if len(source) == 1:
return f"vertex:{source[0]}"
return "cycle:{" + ",".join(map(str, source)) + "}"
def graphs_to_check(n: int, max_graphs: int | None):
for idx, g in enumerate(graphs.triangulations(n)):
if max_graphs is not None and idx >= max_graphs:
break
yield idx, cast(Graph, g)
def run(args: argparse.Namespace) -> None:
total_cases = 0
skipped_no_cycles = 0
witnesses = 0
failures = []
inconclusive = []
for n in range(args.n_min, args.n_max + 1):
print(f"n={n}")
for graph_idx, g in graphs_to_check(n, args.max_graphs_per_n):
m = medial_graph(g)
sources = list(level_sources(g, args.sources, args.max_cycle_source_size))
if args.max_sources_per_graph is not None:
sources = sources[: args.max_sources_per_graph]
for source in sources:
cycles = annular_medial_cycles(g, source)
if not cycles:
skipped_no_cycles += 1
continue
total_cases += 1
witness, checked, exhausted, violation, last_coloring = coloring_witness(
m, cycles, args.max_colorings
)
if witness is not None:
witnesses += 1
if args.verbose:
print(
f" graph={graph_idx} source={source_label(source)} "
f"cycles={len(cycles)} witness_after={checked}"
)
continue
record = {
"n": n,
"graph_idx": graph_idx,
"graph_edges": sorted(edge_key(u, v) for u, v, _ in g.edge_iterator()),
"source": source_label(source),
"cycles": len(cycles),
"checked": checked,
"exhausted": exhausted,
"violation_size": len(violation) if violation else None,
}
if args.failure_details and violation is not None and last_coloring is not None:
record["violation_cycle"] = sorted(violation)
record["violation_counts"] = color_counts(violation, last_coloring)
record["violation_coloring"] = {
edge: last_coloring[edge] for edge in sorted(violation)
}
if exhausted:
failures.append(record)
print(" FAILURE", record)
if args.stop_on_failure:
print_summary(total_cases, skipped_no_cycles, witnesses, failures, inconclusive)
return
else:
inconclusive.append(record)
print(" INCONCLUSIVE", record)
print_summary(total_cases, skipped_no_cycles, witnesses, failures, inconclusive)
def print_summary(
total_cases: int,
skipped_no_cycles: int,
witnesses: int,
failures: Sequence[dict],
inconclusive: Sequence[dict],
) -> None:
print()
print("summary")
print(f" checked source decompositions with annular cycles: {total_cases}")
print(f" skipped source decompositions with no annular cycles: {skipped_no_cycles}")
print(f" witnesses found: {witnesses}")
print(f" failures: {len(failures)}")
print(f" inconclusive: {len(inconclusive)}")
if failures:
print(f" first failure: {failures[0]}")
if inconclusive:
print(f" first inconclusive: {inconclusive[0]}")
def main() -> None:
parser = argparse.ArgumentParser()
parser.add_argument("--n-min", type=int, default=4)
parser.add_argument("--n-max", type=int, default=8)
parser.add_argument("--sources", choices=("vertex", "cycle", "all"), default="vertex")
parser.add_argument("--max-cycle-source-size", type=int, default=6)
parser.add_argument("--max-graphs-per-n", type=int)
parser.add_argument("--max-sources-per-graph", type=int)
parser.add_argument("--max-colorings", type=int)
parser.add_argument("--stop-on-failure", action="store_true")
parser.add_argument("--failure-details", action="store_true")
parser.add_argument("--verbose", action="store_true")
run(parser.parse_args())
if __name__ == "__main__":
main()
papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_medial_face_parity.py
+181
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"""Probe the parity mechanism behind Remark 5.8.
Claim X: in a proper 3-colouring of the 4-regular medial graph M(G) of a plane
triangulation G, for every face f of M(G) and every colour pair P = {a,b}, the
number of vertices on the boundary of f coloured a or b is even.
M(G) has two kinds of faces: a "vertex-face" per vertex v of G (the cyclic
sequence of edges around v) and a "face-face" per triangular face of G (its
three edges). The face-faces are triangles, trivially even (count 2); the
vertex-faces are the non-obvious case.
We build M(G) from a planar embedding's rotation system, enumerate proper
3-colourings of M(G), and check Claim X on every face / pair.
"""
from __future__ import annotations
import itertools
import networkx as nx
# --- a handful of small plane triangulations (maximal planar graphs) ---------
def tetrahedron() -> nx.Graph:
return nx.complete_graph(4)
def octahedron() -> nx.Graph:
# K_{2,2,2}: antipodal pairs non-adjacent
g = nx.Graph()
pairs = [(0, 1), (2, 3), (4, 5)]
nonadj = set(map(frozenset, pairs))
for u in range(6):
for v in range(u + 1, 6):
if frozenset((u, v)) not in nonadj:
g.add_edge(u, v)
return g
def stacked(levels: int) -> nx.Graph:
"""Apollonian-style: repeatedly insert a vertex in a triangular face."""
g = nx.Graph()
g.add_edges_from([(0, 1), (1, 2), (0, 2)])
faces = [(0, 1, 2)]
nxt = 3
for _ in range(levels):
a, b, c = faces.pop(0)
v = nxt
nxt += 1
g.add_edges_from([(v, a), (v, b), (v, c)])
faces += [(a, b, v), (b, c, v), (a, c, v)]
return g
def icosahedron() -> nx.Graph:
return nx.icosahedral_graph()
def double_wheel(rim: int) -> nx.Graph:
"""Two apexes over a rim cycle: a simple triangulated 'tire' with caps."""
g = nx.Graph()
g.add_cycle = None
for i in range(rim):
g.add_edge(i, (i + 1) % rim)
g.add_edge(i, "N")
g.add_edge(i, "S")
return g
# --- medial graph from a rotation system -------------------------------------
def rotation_system(g: nx.Graph) -> dict:
ok, emb = nx.check_planarity(g)
if not ok:
raise ValueError("graph is not planar")
return {v: list(emb.neighbors_cw_order(v)) for v in g.nodes()}, emb
def medial_graph(g: nx.Graph):
"""Return (M, vertex_faces, face_faces) built from the rotation system.
Medial vertices are edges of g (as sorted tuples). Around each vertex the
incident edges form a face cycle (vertex-face); around each triangular face
of g its three edges form a face cycle (face-face).
"""
rot, emb = rotation_system(g)
def ekey(u, v):
return (u, v) if u <= v else (v, u)
M = nx.Graph()
M.add_nodes_from(ekey(u, v) for u, v in g.edges())
vertex_faces = []
for v, order in rot.items():
edges = [ekey(v, w) for w in order]
vertex_faces.append(edges)
for i in range(len(edges)):
M.add_edge(edges[i], edges[(i + 1) % len(edges)])
# face-faces: traverse each face of the embedding once
seen = set()
face_faces = []
for u, v in list(emb.edges()):
if (u, v) in seen:
continue
face = emb.traverse_face(u, v, mark_half_edges=seen)
edges = [ekey(face[i], face[(i + 1) % len(face)]) for i in range(len(face))]
face_faces.append(edges)
return M, vertex_faces, face_faces
# --- proper 3-colourings of M(G) ---------------------------------------------
def proper_3_colorings(M: nx.Graph, limit: int | None = None):
nodes = list(M.nodes())
adj = {v: set(M.neighbors(v)) for v in nodes}
coloring: dict = {}
out = []
def rec(i):
if limit is not None and len(out) >= limit:
return
if i == len(nodes):
out.append(dict(coloring))
return
v = nodes[i]
used = {coloring[w] for w in adj[v] if w in coloring}
for c in (0, 1, 2):
if c not in used:
coloring[v] = c
rec(i + 1)
del coloring[v]
rec(0)
return out
def check_claim_x(name: str, g: nx.Graph, color_limit: int = 200):
M, vfaces, ffaces = medial_graph(g)
colorings = proper_3_colorings(M, limit=color_limit)
if not colorings:
print(f"{name}: M(G) has no proper 3-colouring (skip)")
return
faces = [("vertex", f) for f in vfaces] + [("face", f) for f in ffaces]
violations = 0
odd_vertex_faces = 0
for col in colorings:
for kind, face in faces:
for pair in ((0, 1), (0, 2), (1, 2)):
cnt = sum(1 for v in face if col[v] in pair)
if cnt % 2 != 0:
violations += 1
if kind == "vertex":
odd_vertex_faces += 1
deg = sorted({len(f) for f in vfaces})
print(f"{name}: |V(G)|={g.number_of_nodes()} |M|={M.number_of_nodes()} "
f"colourings tested={len(colorings)} vertex-face sizes={deg}")
print(f" Claim X violations: {violations} "
f"(vertex-face violations: {odd_vertex_faces})")
def main():
cases = [
("tetrahedron", tetrahedron()),
("octahedron", octahedron()),
("stacked-3", stacked(3)),
("stacked-6", stacked(6)),
("double_wheel-5", double_wheel(5)),
("double_wheel-6", double_wheel(6)),
("double_wheel-7", double_wheel(7)),
("icosahedron", icosahedron()),
]
for name, g in cases:
# ensure it is a triangulation (every face a triangle)
check_claim_x(name, g)
if __name__ == "__main__":
main()
papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_remark58_bite.py
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"""Directly test Remark 5.8 on a genuine tire piece that contains a BITE.
A bite arises when the inner outerplanar graph O has a bridge: the bridge edge
is traversed twice by the outer-face walk, so it borders two tread triangles and
its medial vertex is adjacent to four annular medial vertices.
Minimal construction. Outer 4-cycle o0,o1,o2,o3; two interior vertices u,w
joined by a bridge u-w (V_in = {u,w}). Triangulate the disk so that u-w lies in
two tread triangles:
(o0,o1,u) (o0,u,o3) (o1,w,u) (o1,o2,w) (o2,o3,w) (o3,u,w)
Cap the outer cycle with an apex N (the bridge bounds no inner hole, so no inner
cap is needed). The result G is a closed plane triangulation; M(G) is 4-regular.
Edge classification (by endpoints): annular = one endpoint outer & one inner;
up tooth = both endpoints outer (outer-cycle edge); down tooth = both endpoints
inner (here only the bridge u-w). The bridge's medial vertex is the bite apex.
Remark 5.8 predicts every proper 3-colouring of M(G) restricts to a
Kempe-balanced colouring. Here the only non-trivial condition is the outer face
(the four up apexes), since the single bite contributes no singleton down teeth.
"""
from __future__ import annotations
import networkx as nx
from check_remark58_bitefree import ekey, medial_graph, proper_3_colorings
PAIRS = ((0, 1), (0, 2), (1, 2))
OUTER = ["o0", "o1", "o2", "o3"]
INNER = ["u", "w"]
TREAD_TRIANGLES = [
("o0", "o1", "u"),
("o0", "u", "o3"),
("o1", "w", "u"),
("o1", "o2", "w"),
("o2", "o3", "w"),
("o3", "u", "w"),
]
def build():
g = nx.Graph()
for tri in TREAD_TRIANGLES:
a, b, c = tri
g.add_edges_from([(a, b), (b, c), (a, c)])
# outer cap
for i in range(4):
g.add_edges_from([("N", OUTER[i]), ("N", OUTER[(i + 1) % 4])])
return g
def classify_tread_edges(g):
out = set(OUTER)
inn = set(INNER)
tread_edges = set()
for tri in TREAD_TRIANGLES:
a, b, c = tri
tread_edges |= {ekey(a, b), ekey(b, c), ekey(a, c)}
annular, up, down = [], [], []
for e in tread_edges:
a, b = e
ao, bo = a in out, b in out
ai, bi = a in inn, b in inn
if (ao and bi) or (ai and bo):
annular.append(e)
elif ao and bo:
up.append(e)
elif ai and bi:
down.append(e)
return annular, up, down
def run():
g = build()
assert nx.check_planarity(g)[0]
M = medial_graph(g)
annular, up, down = classify_tread_edges(g)
annular_set = set(annular)
# confirm there is a bite: a down edge whose medial vertex has 4 annular nbrs
bites = [e for e in down if sum(1 for nb in M.neighbors(e) if nb in annular_set) == 4]
print(f"tread: annular={len(annular)} up={len(up)} down={len(down)} "
f"bite apexes={len(bites)} (bite edge: {bites})")
colorings = proper_3_colorings(M, limit=20000)
balanced = 0
bad = []
for col in colorings:
ok = all(
sum(1 for e in up if col[e] in pair) % 2 == 0
for pair in PAIRS
)
if ok:
balanced += 1
else:
bad.append(col)
print(f"|V(G)|={g.number_of_nodes()} |M(G)|={M.number_of_nodes()} "
f"colourings tested={len(colorings)}")
print(f" outer-face (up-apex) balanced={balanced} UNBALANCED={len(bad)}")
if bad:
print(f" first unbalanced up colours: {[bad[0][e] for e in up]}")
print()
print("Remark 5.8 holds on this bite tread"
if not bad else
"Remark 5.8 FAILS on this bite tread")
return len(bad)
if __name__ == "__main__":
run()
papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_remark58_bite_rich.py
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"""Test Remark 5.8 on a bite tread that also has singleton down teeth in the
bite's inner-gap face -- the subtle case of the condition.
Inner outerplanar graph O = triangle (a,b,c) plus a pendant bridge a-d. Its
outer-face walk is the cyclic sequence W = [d, a, b, c, a]: the bridge a-d is
traversed twice (-> a bite), the triangle edges a-b, b-c, c-a once each (-> three
singleton down teeth, all sitting in the bite's inner-gap face).
We triangulate the annulus between an outer m-cycle and W by the lattice-path
method, searching interleavings for one giving a simple closed triangulation
after capping the outer cycle with an apex N. Then we test Remark 5.8: every
proper 3-colouring of M(G) restricts to a Kempe-balanced colouring, i.e.
* the up apexes (outer edges) are even per colour pair, and
* the three singleton down apexes (a-b, b-c, c-a), which share the bite-gap
face, are even per colour pair (equivalently: a rainbow).
"""
from __future__ import annotations
import itertools
import networkx as nx
from check_remark58_bitefree import ekey, medial_graph, proper_3_colorings
PAIRS = ((0, 1), (0, 2), (1, 2))
INNER_WALK = ["d", "a", "b", "c", "a"] # bridge a-d traversed twice
SINGLETON_DOWN = [ekey("a", "b"), ekey("b", "c"), ekey("c", "a")]
BITE_EDGE = ekey("a", "d")
def build_tread(m: int, path: str):
"""Build the annular triangulation for a given lattice path (m 'O', L 'I')."""
outer = [f"o{t}" for t in range(m)]
W = INNER_WALK
L = len(W)
g = nx.Graph()
g.add_edge(outer[0], W[0]) # anchor
i = j = 0
tread_triangles = []
for mv in path:
if mv == "O":
tri = (outer[i % m], W[j % L], outer[(i + 1) % m])
i += 1
else:
tri = (outer[i % m], W[j % L], W[(j + 1) % L])
j += 1
a, b, c = tri
g.add_edges_from([(a, b), (b, c), (a, c)])
tread_triangles.append(tri)
if (i, j) != (m, L):
return None
return g, outer, tread_triangles
def cap_and_validate(g, outer):
"""Cap the outer cycle with apex N; require a simple closed triangulation."""
h = g.copy()
for t in range(len(outer)):
h.add_edges_from([("N", outer[t]), ("N", outer[(t + 1) % len(outer)])])
if not nx.check_planarity(h)[0]:
return None
V, E = h.number_of_nodes(), h.number_of_edges()
if E != 3 * V - 6: # maximal planar == triangulation
return None
return h
def find_construction(m: int):
L = len(INNER_WALK)
for combo in itertools.combinations(range(m + L), L):
path = "".join("I" if t in combo else "O" for t in range(m + L))
built = build_tread(m, path)
if built is None:
continue
g, outer, tris = built
h = cap_and_validate(g, outer)
if h is not None:
return h, outer, tris, path
return None
def run():
for m in (4, 5, 6, 7):
found = find_construction(m)
if found:
break
if not found:
print("no valid bite-with-singletons triangulation found")
return 1
h, outer, tris, path = found
M = medial_graph(h)
annular = set()
for tri in tris:
a, b, c = tri
for e in (ekey(a, b), ekey(b, c), ekey(a, c)):
x, y = e
xo, yo = x in outer, y in outer
if (xo and not yo) or (yo and not xo):
annular.add(e)
n_bite_nbrs = sum(1 for nb in M.neighbors(BITE_EDGE) if nb in annular)
up = [ekey(outer[t], outer[(t + 1) % len(outer)]) for t in range(len(outer))]
up = [e for e in up if e in M]
print(f"m={len(outer)} path={path} |V(G)|={h.number_of_nodes()} "
f"|M(G)|={M.number_of_nodes()}")
print(f"bite edge {BITE_EDGE}: annular neighbours={n_bite_nbrs} (4 => bite)")
print(f"up apexes={len(up)} singleton down apexes={SINGLETON_DOWN}")
colorings = proper_3_colorings(M, limit=50000)
bad_outer = bad_bitegap = 0
for col in colorings:
if any(sum(1 for e in up if col[e] in p) % 2 for p in PAIRS):
bad_outer += 1
if any(sum(1 for e in SINGLETON_DOWN if col[e] in p) % 2 for p in PAIRS):
bad_bitegap += 1
print(f"colourings tested={len(colorings)}")
print(f" outer face unbalanced: {bad_outer}")
print(f" bite-gap face (3 singletons) unbalanced: {bad_bitegap}")
print()
ok = (bad_outer == 0 and bad_bitegap == 0)
print("Remark 5.8 holds on this bite-with-singletons tread"
if ok else "Remark 5.8 FAILS on this bite-with-singletons tread")
return 0 if ok else 1
if __name__ == "__main__":
run()
papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_remark58_bitefree.py
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"""Directly test Remark 5.8 on genuine (bite-free) tire pieces.
Construction. Build a triangulated annulus (an antiprism band) between an outer
p-cycle O = o_0..o_{p-1} and an inner p-cycle I = i_0..i_{p-1}, with the 2p
triangles
(o_k, o_{k+1}, i_k) and (o_{k+1}, i_k, i_{k+1}).
Cap the outer disk with an apex N joined to all o_k and the inner disk with an
apex S joined to all i_k. The result G is a closed plane triangulation, so its
medial graph M(G) is 4-regular.
The tread T is the annulus; its full medial tire graph M(T) is the subgraph of
M(G) on the medial vertices of the tread edges (outer, inner and annular edges).
This tread has simple boundaries, hence no bites: the up teeth are the outer
edges, the down teeth the inner edges, and the only valid faces are the outer
face (up apexes) and the root face (down apexes).
Remark 5.8 predicts: every proper 3-colouring of M(G), restricted to M(T), is
Kempe-balanced, i.e. for each colour pair P the up apexes coloured in P are even
in number, and likewise the down apexes. We enumerate colourings of M(G) and
check this.
"""
from __future__ import annotations
import itertools
import networkx as nx
PAIRS = ((0, 1), (0, 2), (1, 2))
def ekey(u, v):
return (u, v) if (str(u), u) <= (str(v), v) else (v, u)
def build_capped_annulus(p: int):
g = nx.Graph()
O = [("o", k) for k in range(p)]
I = [("i", k) for k in range(p)]
outer_edges, inner_edges, annular_edges = [], [], []
for k in range(p):
o, on = O[k], O[(k + 1) % p]
i, ino = I[k], I[(k + 1) % p]
outer_edges.append(ekey(o, on))
inner_edges.append(ekey(i, ino))
annular_edges += [ekey(o, i), ekey(on, i)]
# tread triangles
g.add_edges_from([(o, on), (on, i), (o, i)]) # (o_k,o_{k+1},i_k)
g.add_edges_from([(on, i), (i, ino), (on, ino)]) # (o_{k+1},i_k,i_{k+1})
# caps
for k in range(p):
g.add_edges_from([("N", O[k]), ("N", O[(k + 1) % p])])
g.add_edges_from([("S", I[k]), ("S", I[(k + 1) % p])])
meta = {
"outer_edges": [ekey(*e) for e in outer_edges],
"inner_edges": [ekey(*e) for e in inner_edges],
"annular_edges": [ekey(*e) for e in annular_edges],
}
return g, meta
def medial_graph(g: nx.Graph) -> nx.Graph:
ok, emb = nx.check_planarity(g)
if not ok:
raise ValueError("not planar")
M = nx.Graph()
M.add_nodes_from(ekey(u, v) for u, v in g.edges())
for v in g.nodes():
order = list(emb.neighbors_cw_order(v))
edges = [ekey(v, w) for w in order]
for a in range(len(edges)):
M.add_edge(edges[a], edges[(a + 1) % len(edges)])
return M
def proper_3_colorings(M: nx.Graph, limit: int):
nodes = list(M.nodes())
adj = {v: set(M.neighbors(v)) for v in nodes}
coloring: dict = {}
out = []
def rec(i):
if len(out) >= limit:
return
if i == len(nodes):
out.append(dict(coloring))
return
v = nodes[i]
used = {coloring[w] for w in adj[v] if w in coloring}
for c in (0, 1, 2):
if c not in used:
coloring[v] = c
rec(i + 1)
del coloring[v]
rec(0)
return out
def is_kempe_balanced(coloring, up_apexes, down_apexes):
for face in (up_apexes, down_apexes):
for pair in PAIRS:
if sum(1 for e in face if coloring[e] in pair) % 2 != 0:
return False, face is down_apexes
return True, None
def run(p: int, limit: int = 4000):
g, meta = build_capped_annulus(p)
M = medial_graph(g)
up = meta["outer_edges"]
down = meta["inner_edges"]
colorings = proper_3_colorings(M, limit)
balanced = 0
unbalanced = []
for col in colorings:
ok, _ = is_kempe_balanced(col, up, down)
if ok:
balanced += 1
else:
unbalanced.append(col)
n_ann = len(meta["annular_edges"])
print(f"p={p}: |V(G)|={g.number_of_nodes()} |M(G)|={M.number_of_nodes()} "
f"|A(T)|={n_ann} up={len(up)} down={len(down)}")
print(f" colourings tested={len(colorings)} (cap {limit}) "
f"balanced={balanced} UNBALANCED={len(unbalanced)}")
if unbalanced:
col = unbalanced[0]
upc = [col[e] for e in up]
dnc = [col[e] for e in down]
print(f" first unbalanced restriction: up colours={upc} down colours={dnc}")
return len(unbalanced)
def main():
total_bad = 0
for p in (3, 4, 5, 6):
total_bad += run(p)
print()
print("Remark 5.8 (bite-free) holds on all tested colourings"
if total_bad == 0 else
f"Remark 5.8 (bite-free) FAILS: {total_bad} unbalanced restrictions found")
if __name__ == "__main__":
main()
papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py
+350
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"""Compare full and reduced medial tire graphs on generated tires.
The new medial decomposition paper defines:
* full medial tire graph: the subgraph of M(G) induced by medial
vertices corresponding to edges incident to tread triangles;
* reduced medial tire graph: delete same-boundary medial edges and
chord-only medial edges.
For a tire tread inside an ambient triangulation, the medial edges
visible in the tread come from annular triangular faces. This script
checks whether any same-boundary medial edges are actually present in
that model. It also compares against the older standalone drawing
model, which added artificial outer/inner boundary faces.
"""
from __future__ import annotations
import argparse
import itertools
import random
from collections import Counter
Edge = tuple[int, int]
MedialEdge = tuple[Edge, Edge]
def random_tire(m: int, k: int, n_chords: int = 0, seed: int | None = None) -> dict:
"""Generate the same labelled annular tires used in earlier experiments."""
rng = random.Random(seed)
outer = list(range(m))
inner = list(range(m, m + k))
edges: set[Edge] = set()
for i in range(m):
edges.add(edge_key(outer[i], outer[(i + 1) % m]))
for j in range(k):
edges.add(edge_key(inner[j], inner[(j + 1) % k]))
inner_chords = set()
candidates = []
for a in range(k):
for b in range(a + 2, k):
if not (a == 0 and b == k - 1):
candidates.append((a, b))
rng.shuffle(candidates)
for a, b in candidates:
if len(inner_chords) >= n_chords:
break
if any((a < a2 < b < b2) or (a2 < a < b2 < b) for a2, b2 in inner_chords):
continue
inner_chords.add((a, b))
edges.add(edge_key(inner[a], inner[b]))
edges.add(edge_key(outer[0], inner[0]))
moves = ["O"] * m + ["I"] * k
rng.shuffle(moves)
triangles = []
i, j = 0, 0
for move in moves:
if move == "O":
tri = (outer[i % m], inner[j % k], outer[(i + 1) % m])
triangles.append(tri)
edges.add(edge_key(inner[j % k], outer[(i + 1) % m]))
i += 1
else:
tri = (outer[i % m], inner[j % k], inner[(j + 1) % k])
triangles.append(tri)
edges.add(edge_key(outer[i % m], inner[(j + 1) % k]))
j += 1
return {
"m": m,
"k": k,
"n_chords": len(inner_chords),
"outer": outer,
"inner": inner,
"edges": sorted(edges),
"triangles": triangles,
"inner_chords": sorted(inner_chords),
"lattice_path": "".join(moves),
"seed": seed,
}
def tire_from_path(m: int, k: int, chords: tuple[tuple[int, int], ...], path: str) -> dict:
outer = list(range(m))
inner = list(range(m, m + k))
edges: set[Edge] = set()
for i in range(m):
edges.add(edge_key(outer[i], outer[(i + 1) % m]))
for j in range(k):
edges.add(edge_key(inner[j], inner[(j + 1) % k]))
for a, b in chords:
edges.add(edge_key(inner[a], inner[b]))
edges.add(edge_key(outer[0], inner[0]))
triangles = []
i, j = 0, 0
for move in path:
if move == "O":
tri = (outer[i % m], inner[j % k], outer[(i + 1) % m])
triangles.append(tri)
edges.add(edge_key(inner[j % k], outer[(i + 1) % m]))
i += 1
else:
tri = (outer[i % m], inner[j % k], inner[(j + 1) % k])
triangles.append(tri)
edges.add(edge_key(outer[i % m], inner[(j + 1) % k]))
j += 1
return {
"m": m,
"k": k,
"n_chords": len(chords),
"outer": outer,
"inner": inner,
"edges": sorted(edges),
"triangles": triangles,
"inner_chords": sorted(chords),
"lattice_path": path,
"seed": None,
}
def chord_crosses(c1: tuple[int, int], c2: tuple[int, int]) -> bool:
a, b = c1
c, d = c2
return (a < c < b < d) or (c < a < d < b)
def chord_sets(k: int, max_chords: int) -> list[tuple[tuple[int, int], ...]]:
candidates = []
for a in range(k):
for b in range(a + 2, k):
if not (a == 0 and b == k - 1):
candidates.append((a, b))
out = [()]
def rec(start: int, chosen: tuple[tuple[int, int], ...]) -> None:
if len(chosen) >= max_chords:
return
for idx in range(start, len(candidates)):
chord = candidates[idx]
if any(chord_crosses(chord, old) for old in chosen):
continue
nxt = chosen + (chord,)
out.append(nxt)
rec(idx + 1, nxt)
rec(0, ())
return out
def lattice_paths(m: int, k: int):
for o_positions in itertools.combinations(range(m + k), m):
o_set = set(o_positions)
yield "".join("O" if idx in o_set else "I" for idx in range(m + k))
def edge_key(u: int, v: int) -> Edge:
return tuple(sorted((u, v)))
def face_edges(face: tuple[int, ...]) -> list[Edge]:
return [edge_key(face[i], face[(i + 1) % len(face)]) for i in range(len(face))]
def is_cycle_edge(edge: Edge, cycle: list[int]) -> bool:
cycle_set = set(cycle)
if not set(edge) <= cycle_set:
return False
n = len(cycle)
idx = {v: i for i, v in enumerate(cycle)}
a, b = idx[edge[0]], idx[edge[1]]
return (a - b) % n in (1, n - 1)
def is_inner_chord(edge: Edge, m: int, k: int) -> bool:
u, v = edge
if not (m <= u < m + k and m <= v < m + k):
return False
a, b = u - m, v - m
d = abs(a - b)
return min(d, k - d) != 1
def suppress_reason(e1: Edge, e2: Edge, tire: dict) -> str | None:
outer = tire["outer"]
inner = tire["inner"]
if is_cycle_edge(e1, outer) and is_cycle_edge(e2, outer):
return "outer_boundary"
if is_cycle_edge(e1, inner) and is_cycle_edge(e2, inner):
return "inner_boundary"
m, k = tire["m"], tire["k"]
if is_inner_chord(e1, m, k) or is_inner_chord(e2, m, k):
return "inner_chord"
return None
def medial_from_faces(faces: list[tuple[int, ...]], retained: set[Edge]) -> set[MedialEdge]:
medial_edges: set[MedialEdge] = set()
for face in faces:
boundary = [e for e in face_edges(face) if e in retained]
if len(boundary) < 2:
continue
for i, e in enumerate(boundary):
nxt = boundary[(i + 1) % len(boundary)]
if e != nxt:
medial_edges.add(tuple(sorted((e, nxt))))
return medial_edges
def compare_tire(tire: dict, *, standalone_boundary_faces: bool) -> dict:
annular_faces = [tuple(tri) for tri in tire["triangles"]]
faces = list(annular_faces)
if standalone_boundary_faces:
faces.append(tuple(tire["outer"]))
faces.append(tuple(reversed(tire["inner"])))
# Definition 3.1 includes edges incident to at least one tread triangle.
retained = {e for face in annular_faces for e in face_edges(face)}
full_edges = medial_from_faces(faces, retained)
removed = {me for me in full_edges if suppress_reason(me[0], me[1], tire)}
reduced_edges = full_edges - removed
reasons = Counter(suppress_reason(me[0], me[1], tire) for me in removed)
reasons.pop(None, None)
return {
"vertices": len(retained),
"full_edges": len(full_edges),
"reduced_edges": len(reduced_edges),
"removed": len(removed),
"reasons": reasons,
"examples": sorted(removed)[:5],
}
def run_sweep(args: argparse.Namespace) -> None:
ambient_cases = 0
ambient_differ = []
standalone_cases = 0
standalone_differ = []
ambient_reasons: Counter[str] = Counter()
standalone_reasons: Counter[str] = Counter()
max_chords = args.max_chords
for m in range(args.min_cycle, args.max_cycle + 1):
for k in range(args.min_cycle, args.max_cycle + 1):
for chords in range(max_chords + 1):
for seed in range(args.seeds):
tire = random_tire(m=m, k=k, n_chords=chords, seed=seed)
ambient = compare_tire(tire, standalone_boundary_faces=False)
ambient_cases += 1
ambient_reasons.update(ambient["reasons"])
if ambient["removed"]:
ambient_differ.append((m, k, chords, seed, tire, ambient))
standalone = compare_tire(tire, standalone_boundary_faces=True)
standalone_cases += 1
standalone_reasons.update(standalone["reasons"])
if standalone["removed"]:
standalone_differ.append((m, k, chords, seed, tire, standalone))
print("ambient tread-face model")
print(f" cases checked: {ambient_cases}")
print(f" cases where full != reduced: {len(ambient_differ)}")
print(f" removed-edge reasons: {dict(sorted(ambient_reasons.items()))}")
if ambient_differ:
m, k, chords, seed, tire, result = ambient_differ[0]
print(" first difference:")
print(f" m={m} k={k} requested_chords={chords} seed={seed}")
print(f" path={tire['lattice_path']} chords={tire['inner_chords']}")
print(f" removed examples={result['examples']}")
print()
print("standalone tire-with-boundary-faces model")
print(f" cases checked: {standalone_cases}")
print(f" cases where full != reduced: {len(standalone_differ)}")
print(f" removed-edge reasons: {dict(sorted(standalone_reasons.items()))}")
if standalone_differ:
m, k, chords, seed, tire, result = standalone_differ[0]
print(" first difference:")
print(f" m={m} k={k} requested_chords={chords} seed={seed}")
print(f" path={tire['lattice_path']} chords={tire['inner_chords']}")
print(f" full_edges={result['full_edges']} reduced_edges={result['reduced_edges']}")
print(f" removed examples={result['examples']}")
def run_exhaustive(args: argparse.Namespace) -> None:
ambient_cases = 0
ambient_differ = []
standalone_cases = 0
standalone_differ = []
for m in range(args.min_cycle, args.max_cycle + 1):
for k in range(args.min_cycle, args.max_cycle + 1):
for chords in chord_sets(k, args.max_chords):
for path in lattice_paths(m, k):
tire = tire_from_path(m, k, chords, path)
ambient = compare_tire(tire, standalone_boundary_faces=False)
ambient_cases += 1
if ambient["removed"]:
ambient_differ.append((m, k, chords, path, ambient))
standalone = compare_tire(tire, standalone_boundary_faces=True)
standalone_cases += 1
if standalone["removed"]:
standalone_differ.append((m, k, chords, path, standalone))
print("exhaustive ambient tread-face model")
print(f" cases checked: {ambient_cases}")
print(f" cases where full != reduced: {len(ambient_differ)}")
if ambient_differ:
m, k, chords, path, result = ambient_differ[0]
print(" first difference:")
print(f" m={m} k={k} chords={chords} path={path}")
print(f" removed examples={result['examples']}")
print()
print("exhaustive standalone tire-with-boundary-faces model")
print(f" cases checked: {standalone_cases}")
print(f" cases where full != reduced: {len(standalone_differ)}")
if standalone_differ:
m, k, chords, path, result = standalone_differ[0]
print(" first difference:")
print(f" m={m} k={k} chords={chords} path={path}")
print(f" full_edges={result['full_edges']} reduced_edges={result['reduced_edges']}")
print(f" removed examples={result['examples']}")
def main() -> None:
parser = argparse.ArgumentParser()
parser.add_argument("--min-cycle", type=int, default=3)
parser.add_argument("--max-cycle", type=int, default=8)
parser.add_argument("--max-chords", type=int, default=3)
parser.add_argument("--seeds", type=int, default=50)
parser.add_argument("--exhaustive", action="store_true")
args = parser.parse_args()
if args.exhaustive:
run_exhaustive(args)
else:
run_sweep(args)
if __name__ == "__main__":
main()
papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_evened_leaf.py
+144
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"""Picture: evening a terminal (leaf) triangle by the two-vertex operation:
add y at the midpoint of uv and z at the centroid of uvt, delete uv, add edges
xy, uy, vy, zy, zu, zv, zt. The leaf becomes a 4-wheel tread with hub z.
Panels:
A before: terminal face uvt, level cycle C_k = the 3-cycle (odd seam)
B after: seam u-y-v-t (length 4, even); leaf = 4-wheel with hub z (level k+1)
C medial overlay with the canonical colouring: seam apexes mono-3,
leaf annular 4-cycle alternating 1,2 -- proper, no ears, no defect.
"""
import os
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
HERE = os.path.dirname(os.path.abspath(__file__))
PAL = {0: "#e6550d", 1: "#3182bd", 2: "#31a354"} # colours "1","2","3"
GRAY = "#999999"
u, v, t, x = (-1.0, 0.0), (1.0, 0.0), (0.0, -1.6), (0.0, 1.0)
y = (0.0, 0.0) # midpoint of uv
z = (0.0, -1.6 / 3) # centroid of uvt
def mid(a, b):
return ((a[0] + b[0]) / 2, (a[1] + b[1]) / 2)
def vertex(ax, p, name, dx, dy, color="black"):
ax.plot(*p, "o", color=color, ms=5.5, zorder=6)
ax.annotate(name, p, textcoords="offset points", xytext=(dx, dy),
fontsize=10, fontweight="bold", zorder=6)
def panel_before(ax):
ax.fill([u[0], v[0], t[0]], [u[1], v[1], t[1]], color="#fde9d9")
for a, b in [(u, v), (v, t), (t, u)]:
ax.plot([a[0], b[0]], [a[1], b[1]], color="black", lw=2.4)
for p in (u, v):
ax.plot([x[0], p[0]], [x[1], p[1]], color=GRAY, lw=0.9)
ax.annotate("terminal face\n(leaf of tire tree)", (0, -0.62), ha="center",
fontsize=8, color="#b06030")
vertex(ax, u, "u", -12, -2); vertex(ax, v, "v", 8, -2)
vertex(ax, t, "t", 0, -14); vertex(ax, x, "x", 8, 2)
ax.annotate("(apex in tread above)", x, textcoords="offset points",
xytext=(16, -2), fontsize=7, color=GRAY)
def panel_after(ax, faint=1.0):
# wheel faces
for tri, c in [((u, y, z), "#fde9d9"), ((y, v, z), "#fdf3d9"),
((v, t, z), "#fde9d9"), ((t, u, z), "#fdf3d9")]:
ax.fill([p[0] for p in tri], [p[1] for p in tri], color=c, alpha=faint)
# seam (level cycle) bold: u-y, y-v, v-t, t-u
for a, b in [(u, y), (y, v), (v, t), (t, u)]:
ax.plot([a[0], b[0]], [a[1], b[1]], color="black", lw=2.4, alpha=faint)
# parent spokes xu, xy, xv
for p in (u, y, v):
ax.plot([x[0], p[0]], [x[1], p[1]], color=GRAY, lw=0.9, alpha=faint)
# leaf spokes zu, zy, zv, zt
for p in (u, y, v, t):
ax.plot([z[0], p[0]], [z[1], p[1]], color="black", lw=1.0, alpha=faint)
vertex(ax, u, "u", -12, -2); vertex(ax, v, "v", 8, -2)
vertex(ax, t, "t", 0, -14); vertex(ax, x, "x", 8, 2)
vertex(ax, y, "y", 6, 6); vertex(ax, z, "z", 7, -4)
def panel_medial(ax):
panel_after(ax, faint=0.35)
apexes = {"uy": mid(u, y), "yv": mid(y, v), "vt": mid(v, t), "tu": mid(t, u)}
leaf_ann = {"zu": mid(z, u), "zy": mid(z, y), "zv": mid(z, v), "zt": mid(z, t)}
par_ann = {"ux": mid(u, x), "xy": mid(x, y), "xv": mid(x, v)}
# leaf annular 4-cycle (faces uyz, yvz, vtz, tuz)
ring = ["zu", "zy", "zv", "zt"]
for i in range(4):
a, b = leaf_ann[ring[i]], leaf_ann[ring[(i + 1) % 4]]
ax.plot([a[0], b[0]], [a[1], b[1]], color="#555555", lw=1.6, zorder=4)
# parent annular path m_ux - m_xy - m_xv (faces xuy, xyv)
for a, b in [("ux", "xy"), ("xy", "xv")]:
ax.plot([par_ann[a][0], par_ann[b][0]], [par_ann[a][1], par_ann[b][1]],
color="#555555", lw=1.6, zorder=4)
# apex spokes: each seam apex to its two leaf-annular and two parent nbrs
spokes = [("uy", "zu"), ("uy", "zy"), ("yv", "zy"), ("yv", "zv"),
("vt", "zv"), ("vt", "zt"), ("tu", "zt"), ("tu", "zu")]
for a, b in spokes:
pa, pb = apexes[a], leaf_ann[b]
ax.plot([pa[0], pb[0]], [pa[1], pb[1]], color="#aaaaaa", lw=1.0, zorder=3)
for a, b in [("uy", "ux"), ("uy", "xy"), ("yv", "xy"), ("yv", "xv")]:
pa, pb = apexes[a], par_ann[b]
ax.plot([pa[0], pb[0]], [pa[1], pb[1]], color="#aaaaaa", lw=1.0, zorder=3)
# off-picture parent stubs for m_vt, m_tu
for k, d in [("vt", (0.25, -0.18)), ("tu", (-0.25, -0.18))]:
p = apexes[k]
ax.plot([p[0], p[0] + d[0]], [p[1], p[1] + d[1]], color="#cccccc",
lw=0.9, linestyle=":", zorder=2)
# colours: apexes mono-3 (green); leaf ring alternating 1,2; parent 1,2,1
col = {}
for k in apexes: col[("a", k)] = 2
for k, c in zip(ring, (0, 1, 0, 1)): col[("l", k)] = c
for k, c in zip(("ux", "xy", "xv"), (0, 1, 0)): col[("p", k)] = c
for k, p in apexes.items():
ax.plot(*p, "o", color=PAL[2], ms=11, markeredgecolor="black", zorder=5)
for k, p in leaf_ann.items():
ax.plot(*p, "o", color=PAL[col[("l", k)]], ms=9,
markeredgecolor="black", zorder=5)
for k, p in par_ann.items():
ax.plot(*p, "o", color=PAL[col[("p", k)]], ms=9,
markeredgecolor="black", zorder=5)
lbl = {"uy": (-26, 4), "yv": (12, 4), "vt": (12, -2), "tu": (-30, -2)}
for k, p in apexes.items():
ax.annotate(f"m_{k}", p, textcoords="offset points", xytext=lbl[k],
fontsize=7, zorder=6)
fig, axes = plt.subplots(1, 3, figsize=(14, 4.8))
for ax in axes:
ax.set_xlim(-1.7, 1.7)
ax.set_ylim(-2.1, 1.45)
ax.set_aspect("equal")
ax.axis("off")
panel_before(axes[0])
axes[0].set_title("A. before: leaf = terminal face uvt\nseam C_k = 3-cycle (odd)",
fontsize=9)
panel_after(axes[1])
axes[1].set_title("B. add y = mid(uv), z = centroid; delete uv;\n"
"add xy, uy, vy, zy, zu, zv, zt\n"
"seam u-y-v-t (even); leaf = 4-wheel, hub z (level k+1)",
fontsize=9)
panel_medial(axes[2])
axes[2].set_title("C. medial + canonical colouring:\nseam apexes all 3 (green), "
"leaf ring alternates 1,2 — proper, no ears", fontsize=9)
fig.suptitle(
"Evening a terminal leaf with the two-vertex operation (y splits uv under x; "
"z stellates the leaf as a wheel hub).\n"
"Both new vertices have degree 4; the leaf tread is a 4-wheel with an even "
"annular cycle, so the monochromatic-3 seam is VALID — no leaf defect.",
fontsize=10)
fig.tight_layout(rect=(0, 0, 1, 0.86))
out = os.path.join(HERE, "evened_leaf.png")
fig.savefig(out, dpi=170)
print("wrote", out)
papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_failing_graph.py
+145
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"""Straight-line planar drawing of the minimal genuine obstruction found by the
even-level-cycle programme: the ring triangulation sizes=[3,6,3], leaf='face'
(generator random.Random(2), the 27th graph), 12 vertices. It survives
exhausting insertion sites x tread phases x root colour-orders (residue_phase_
sweep.py: 24 settings, 0 ok) and fails at the leaf-gadget removal step.
Embedding: networkx planar_layout (a canonical-ordering straight-line embedding
of a planar graph), recentred. We additionally VERIFY no two non-incident edges
cross before drawing. Every triangulation on >=4 vertices is 3-connected, so a
crossing-free straight-line embedding is guaranteed to exist.
"""
import os, random
import numpy as np
import networkx as nx
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
from matplotlib.patches import Polygon
import kempe_even_program_harness as H
HERE = os.path.dirname(os.path.abspath(__file__))
LEVCOL = {0: "#d9d9d9", 1: "#9ecae1", 2: "#fc9272"} # by BFS level
def reconstruct(seed, idx):
rng = random.Random(seed)
for _ in range(idx + 1):
sizes, leaf = H.random_profile(rng)
g, outer = H.ring_triangulation(sizes, leaf, rng)
return g, outer
def planar_pos(g):
nxg = nx.Graph()
nxg.add_nodes_from(g.rot)
for ed in g.edges():
a, b = tuple(ed); nxg.add_edge(a, b)
ok, _ = nx.check_planarity(nxg)
assert ok, "graph is not planar?!"
pos = nx.planar_layout(nxg)
pts = np.array([pos[v] for v in g.rot])
c = pts.mean(axis=0); s = np.abs(pts - c).max()
return {v: ((pos[v][0] - c[0]) / s, (pos[v][1] - c[1]) / s) for v in g.rot}
def seg_cross(p, q, r, s):
def o(a, b, c):
return (b[0]-a[0])*(c[1]-a[1]) - (b[1]-a[1])*(c[0]-a[0])
d1, d2, d3, d4 = o(r, s, p), o(r, s, q), o(p, q, r), o(p, q, s)
return ((d1 > 0) != (d2 > 0)) and ((d3 > 0) != (d4 > 0))
def verify_planar(g, pos):
edges = [tuple(e) for e in g.edges()]
bad = []
for i in range(len(edges)):
a, b = edges[i]
for j in range(i + 1, len(edges)):
c, d = edges[j]
if len({a, b, c, d}) < 4:
continue
if seg_cross(pos[a], pos[b], pos[c], pos[d]):
bad.append((edges[i], edges[j]))
return bad
g, outer = reconstruct(2, 26)
g.check()
an = H.Analysis(g.copy(), outer)
pos = planar_pos(g)
bad = verify_planar(g, pos)
print("crossing edge-pairs:", bad if bad else "NONE -- valid straight-line planar embedding")
assert not bad, "embedding has crossings"
terminal = tuple(an.terminal[0])
odd_seam = [c for k, c in an.seams if len(c) % 2][0]
faces = [tuple(f) for f in g.faces()]
outer_set = frozenset(outer)
fig, axes = plt.subplots(1, 2, figsize=(13.5, 6.8))
xs = [p[0] for p in pos.values()]; ys = [p[1] for p in pos.values()]
mx = max(abs(min(xs)), abs(max(xs))); my = max(abs(min(ys)), abs(max(ys)))
for ax in axes:
ax.set_aspect("equal"); ax.axis("off")
ax.set_xlim(-mx - 0.25, mx + 0.25); ax.set_ylim(-my - 0.25, my + 0.35)
def draw_faces(ax):
for f in faces:
if frozenset(f) == outer_set:
continue
ax.add_patch(Polygon([pos[v] for v in f], closed=True,
facecolor="#fbfbfb", edgecolor="none", zorder=0))
def draw_edges(ax, bold=None):
bold = bold or set()
for ed in g.edges():
a, b = tuple(ed)
hot = frozenset((a, b)) in bold
ax.plot([pos[a][0], pos[b][0]], [pos[a][1], pos[b][1]],
color="#cc2222" if hot else "#7a7a7a",
lw=2.8 if hot else 1.1, zorder=2)
def draw_verts(ax, by_level=False):
for v, p in pos.items():
fc = LEVCOL[an.level[v]] if by_level else "white"
ax.plot(*p, "o", ms=20, mfc=fc, mec="#222222", mew=1.6, zorder=4)
ax.annotate(str(v), p, ha="center", va="center", fontsize=10,
fontweight="bold", zorder=5)
draw_faces(axes[0]); draw_edges(axes[0]); draw_verts(axes[0])
axes[0].set_title("A. ring [3,6,3] + face leaf, 12 vertices\n"
"straight-line planar embedding (verified crossing-free)",
fontsize=10)
draw_faces(axes[1])
seam_edges = {frozenset((odd_seam[i], odd_seam[(i+1) % len(odd_seam)]))
for i in range(len(odd_seam))}
axes[1].add_patch(Polygon([pos[v] for v in terminal], closed=True,
facecolor="#fee0d2", edgecolor="none", zorder=1))
draw_edges(axes[1], bold=seam_edges)
draw_verts(axes[1], by_level=True)
tc = np.mean([pos[v] for v in terminal], axis=0)
axes[1].annotate("terminal triangle " + "-".join(map(str, terminal)) +
"\n(level-2 odd seam; carries the leaf\ngadget whose removal "
"STILL FAILS)",
xy=tc, xytext=(0.02, 0.99), textcoords="axes fraction",
ha="left", va="top", fontsize=8, color="#a63603",
arrowprops=dict(arrowstyle="->", color="#a63603", lw=1.2),
zorder=6)
axes[1].set_title("B. BFS levels from source 0-1-2 "
"(grey 0 / blue 1 / red 2)\nodd level-2 seam "
+ "-".join(map(str, odd_seam)) + " bold red",
fontsize=10)
fig.suptitle("Minimal genuine obstruction (seed2 #26): the programme fails here "
"even after exhausting\nsites x tread-phases x root colour-orders "
"(24 settings, 0 ok) -- a face-leaf / gadget case.", fontsize=10)
fig.tight_layout(rect=(0, 0, 1, 0.9))
out = os.path.join(HERE, "failing_graph_seed2_26.png")
fig.savefig(out, dpi=160)
print("wrote", out)
papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_tire_realization.py
+166
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"""Draw every full medial tire graph from the seed-1 analysis, one note each.
For each M(T) found by tire_realization_analysis.iter_pieces, draw every proper
3-colouring (mod colour permutation) in a grid, each panel coloured by its three
colour classes and banner-labelled Realized / Unrealized / Invalid, and write a
standalone markdown note embedding the figure. Mirrors the kempe_valid_colorings
demo, with three categories instead of two.
"""
from __future__ import annotations
import math
import os
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
from tire_realization_analysis import iter_pieces
HERE = os.path.dirname(os.path.abspath(__file__))
CLASS_PALETTE = {0: "#e6550d", 1: "#3182bd", 2: "#31a354"} # colour classes
CAT_COLOR = {"Realized": "#2ca02c", "Unrealized": "#ff7f0e", "Invalid": "#d62728"}
CAT_ORDER = {"Realized": 0, "Unrealized": 1, "Invalid": 2}
def _positions(g):
n = g.n
matched = g.bite_edges
def ann(k):
a = math.pi / 2 - 2 * math.pi * k / n
return math.cos(a), math.sin(a)
def mid(i):
return math.pi / 2 - 2 * math.pi * (i + 0.5) / n
pos = {f"a{k}": ann(k) for k in range(n)}
for i, t in enumerate(g.tooth_word):
if t == "U":
pos[f"u{i}"] = (1.42 * math.cos(mid(i)), 1.42 * math.sin(mid(i)))
elif i not in matched:
pos[f"d{i}"] = (0.58 * math.cos(mid(i)), 0.58 * math.sin(mid(i)))
for i, j in sorted(g.bites):
corners = [ann(i), ann((i + 1) % n), ann(j), ann((j + 1) % n)]
cx = sum(p[0] for p in corners) / 4.0
cy = sum(p[1] for p in corners) / 4.0
pos[f"p{i}_{j}"] = (cx * 0.82, cy * 0.82)
return pos
def _draw(ax, g, pos, coloring, category):
n = g.n
for u, v in g.edges():
ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]],
color="#cccccc", lw=0.4, zorder=1)
for k in range(n):
a, b = f"a{k}", f"a{(k + 1) % n}"
ax.plot([pos[a][0], pos[b][0]], [pos[a][1], pos[b][1]],
color="#777777", lw=0.8, zorder=2)
for v, (x, y) in pos.items():
big = v.startswith("p")
ax.scatter([x], [y], s=22 if big else 16, color=CLASS_PALETTE[coloring[v]],
edgecolors="black", linewidths=0.3, zorder=3)
ax.set_xlim(-1.6, 1.6)
ax.set_ylim(-1.7, 1.6)
ax.set_aspect("equal")
ax.axis("off")
ax.set_title(category, fontsize=6, color=CAT_COLOR[category], pad=1.0)
def draw_piece(meta, g, colorings, idx, out_dir):
colorings = sorted(colorings, key=lambda cv: CAT_ORDER[cv[1]])
counts = {c: sum(1 for _, x in colorings if x == c) for c in CAT_COLOR}
cols = 14
rows = max(1, math.ceil(len(colorings) / cols))
fig, axes = plt.subplots(rows, cols, figsize=(cols * 1.15, rows * 1.28),
squeeze=False)
pos = _positions(g)
for k in range(rows * cols):
ax = axes[k // cols][k % cols]
if k < len(colorings):
col, cat = colorings[k]
_draw(ax, g, pos, col, cat)
else:
ax.axis("off")
bites = ",".join(f"({i},{j})" for i, j in sorted(g.bites)) or "none"
fig.suptitle(
f"M(T) from source {meta['source']}, tread T{meta['tread']}: "
f"|A(T)|={g.n}, word={g.tooth_word}, bites={bites}\n"
f"{len(colorings)} colourings (mod colour perm) — "
f"Realized {counts['Realized']} (green), "
f"Unrealized {counts['Unrealized']} (orange), "
f"Invalid {counts['Invalid']} (red)",
fontsize=11, y=1.0 - 0.0,
)
fig.tight_layout(rect=(0, 0, 1, 0.985))
base = f"piece_{idx:02d}_src{meta['source']}_T{meta['tread']}"
png = os.path.join(out_dir, base + ".png")
pdf = os.path.join(out_dir, base + ".pdf")
fig.savefig(png, dpi=110)
fig.savefig(pdf)
plt.close(fig)
note = os.path.join(out_dir, base + ".md")
with open(note, "w") as fh:
fh.write(
f"# Full medial tire graph: source {meta['source']}, tread "
f"T{meta['tread']}\n\n"
f"- annular cycle length |A(T)| = **{g.n}**\n"
f"- tooth word: `{g.tooth_word}` "
f"({len(g.up_edges)} up, {len(g.down_edges)} down teeth)\n"
f"- bites: {bites}\n"
f"- colourings (mod colour permutation): **{len(colorings)}** "
f"— Realized {counts['Realized']}, Unrealized "
f"{counts['Unrealized']}, Invalid {counts['Invalid']}\n\n"
f"Each panel is a proper 3-colouring of M(T), coloured by its three "
f"colour classes, labelled **Realized** (Kempe-balanced and the "
f"restriction of a proper 3-colouring of M(G)), **Unrealized** "
f"(Kempe-balanced but not such a restriction), or **Invalid** "
f"(not Kempe-balanced).\n\n"
f"![colourings]({base}.png)\n\n"
f"Vector copy: [`{base}.pdf`]({base}.pdf).\n"
)
return base, counts
def main(seed: int = 1):
out_dir = os.path.join(HERE, f"tire_realization_seed{seed}")
os.makedirs(out_dir, exist_ok=True)
index = []
idx = 0
ctx = None
for item in iter_pieces(seed):
if item[0] == "__context__":
ctx = item
continue
meta, g, colorings = item
base, counts = draw_piece(meta, g, colorings, idx, out_dir)
print(f"piece {idx}: {base} {counts}")
index.append((idx, meta, g, counts, base))
idx += 1
_, G, M, n_global = ctx
with open(os.path.join(out_dir, "README.md"), "w") as fh:
fh.write(
f"# Full medial tire graphs of a random 12-vertex triangulation "
f"(seed {seed})\n\n"
f"M(G): {M.number_of_nodes()} medial vertices, {n_global} proper "
f"3-colourings. {len(index)} full medial tire graphs, one note each "
f"below.\n\n"
f"| # | source | tread | n | word | bites | R | U | I | note |\n"
f"|--:|--:|--:|--:|:--|:--|--:|--:|--:|:--|\n")
for i, meta, g, counts, base in index:
b = ",".join(f"({x},{y})" for x, y in sorted(g.bites)) or "-"
fh.write(
f"| {i} | {meta['source']} | T{meta['tread']} | {g.n} | "
f"`{g.tooth_word}` | {b} | {counts['Realized']} | "
f"{counts['Unrealized']} | {counts['Invalid']} | "
f"[{base}.md]({base}.md) |\n")
print(f"wrote {len(index)} notes to {out_dir}")
if __name__ == "__main__":
main()
papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_walkthrough.py
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"""Step-by-step picture of the even-level-cycle programme on the smallest clean
example: the ring triangulation sizes=[3,5], leaf='hub' (rng seed 0), 9 vertices.
One odd level cycle (level 1, the 5-cycle 3-4-5-6-7), no terminal triangles, so
the only surgery is a single DIAMOND. We walk the FIRST successful choice-set
found by the sweep: insertion site = edge (3,4); colour phase = (0,); root DFS
colour order = (1,0,2). Panels:
A G with its odd level-5 seam (BFS levels from the outer triangle 0-1-2).
B G' = G + diamond w(=9) on edge (3,4): seam is now an even 6-cycle; the
diamond quad 3-0-4-8 (restored diagonal 3-4) shaded.
C medial M(G') with the canonical colouring BEFORE any switch: the four quad
medials m(0,3),m(0,4),m(4,8),m(3,8) are ALL colour 1 -> diamond_condition
fails (the obstruction).
D after one {1,2}-Kempe switch on the component through m(0,3)
{(0,3),(3,7),(3,8),(3,9)}: quad medials become 2,1,1,2 -> reducible;
remove w, restored diagonal (3,4) takes the third colour 0.
Embedding: networkx planar_layout (canonical-ordering straight-line embedding),
recentred, with EVERY panel verified crossing-free before drawing -- G, G', and
the medial M(G') drawn at edge midpoints. G' is embedded once and reused for
panels B/C/D; G reuses it (minus w, with the diagonal 3-4 restored) when that is
still crossing-free, else it is embedded independently. Run with the repo venv
python (numpy + matplotlib + networkx).
"""
import os, random
import numpy as np
import networkx as nx
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
from matplotlib.patches import Polygon
import kempe_even_program_harness as H
HERE = os.path.dirname(os.path.abspath(__file__))
PAL = {0: "#e6550d", 1: "#3182bd", 2: "#31a354"} # colours "1","2","3"(=2)
def planar_pos(g):
nxg = nx.Graph()
nxg.add_nodes_from(g.rot)
for ed in g.edges():
a, b = tuple(ed); nxg.add_edge(a, b)
ok, _ = nx.check_planarity(nxg)
assert ok, "graph not planar?!"
pos = nx.planar_layout(nxg)
pts = np.array([pos[v] for v in g.rot]); c = pts.mean(axis=0)
s = np.abs(pts - c).max()
return {v: ((pos[v][0]-c[0])/s, (pos[v][1]-c[1])/s) for v in g.rot}
def seg_cross(p, q, r, s):
def o(a, b, c):
return (b[0]-a[0])*(c[1]-a[1]) - (b[1]-a[1])*(c[0]-a[0])
d1, d2, d3, d4 = o(r, s, p), o(r, s, q), o(p, q, r), o(p, q, s)
return ((d1 > 0) != (d2 > 0)) and ((d3 > 0) != (d4 > 0))
def crossings(edges, pos):
bad = []
for i in range(len(edges)):
a, b = edges[i]
for j in range(i+1, len(edges)):
c, d = edges[j]
if len({a, b, c, d}) < 4:
continue
if seg_cross(pos[a], pos[b], pos[c], pos[d]):
bad.append((edges[i], edges[j]))
return bad
def mid(p, q): return ((p[0]+q[0])/2, (p[1]+q[1])/2)
# ---- build G and G' ------------------------------------------------------
rng = random.Random(0)
g, outer = H.ring_triangulation([3, 5], 'hub', rng)
an = H.Analysis(g.copy(), outer)
ring = [c for k, c in an.seams if k == 1][0]
prep = H._prep_gadgets(g.copy(), outer)
template, an_g, gadgets = prep
gg = template.copy()
w, u, v, x, t = gg.insert_diamond(3, 4)
an2 = H.Analysis(gg, outer)
ring2 = [c for k, c in an2.seams if k == 1][0]
quad = H.quad_of(gg, w, u, v) # (3,0,4,8)
# embed G' (verified), reuse for G if still crossing-free else embed G alone
posGp = planar_pos(gg)
edgesGp = [tuple(e) for e in gg.edges()]
badGp = crossings(edgesGp, posGp)
print("G' crossings:", badGp if badGp else "NONE")
assert not badGp
posG = {vv: posGp[vv] for vv in g.rot}
edgesG = [tuple(e) for e in g.edges()]
badG = crossings(edgesG, posG)
if badG:
print("G reuse crossed; embedding G independently")
posG = planar_pos(g)
badG = crossings([tuple(e) for e in g.edges()], posG)
print("G crossings:", badG if badG else "NONE")
assert not badG
# medial drawn at edge midpoints. The medial drawing at midpoints is planar
# EXCEPT for the medial triangle of whichever face is geometrically OUTER
# (its three midpoint-chords would cut straight across the unbounded region), so
# we omit exactly those three edges, then verify the rest are crossing-free.
def convex_hull(points):
pts = sorted(points)
def cross(o, a, b):
return (a[0]-o[0])*(b[1]-o[1]) - (a[1]-o[1])*(b[0]-o[0])
lo = []
for p in pts:
while len(lo) >= 2 and cross(lo[-2], lo[-1], p) <= 0:
lo.pop()
lo.append(p)
up = []
for p in reversed(pts):
while len(up) >= 2 and cross(up[-2], up[-1], p) <= 0:
up.pop()
up.append(p)
return lo[:-1] + up[:-1]
adjm = H.medial_adj(gg)
mpos = {m: mid(posGp[tuple(m)[0]], posGp[tuple(m)[1]]) for m in adjm}
hull = convex_hull(list(posGp.values()))
pos2v = {tuple(p): v for v, p in posGp.items()}
outer_face = {pos2v[tuple(p)] for p in hull}
print("geometric outer face (hull):", sorted(outer_face))
medges = []
seen = set()
for m in adjm:
for b in adjm[m]:
k = frozenset((m, b))
if k in seen:
continue
seen.add(k)
# skip the medial edge joining two edges of the outer face
if set(m) <= outer_face and set(b) <= outer_face:
continue
medges.append((m, b))
badM = crossings(medges, mpos)
print("M(G') crossings (outer-face medial triangle omitted):",
badM if badM else "NONE")
assert not badM
# ---- colourings ----------------------------------------------------------
col0, _ = H.canonical_coloring_explicit(gg, an2.level, outer, (0,), [1, 0, 2])
col1 = dict(col0)
comp = H.kempe_component(col1, adjm, H.e(0, 3), (1, 2))
H.switch(col1, comp, (1, 2))
third = H.diamond_condition(col1, quad)
col1[H.e(3, 4)] = third
# ---- drawing -------------------------------------------------------------
def lims(ax, pos):
xs = [p[0] for p in pos.values()]; ys = [p[1] for p in pos.values()]
ax.set_xlim(min(xs)-0.25, max(xs)+0.25); ax.set_ylim(min(ys)-0.25, max(ys)+0.3)
def draw_graph(ax, gr, pos, level=None, bold_cycle=None, shade_quad=None, wvert=None):
if shade_quad:
ax.add_patch(Polygon([pos[vv] for vv in shade_quad], closed=True,
color="#ffe2bf", zorder=0))
bold = set()
if bold_cycle:
for i in range(len(bold_cycle)):
bold.add(frozenset((bold_cycle[i], bold_cycle[(i+1) % len(bold_cycle)])))
for ed in gr.edges():
a, b = tuple(ed); pa, pb = pos[a], pos[b]
hot = ed in bold
ax.plot([pa[0], pb[0]], [pa[1], pb[1]],
color="#d62728" if hot else "#888888",
lw=2.8 if hot else 1.1, zorder=2)
for vv, p in pos.items():
c = "#d62728" if (wvert is not None and vv == wvert) else "#222222"
ax.plot(*p, "o", ms=18, mfc="white", mec=c, mew=1.7, zorder=5)
ax.annotate(str(vv), p, ha="center", va="center", fontsize=9,
fontweight="bold", color=c, zorder=6)
if level is not None:
ax.annotate(f"L{level[vv]}", p, textcoords="offset points",
xytext=(11, 10), fontsize=6.5, color="#999999", zorder=6)
def draw_medial(ax, pos, col, halo=None, restored=None):
# medial graph only -- no base graph underneath
for m, b in medges:
pa, pb = mpos[m], mpos[b]
ax.plot([pa[0], pb[0]], [pa[1], pb[1]], color="#c6c6c6", lw=0.8, zorder=1)
if restored is not None:
a, b = restored
ax.plot(*mid(pos[a], pos[b]), "s", color=PAL[col[H.e(a, b)]], ms=13,
mec="#d62728", mew=2.0, zorder=7)
halo = halo or set()
for m, p in mpos.items():
if m not in col:
continue
if m in halo:
ax.plot(*p, "o", color="#000000", ms=16, zorder=5)
ax.plot(*p, "o", color=PAL[col[m]], ms=10.5, mec="black", mew=0.8, zorder=6)
fig, axes = plt.subplots(1, 4, figsize=(19, 5.4))
for ax in axes:
ax.set_aspect("equal"); ax.axis("off")
draw_graph(axes[0], g, posG, level=an.level, bold_cycle=ring)
lims(axes[0], posG)
axes[0].set_title("A. G (BFS levels from source triangle 0-1-2)\n"
"odd level-1 seam = 5-cycle 3-4-5-6-7 (red)\n"
"verified straight-line planar embedding", fontsize=9)
draw_graph(axes[1], gg, posGp, level=an2.level, bold_cycle=ring2,
shade_quad=quad, wvert=w)
lims(axes[1], posGp)
axes[1].set_title("B. G' = G + diamond w=9 on edge (3,4)\n"
"seam now even 6-cycle; quad 3-0-4-8 shaded", fontsize=9)
quad_med = {H.e(quad[i], quad[(i+1) % 4]) for i in range(4)}
draw_medial(axes[2], posGp, col0, halo=quad_med)
lims(axes[2], posGp)
axes[2].set_title("C. M(G') canonical colour (phase 0, DFS order 1,0,2)\n"
"quad medials m(0,3)m(0,4)m(4,8)m(3,8) ALL =1 (haloed)\n"
"-> diamond_condition FAILS", fontsize=9)
draw_medial(axes[3], posGp, col1, halo=comp, restored=(3, 4))
lims(axes[3], posGp)
axes[3].set_title("D. after {1,2}-Kempe switch on comp through m(0,3)\n"
"{(0,3),(3,7),(3,8),(3,9)} (haloed): quad -> 2,1,1,2\n"
f"remove w; restored edge (3,4)=square takes colour {third}",
fontsize=9)
fig.suptitle("Even-level-cycle programme, worked example (ring [3,5]+hub, 9 "
"vertices): one odd seam -> one diamond -> one Kempe switch -> "
"proper 3-colouring of M(G). Colours: 1=orange(0), 2=blue(1), "
"3=green(2).", fontsize=10)
fig.tight_layout(rect=(0, 0, 1, 0.9))
out = os.path.join(HERE, "even_program_walkthrough.png")
fig.savefig(out, dpi=160)
print("wrote", out)
papers/medial_tire_decompositions_of_plane_triangulations/experiments/dump_walkthrough.py
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"""Print every stage of the even-level-cycle programme on the smallest clean
example (ring sizes=[3,5], leaf='hub', rng seed 0; 9 vertices) for the first
choice-set the sweep succeeds on: site (3,4), phase (0,), DFS order (1,0,2).
This is the textual companion to even_program_walkthrough.md / .png.
"""
import random
import kempe_even_program_harness as H
def fz(m): # pretty-print an edge-medial
return tuple(sorted(tuple(m)))
def main():
rng = random.Random(0)
g, outer = H.ring_triangulation([3, 5], 'hub', rng)
print("OUTER (source/root triangle):", outer)
print("\n# STEP 1: graph G (rotation system: vertex -> embedding-order neighbours)")
for v in sorted(g.rot):
print(f" {v}: {g.rot[v]}")
print("faces:", [tuple(f) for f in g.faces()])
print("\n# STEP 2: levels (BFS from the source triangle) + seams")
an = H.Analysis(g.copy(), outer)
for v in sorted(an.level):
print(f" v{v}: level {an.level[v]}")
for k, cyc in an.seams:
print(f" seam level {k}: {cyc} (len {len(cyc)}, "
f"{'ODD' if len(cyc) % 2 else 'even'})")
print(" terminal triangles (need leaf gadget):", an.terminal)
print("\n# STEP 3: diamond sites + chosen edge")
template, an_g, gadgets = H._prep_gadgets(g.copy(), outer)
sites = H._candidate_sites(an_g)
print(" gadgets inserted:", gadgets)
print(" candidate diamond edges (odd seam):", sites)
combo = ((3, 4),)
print(" chosen combo (first successful):", combo)
gg = template.copy()
dia = [gg.insert_diamond(a, b) for (a, b) in combo]
print(" inserted (w,u,v,x,t):", dia)
an2 = H.Analysis(gg, outer)
print(" rot[w]:", gg.rot[dia[0][0]], " level[w]:", an2.level[dia[0][0]])
for k, cyc in an2.seams:
print(f" seam level {k} now: len {len(cyc)} "
f"{'ODD' if len(cyc) % 2 else 'even'} {cyc}")
print("\n# STEP 4: medial graph M(G') (one vertex per edge of G')")
adj = H.medial_adj(gg)
print(f" |V(M)| = {len(adj)}")
for m in sorted(adj, key=fz):
print(f" m{fz(m)}: {sorted(fz(b) for b in adj[m])}")
print("\n# STEP 5: canonical colouring phases=(0,) colorder=(1,0,2)")
phases, colorder = (0,), [1, 0, 2]
sk, _ = H.coloring_skeleton(gg, an2.level, outer)
for i, cyc in enumerate(sk['nonroot']):
print(f" non-root annulus #{i} (len {len(cyc)}): {[fz(m) for m in cyc]}")
print(" root annulus:", [fz(m) for m in sk['root']])
print(" outer-trio (free/DFS):", [fz(m) for m in sk['outer_es']])
col, _ = H.canonical_coloring_explicit(gg, an2.level, outer, phases, colorder)
for m in sorted(col, key=fz):
print(f" m{fz(m)} = {col[m]}")
print("\n# STEP 6: Kempe switch + diamond collapse")
w, u, v, x, t = dia[0]
quad = H.quad_of(gg, w, u, v)
support = [H.e(quad[i], quad[(i + 1) % 4]) for i in range(4)]
print(f" diamond w={w}, quad {quad} (diagonal {u}-{v})")
print(" quad medials:", [fz(s) for s in support])
print(" diamond_condition BEFORE switch:", H.diamond_condition(col, quad),
" support:", {fz(s): col[s] for s in support})
adjm = H.medial_adj(gg)
comp = H.kempe_component(col, adjm, H.e(0, 3), (1, 2))
print(" switch {1,2}-component through m(0,3):", sorted(fz(m) for m in comp))
H.switch(col, comp, (1, 2))
third = H.diamond_condition(col, quad)
print(" diamond_condition AFTER switch:", third,
" support:", {fz(s): col[s] for s in support})
H.collapse_degree4(gg, col, w, u, v)
col[H.e(u, v)] = third
print(f" removed w; restored edge ({u},{v}) takes colour {third}")
print(" proper 3-colouring of M(G)?", H.verify_proper(gg, col))
print(" vertices back to original G?", set(gg.rot) == set(g.rot))
if __name__ == "__main__":
main()
papers/medial_tire_decompositions_of_plane_triangulations/experiments/even_program_findings.md
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# The even-level-cycle colouring program
A constructive route distinct from the `R_T` composition line. Idea: surger a
triangulation `G` so that **every level cycle is even**, take the resulting
*canonical even colouring* of `M(G')` (no 4CT used), then **remove the planted
vertices** by Kempe switches, landing on a proper 3-colouring of `M(G)` — i.e. a
Tait/4CT colouring of `G`.
Scripts: `kempe_even_program_harness.py`, `draw_evened_leaf.py` (`evened_leaf.png`).
## The two surgeries
- **Leaf gadget (two vertices).** On a terminal triangle `uvt` with outer apex
`x`: add `y = mid(uv)` and hub `z`; delete `uv`; add `xy, uy, vy, zy, zu, zv,
zt`. Both new vertices have degree 4; the seam becomes `u-y-v-t` (even) and the
leaf becomes a **4-wheel** with hub `z`. No ears, no chord — the monochromatic-3
seam stays valid, so **leaves create no colouring defect**. (Earlier one-vertex
chord version forced a `{0011,0101}` defect; this is strictly better.)
- **Diamond.** On an odd internal seam edge `uv` with apexes `x,t`: delete `uv`,
add `w ~ u,v,x,t` (degree 4). Flips that seam's parity.
By `n_T = p + Σq_i + 2b`, evening every internal seam makes every annular cycle
even **except the root** (the outer triangle's odd charge `Σ_T n_T ≡ 3 (mod 2)` is
invariant — confirmed; the root is handled as the one unavoidable defect region,
solved by local backtracking).
## Canonical even colouring (constructive, no 4CT)
Every level-edge medial vertex → colour 3; every non-root annular cycle alternates
1,2; the root region solved by DFS. Proper because each apex is forced 3 between
two `{1,2}` pairs and (in the non-degenerate tread model) no two level edges are
consecutive around a vertex or face.
## Removal conditions (degree-4 Kempe reduction — the historically *safe* case)
- **Diamond** `w` (quad `u-x-v-t`, restore diagonal `uv`): removable iff the pair
`(m_ux,m_xv)` is distinct, `(m_vt,m_tu)` is distinct, ≤2 colours total; then
`m_uv` takes the third.
- **Gadget**: collapse `z` then `y` (or `y` then `z`), ending in a degree-3
unstellation needing a rainbow triangle. Two orders = free choice.
## Status (synthetic ring triangulations, the clean-level-structure domain)
Pipeline runs end to end. Surgery, canonical colouring, and gadget removal all
work. The program now lands squarely on the **cycle layer**.
The original `60 random ring triangulations: 39 ok, 21 fail` figure was the
**first-match heuristic** — one diamond per odd seam, placed at the *first*
admissible seam edge, only the colouring phase varied (≤4 random tread phases).
That is one point in the insertion-site design space, not a sweep of it.
**Site sweep (`run_graph` now enumerates every combination of insertion sites,
one per odd seam, ≤4 colour phases each; `--max-combos` caps the product).**
A graph counts `ok` iff *some* placement fully descends:
```
seed 1, 60 graphs: first-match 31 ok / 29 fail -> sweep 54 ok / 6 fail (rescued 23)
seed 2, 60 graphs: first-match 36 ok / 24 fail -> sweep 57 ok / 3 fail (rescued 21)
```
(First-match is seed-sensitive — 31–39 depending on seed; the 39 was one such
seed. What is robust is the *gap*: sweeping insertion sites rescues ~20 of the
~24 first-match failures, leaving a small stubborn residue of ~3–6
`fail:diamond-switch` graphs.) Design space is real but modest: ~50 graphs need
a diamond, ~2900 combos total, max ~900–1200 on a single graph (a handful hit
the cap). So the answer to "did we test every way of adding a diamond?" is:
**now yes** (per odd seam, up to the cap), and most of the apparent failures
were heuristic, not intrinsic.
**Crucial diagnostic:** for a failing case, a simultaneously-removable proper
3-colouring of `M(G')` was shown to **exist** (it must — `M(G)` is 3-colourable).
So `fail:diamond-switch` is **not** non-existence; it is **Kempe-reachability** —
whether switches carry the *canonical even* colouring to a descendable one. That is
exactly the conjecture's core, and the harness has localised the entire program
difficulty to it, with everything upstream constructive.
**Why greedy fails (and what's next).** Diamonds on different odd seams share
*vertical* `{1,3}`-Kempe cycles, so per-diamond local switching cannot satisfy them
simultaneously. The principled solve is joint: vertices = `{1,3}`-Kempe cycles,
one edge per diamond joining its two side cycles; removability for all diamonds at
once = a consistent XOR assignment = **bipartiteness** of that graph (no self-loop =
the side cycles differ; no odd cycle = no three diamonds whose side cycles form a
triangle). Insertion-site choice (which seam edge) and tread phase are the control
knobs. Building this joint solver — and finding the smallest configuration, if any,
forcing a self-loop or odd cycle — is the next step and the exact thing a proof
would need to rule out.
## Exhausting the control knobs over the residue
The site sweep above counts a graph `ok` if some placement works over only **4
random** colour phases per combo. `residue_phase_sweep.py` takes the graphs that
sweep still fails (the residue) and **exhaustively enumerates the colour/tread
phase and the root-DFS colour order** on top of every insertion site:
```
phases in {0,1}^A (A = # non-root annuli; the tread phases)
colorder in perms(0,1,2) (root-region DFS colour priority)
```
over all site combos (cap 512). Result on the seed-1/seed-2 residue
(`residue_phase_sweep_results.txt`):
```
seed1 #16 [3,8,3,5] hub RESCUED (720 settings, 18 ok)
seed1 #51 [3,7,3,3,7] hub RESCUED (42336 settings, 196 ok)
seed1 #3 [3,7,4,6,3] face STILL FAILS (672 settings, 0 ok)
seed1 #4 [3,4,5,5,3] face STILL FAILS (2400 settings, 0 ok)
seed2 #26 [3,6,3] face STILL FAILS (24 settings, 0 ok)
seed2 #30 [3,3,6,7,3] face STILL FAILS (2016 settings, 0 ok)
seed2 #54 [3,3,5,3] face STILL FAILS (720 settings, 0 ok)
```
Two things fall out:
1. **Phase reachability explains part of the residue.** The two `hub` graphs are
*rescued* once the phase/colour-order is enumerated rather than sampled — they
were never genuine obstructions, just unlucky random phases. So the
random-phase `fail` count overstates the true difficulty.
2. **The genuine obstructions are exactly the `face`-leaf graphs.** Every graph
that survives exhausting sites × phases × colour-orders has `leaf='face'` —
i.e. an inner terminal triangle carrying a leaf gadget. The smallest is
`seed2 #26 [3,6,3]` (one site combo, 24 settings, all fail at
`gadget-removal`): a minimal target for the joint solver / an obstruction
hunt. (#26 fails at the gadget step; #3/#4/#30/#54 at `diamond-switch`.)
**Caveat on "STILL FAILS".** `try_establish` is a *bounded* local Kempe search
(≤3 components anchored at the quad support). So `STILL FAILS` means *no (site,
phase, colour-order) lets the bounded search from the canonical-even colouring
reach a descendable one* — not that no Kempe path exists. A descendable colouring
provably exists (M(G) is 3-colourable); whether it is reachable under a principled
(joint, unbounded) switch is the open question, now sharply localised to the
`face`-leaf family.
## Caveats / domain
- Real plantri triangulations mostly `skip:chord-level-edge` under BFS-from-outer
level structure — a reflection of how restrictive the clean nested-tire level
structure is, not a harness bug. The synthetic concentric-ring generator produces
the clean domain the program is stated for.
- Root defect and the (deferred) outer-face handling are localised; the user has a
separate idea for the outer face.
papers/medial_tire_decompositions_of_plane_triangulations/experiments/even_program_walkthrough.md
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# Even-level-cycle programme — a fully worked example
A step-by-step trace of the whole pipeline on the **smallest clean graph**: the
synthetic ring triangulation `sizes=[3,5]`, `leaf='hub'` (generator
`random.Random(0)`), **9 vertices**. It has exactly one odd level cycle and no
terminal triangles, so the only surgery is a **single diamond** — which makes
every stage small enough to print in full.
Everything below is the *actual* state produced by `kempe_even_program_harness.py`
(regenerate the data with the dump at the end of this note; the figure is
`even_program_walkthrough.png`, drawn by `draw_walkthrough.py`). We walk the
**first choice-set the sweep finds that succeeds**:
> insertion site = edge `(3,4)` · colour phase = `(0,)` · root-DFS colour order = `(1,0,2)`
Colour convention throughout: values `{0,1,2}` are Tait colours "1,2,3"; `2` is
the "colour 3" the seam is painted with. In the figure: `0`=orange, `1`=blue,
`2`=green.
---
## Step 1 — Generate the triangulation with a plane embedding *(panel A)*
`ring_triangulation([3,5], 'hub')` builds three concentric rings — an outer
triangle, a 5-ring, and a hub — triangulating each annulus with a random tooth
word and capping the centre with a hub vertex. The result is a genuine plane
triangulation given by its rotation system (neighbours in embedding order):
```
0: [4, 3, 7, 6, 5, 2, 1] 3: [0, 4, 8, 7] 6: [5, 0, 7, 8]
1: [4, 0, 2, 5] 4: [3, 0, 1, 5, 8] 7: [6, 0, 3, 8]
2: [5, 1, 0] 5: [4, 1, 2, 0, 6, 8] 8: [3, 4, 5, 6, 7]
```
The 14 triangular faces (one is the outer/unbounded face) are
```
(4,3,8) (4,0,3) (4,1,0) (4,5,1) (4,8,5) (3,0,7) (3,7,8)
(0,6,7) (0,5,6) (0,2,5) (0,1,2) (1,5,2) (5,8,6) (6,8,7)
```
and the 21 edges are the pairs appearing above. (Euler check: 9 − 21 + 14 = 2.)
The figure draws G (and G′, and the medial M(G′) at edge midpoints) with a
straight-line planar embedding from `networkx.planar_layout`, each **verified
crossing-free** before rendering (the medial triangle of the geometric outer
face is omitted, since its midpoint-chords would otherwise cut across the
unbounded region).
## Step 2 — Pick the source and read off levels *(panel A)*
The **source** is the outer triangle, taken as the unbounded face `(0,1,2)`.
A BFS from those three vertices assigns each vertex its **level** (graph distance
to the source tread):
| level | vertices |
|------:|----------|
| 0 | 0, 1, 2 (the source triangle) |
| 1 | 3, 4, 5, 6, 7 (the ring) |
| 2 | 8 (the hub) |
The **level cycles ("seams")** are the same-level edge cycles at each depth ≥1:
```
level 1: cycle 3-4-5-6-7 length 5 -> ODD
```
There is exactly one seam and it is **odd**. There are **no terminal
triangles**, so the leaf gadget never fires — the only surgery needed is a
diamond on this one odd seam.
## Step 3 — Choose the edge(s) that make the level cycles even *(panel B)*
A diamond can be inserted on any seam edge whose two apexes straddle the
neighbouring levels (`k−1` and `k+1`). For the level-1 seam, **all five** seam
edges qualify:
```
candidate diamond sites: (3,4) (4,5) (5,6) (6,7) (7,3)
```
This is the choice the **site sweep** ranges over (here a 5-element design
space). We take the **first one that leads to a full success: `(3,4)`**.
Insert the diamond on `(3,4)`:
- delete the edge `(3,4)`;
- add a new degree-4 vertex `w = 9` adjacent to `u=3, v=4` and the two apexes
`x=0` (level 0) and `t=8` (level 2), with rotation `rot[9] = [3,0,4,8]`.
`w` lands at level 1, so the level-1 seam becomes the cycle
```
3-9-4-5-6-7 length 6 -> EVEN
```
Every level cycle is now even. The four-cycle `3-0-4-8` around `w` (diagonal the
restored edge `3-4`) is the **diamond quad** we must later collapse — shaded in
panel B.
## Step 4 — Build the medial graph M(G′) *(panels C, D)*
The medial graph has **one vertex per edge of G′** (24 of them) and joins two
edge-medials iff the edges are consecutive around a common face. A 4-colouring
of the triangulation = a proper **3-colouring of M(G′)**. The adjacency (each
medial `m(a,b)` listed with its neighbours):
```
m(0,1): (0,2)(0,4)(1,2)(1,4) m(3,7): (0,3)(0,7)(3,8)(7,8)
m(0,2): (0,1)(0,5)(1,2)(2,5) m(3,8): (3,7)(3,9)(7,8)(8,9)
m(0,3): (0,7)(0,9)(3,7)(3,9) m(3,9): (0,3)(0,9)(3,8)(8,9)
m(0,4): (0,1)(0,9)(1,4)(4,9) m(4,5): (1,4)(1,5)(4,8)(5,8)
m(0,5): (0,2)(0,6)(2,5)(5,6) m(4,8): (4,5)(4,9)(5,8)(8,9)
m(0,6): (0,5)(0,7)(5,6)(6,7) m(4,9): (0,4)(0,9)(4,8)(8,9)
m(0,7): (0,3)(0,6)(3,7)(6,7) m(5,6): (0,5)(0,6)(5,8)(6,8)
m(0,9): (0,3)(0,4)(3,9)(4,9) m(5,8): (4,5)(4,8)(5,6)(6,8)
m(1,2): (0,1)(0,2)(1,5)(2,5) m(6,7): (0,6)(0,7)(6,8)(7,8)
m(1,4): (0,1)(0,4)(1,5)(4,5) m(6,8): (5,6)(5,8)(6,7)(7,8)
m(1,5): (1,2)(1,4)(2,5)(4,5) m(7,8): (3,7)(3,8)(6,7)(6,8)
m(2,5): (0,2)(0,5)(1,2)(1,5) m(8,9): (3,8)(3,9)(4,8)(4,9)
```
## Step 5 — Canonical colouring (no 4CT): seam = 3, annuli alternate, root by DFS *(panel C)*
The canonical colouring is assembled from three deterministic ingredients plus
the two control knobs (phase, DFS order):
1. **Every level-edge medial → colour 3 (=2).** The even seam `3-9-4-5-6-7`
becomes **monochromatic 3**:
`m(3,9)=m(4,9)=m(4,5)=m(5,6)=m(6,7)=m(3,7)=2`.
2. **Each non-root annulus alternates {0,1} with a phase bit.** Here there is one
non-root annulus — the hub spokes between levels 1 and 2:
`[(8,9),(3,8),(7,8),(6,8),(5,8),(4,8)]` (length 6). With **phase 0** it is
coloured `0,1,0,1,0,1`:
`m(8,9)=0, m(3,8)=1, m(7,8)=0, m(6,8)=1, m(5,8)=0, m(4,8)=1`.
3. **The root region** — the level-0↔1 spokes plus the three outer-triangle
medials `m(0,1),m(0,2),m(1,2)` — is solved by a small DFS using colour
priority **`(1,0,2)`**.
The resulting proper colouring of M(G′):
```
m(0,1)=2 m(0,2)=1 m(0,3)=1 m(0,4)=1 m(0,5)=0 m(0,6)=1 m(0,7)=0 m(0,9)=0
m(1,2)=0 m(1,4)=0 m(1,5)=1 m(2,5)=2 m(3,7)=2 m(3,8)=1 m(3,9)=2 m(4,5)=2
m(4,8)=1 m(4,9)=2 m(5,6)=2 m(5,8)=0 m(6,7)=2 m(6,8)=1 m(7,8)=0 m(8,9)=0
```
This is the "no-4CT" colouring of the **evened** graph — proper because the seam
is even (a monochromatic-3 cycle around even-length annuli is consistent). The
only thing standing between it and a colouring of the *original* G is the
diamond.
## Step 6 — Kempe switch, then collapse the diamond *(panel D)*
To remove `w=9` we restore the diagonal `(3,4)` and recolour. The **degree-4
removal condition** on the quad `3-0-4-8` reads: the opposite-corner medial pairs
`(m_{30}, m_{04})` and `(m_{48}, m_{83})` must each be *distinct*, using ≤2
colours total; the restored edge then takes the third.
Read the four quad medials off the canonical colouring:
```
m(0,3)=1 m(0,4)=1 m(4,8)=1 m(3,8)=1 ALL EQUAL
```
The first pair `(m_{30},m_{04}) = (1,1)` is **not** distinct → `diamond_condition`
returns `None`. **This is the obstruction** the bare canonical colouring hits
(haloed in panel C). It is *not* non-existence — a removable colouring exists; we
just have to reach one by a Kempe switch.
**The switch.** The bounded search picks the **`{1,2}`-Kempe component through
`m(0,3)`**:
```
component = { m(0,3), m(3,7), m(3,8), m(3,9) } (pair {1,2})
```
Swapping colours `1↔2` on this component flips `m(0,3): 1→2` and `m(3,8): 1→2`
(the other two are already in `{1,2}` and toggle within the class). The quad
medials become:
```
m(0,3)=2 m(0,4)=1 m(4,8)=1 m(3,8)=2
```
Now `(m_{30},m_{04}) = (2,1)` distinct ✓ and `(m_{48},m_{83}) = (1,2)` distinct ✓,
two colours `{1,2}` used → `diamond_condition` returns the **third colour 0**.
**Collapse.** Delete `w`, restore edge `(3,4)`, and colour the restored medial
`m(3,4) = 0` (the orange square in panel D). The result is verified to be a
**proper 3-colouring of M(G)** on exactly the original 9 vertices — i.e. a
Tait/4CT colouring of the original triangulation, obtained with no appeal to the
4CT.
---
## Reproduce
```bash
python3 dump_walkthrough.py # prints every step's data verbatim
../../../.venv/bin/python draw_walkthrough.py # the 4-panel figure (repo venv: numpy+matplotlib)
```
The reduction here genuinely exercises a Kempe switch. For larger graphs the
same six steps run with more diamonds (one per odd seam, swept over all sites)
and more phase/colour-order choices; the open difficulty is purely whether some
(site, phase) choice lets every diamond's quad become reducible simultaneously
— see `even_program_findings.md`.
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"""Exhaustive generator for full medial tire graphs, indexed by |A(T)|.
Model (Definitions/Remarks 3.1--3.9 of the medial tire decompositions paper).
* The annular medial vertices induce a cycle A(T), the *annular cycle*
(Theorem 3.3). Write n = |A(T)| for its number of vertices = number of
annular faces = number of annular edges e_0,...,e_{n-1}.
* Each edge e_i of A(T) carries exactly one tooth (a triangle of M(T))
whose third vertex is a non-annular apex (Definition 3.4). A tooth is an
*up tooth* (apex in the outer region) or a *down tooth* (apex in the inner
region). We record the tooth types as a word in {U, D}^n.
* No two up teeth share an apex; at most two down teeth share an apex
(Remark 3.5). Two down teeth sharing an apex form a *bite* (Definition
3.7). So the down teeth are partitioned into singletons and bite pairs.
A bite pairs two down-edges and is drawn as an apex inside the disk with
spokes to the four endpoints; bites must be mutually non-crossing, i.e.
the bite pairs form a non-crossing (laminar) matching of the down-edges.
The two annular edges of a bite must be non-incident (Definition 3.7):
they share no annular vertex, so cyclically adjacent edges cannot pair.
* There are at least three up teeth (Remark 3.6).
* Bite-face condition (Remark 3.8). Let B(T) = A(T) together with the bite
apexes. Its interior non-tooth faces are the root face plus one inner-gap
face per bite. A singleton down tooth lies in the innermost bite enclosing
its edge (or in the root face if none). For every interior non-tooth face
the number of down-tooth apexes lying in that face must be 0 or at least 3.
Equivalently: no face holds exactly one or two singleton down teeth.
The generator enumerates, for a given n, every (tooth word, bite matching)
pair satisfying these properties and emits the resulting full medial tire
graph as an explicit vertex/edge structure. Configurations may optionally be
reduced modulo the dihedral symmetry of the cycle.
"""
from __future__ import annotations
import argparse
import itertools
from collections import defaultdict
from dataclasses import dataclass
from functools import lru_cache
from typing import Iterator
# A bite is an unordered pair of down-edge indices (i, j) with i < j.
Bite = tuple[int, int]
Matching = frozenset[Bite]
# ---------------------------------------------------------------------------
# Non-crossing (laminar) matchings of the down edges.
# ---------------------------------------------------------------------------
@lru_cache(maxsize=None)
def noncrossing_matchings(positions: tuple[int, ...]) -> tuple[Matching, ...]:
"""All non-crossing partial matchings of ``positions`` (sorted ascending).
Bite pairs drawn inside the disk are non-crossing iff, read in cyclic
order, no two pairs interleave. Cutting the cycle at the gap before the
first edge turns this into ordinary non-crossing interval matchings, which
obey the Catalan recursion below.
"""
if not positions:
return (frozenset(),)
head, *rest = positions
out: list[Matching] = []
# head left unmatched (a singleton down tooth, if its edge is down)
for tail in noncrossing_matchings(tuple(rest)):
out.append(tail)
# head matched with positions[k]; the strictly-enclosed block must be
# matched within itself to stay non-crossing.
for k in range(1, len(positions)):
partner = positions[k]
inside = tuple(positions[1:k])
outside = tuple(positions[k + 1:])
for m_in in noncrossing_matchings(inside):
for m_out in noncrossing_matchings(outside):
out.append(frozenset({(head, partner)}) | m_in | m_out)
return tuple(out)
# ---------------------------------------------------------------------------
# The bite-face condition (Remark 3.8).
# ---------------------------------------------------------------------------
def incident_edges(i: int, j: int, n: int) -> bool:
"""Whether annular edges i and j share an annular vertex on the n-cycle."""
return (j - i) % n == 1 or (i - j) % n == 1
def has_incident_bite(bites: Matching, n: int) -> bool:
"""Whether any bite pairs two incident (cyclically adjacent) edges."""
return any(incident_edges(i, j, n) for i, j in bites)
def innermost_bite(edge: int, bites: Matching) -> Bite | None:
"""The minimal-span bite whose open interval contains ``edge``, or None."""
enclosing = [b for b in bites if b[0] < edge < b[1]]
if not enclosing:
return None
return min(enclosing, key=lambda b: b[1] - b[0])
def face_singleton_counts(
tooth_word: str, bites: Matching
) -> dict[Bite | None, int]:
"""Down-singletons per interior non-tooth face of B(T).
The key ``None`` is the root face; a bite key is that bite's inner-gap
face. Faces with no singletons are simply absent from the result.
"""
matched = {edge for pair in bites for edge in pair}
counts: dict[Bite | None, int] = defaultdict(int)
for edge, tooth in enumerate(tooth_word):
if tooth != "D" or edge in matched:
continue # only singleton down teeth contribute apexes
counts[innermost_bite(edge, bites)] += 1
return dict(counts)
def satisfies_bite_face_condition(tooth_word: str, bites: Matching) -> bool:
"""Remark 3.8: every non-tooth face holds 0 or >=3 down-tooth apexes."""
return all(count >= 3 for count in face_singleton_counts(tooth_word, bites).values())
# ---------------------------------------------------------------------------
# The full medial tire graph as an explicit object.
# ---------------------------------------------------------------------------
@dataclass(frozen=True)
class FullMedialTireGraph:
"""A full medial tire graph M(T) determined by its combinatorial data.
Vertices are named:
a{k} annular medial vertex k (k = 0..n-1), forming A(T);
u{i} apex of the up tooth on edge i;
d{i} apex of the singleton down tooth on edge i;
p{i}_{j} apex of the bite pairing edges i and j (i < j).
"""
n: int
tooth_word: str
bites: Matching
@property
def up_edges(self) -> tuple[int, ...]:
return tuple(i for i, t in enumerate(self.tooth_word) if t == "U")
@property
def down_edges(self) -> tuple[int, ...]:
return tuple(i for i, t in enumerate(self.tooth_word) if t == "D")
@property
def bite_edges(self) -> frozenset[int]:
return frozenset(edge for pair in self.bites for edge in pair)
@property
def singleton_down_edges(self) -> tuple[int, ...]:
bite = self.bite_edges
return tuple(i for i in self.down_edges if i not in bite)
def apex_of_edge(self, edge: int) -> str:
if self.tooth_word[edge] == "U":
return f"u{edge}"
for i, j in self.bites:
if edge in (i, j):
return f"p{i}_{j}"
return f"d{edge}"
def vertices(self) -> list[str]:
verts = [f"a{k}" for k in range(self.n)]
for i in self.up_edges:
verts.append(f"u{i}")
for i in self.singleton_down_edges:
verts.append(f"d{i}")
for i, j in sorted(self.bites):
verts.append(f"p{i}_{j}")
return verts
def edges(self) -> list[tuple[str, str]]:
n = self.n
out: list[tuple[str, str]] = []
# annular cycle A(T)
for k in range(n):
out.append((f"a{k}", f"a{(k + 1) % n}"))
# singleton teeth (up and down): two spokes each
for i in self.up_edges:
out += [(f"u{i}", f"a{i}"), (f"u{i}", f"a{(i + 1) % n}")]
for i in self.singleton_down_edges:
out += [(f"d{i}", f"a{i}"), (f"d{i}", f"a{(i + 1) % n}")]
# bites: a shared apex with four spokes
for i, j in sorted(self.bites):
apex = f"p{i}_{j}"
for edge in (i, j):
out += [(apex, f"a{edge}"), (apex, f"a{(edge + 1) % n}")]
return [tuple(sorted(e)) for e in out]
def canonical_key(self) -> tuple:
"""Representative under the dihedral group of the cycle (rotations and
reflections), so symmetric configurations collapse to one key."""
n = self.n
best: tuple | None = None
for a in (1, -1):
for b in range(n):
relabel = lambda i: (a * i + b) % n
word = [""] * n
for i, t in enumerate(self.tooth_word):
word[relabel(i)] = t
mapped = tuple(sorted(
tuple(sorted((relabel(i), relabel(j)))) for i, j in self.bites
))
key = (tuple(word), mapped)
if best is None or key < best:
best = key
return best
# ---------------------------------------------------------------------------
# Enumeration.
# ---------------------------------------------------------------------------
def generate(
n: int, min_up_teeth: int = 3, dedup: bool = False
) -> Iterator[FullMedialTireGraph]:
"""Yield every full medial tire graph whose annular cycle has size ``n``.
``min_up_teeth`` defaults to 3 (Remark 3.6). With ``dedup`` set, only one
representative per dihedral symmetry class is returned.
"""
seen: set[tuple] = set()
for word_tuple in itertools.product("UD", repeat=n):
tooth_word = "".join(word_tuple)
if tooth_word.count("U") < min_up_teeth:
continue
down = tuple(i for i, t in enumerate(tooth_word) if t == "D")
for bites in noncrossing_matchings(down):
if has_incident_bite(bites, n):
continue
if not satisfies_bite_face_condition(tooth_word, bites):
continue
graph = FullMedialTireGraph(n=n, tooth_word=tooth_word, bites=bites)
if dedup:
key = graph.canonical_key()
if key in seen:
continue
seen.add(key)
yield graph
# ---------------------------------------------------------------------------
# CLI.
# ---------------------------------------------------------------------------
def figure_one() -> FullMedialTireGraph:
"""The example graph of Figure 1 (Remark 3.8): 12 edges, one bite (0,6)."""
return FullMedialTireGraph(
n=12,
tooth_word="DDDDDUDUUUUU", # edges 0-4,6 down; 5,7,8,9,10,11 up
bites=frozenset({(0, 6)}),
)
def describe(graph: FullMedialTireGraph) -> str:
counts = face_singleton_counts(graph.tooth_word, graph.bites)
face_strs = []
for face, c in sorted(counts.items(), key=lambda kv: (kv[0] is not None, kv[0])):
name = "root" if face is None else f"bite{face}"
face_strs.append(f"{name}:{c}")
bites = ",".join(f"({i},{j})" for i, j in sorted(graph.bites)) or "-"
faces = " ".join(face_strs) or "-"
return (
f"word={graph.tooth_word} up={len(graph.up_edges)} "
f"down={len(graph.down_edges)} bites={bites} faces[{faces}]"
)
def run(args: argparse.Namespace) -> None:
if args.check_figure:
g = figure_one()
print("Figure 1 check:")
print(f" {describe(g)}")
ok = satisfies_bite_face_condition(g.tooth_word, g.bites)
print(f" satisfies Remark 3.8: {ok} (expect True; faces 4 and 0)")
print()
for n in range(args.min_n, args.max_n + 1):
graphs = list(generate(n, min_up_teeth=args.min_up, dedup=args.dedup))
label = "classes" if args.dedup else "graphs"
print(f"n={n}: {len(graphs)} {label}")
if args.show:
for g in graphs[: args.show]:
print(f" {describe(g)}")
def main() -> None:
parser = argparse.ArgumentParser(description=__doc__)
parser.add_argument("--min-n", type=int, default=3)
parser.add_argument("--max-n", type=int, default=8)
parser.add_argument("--min-up", type=int, default=3, help="Remark 3.6 bound")
parser.add_argument("--dedup", action="store_true",
help="reduce modulo dihedral symmetry of the cycle")
parser.add_argument("--show", type=int, default=0,
help="print up to this many graphs per n")
parser.add_argument("--check-figure", action="store_true",
help="verify the Figure 1 example against Remark 3.8")
run(parser.parse_args())
if __name__ == "__main__":
main()
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# Atlas of full medial tire graphs with |A(T)| = 9
This note collects every full medial tire graph whose annular cycle `A(T)` has
nine vertices, generated exhaustively from the structural properties in
Definitions/Remarks 3.1–3.9 of `paper.tex`.
## What is being enumerated
A full medial tire graph of size `n = |A(T)|` is determined by:
- a tooth word in `{U, D}^n` — one up (`U`) or down (`D`) tooth per annular
edge (Def. 3.4), with **at least three up teeth** (Rem. 3.6);
- a **non-crossing matching** of the down edges into *bites* — pairs of down
teeth sharing an apex (Rem. 3.5, Def. 3.7); unmatched down teeth are
singletons. The two annular edges of a bite must be **non-incident**
(Def. 3.7): they share no annular vertex, so cyclically adjacent edges
cannot pair;
- subject to the **bite-face condition** (Rem. 3.8): in `B(T) = A(T) + bite
apexes`, every interior non-tooth face must contain `0` or `≥ 3`
down-tooth apexes in its interior (equivalently, no face holds exactly one
or two singleton down teeth).
Graphs are identified up to the dihedral symmetry of the annular cycle
(rotations and reflections), since these give isomorphic plane graphs.
## The atlas
![All full medial tire graphs with |A(T)| = 9](full_medial_tire_n9.png)
High-resolution vector copy: [`full_medial_tire_n9.pdf`](full_medial_tire_n9.pdf).
Full textual index: [`full_medial_tire_n9_index.txt`](full_medial_tire_n9_index.txt).
In each diagram the thick black ring is `A(T)`; **blue** outer apexes are up
teeth, **red** inner apexes are singleton down teeth, and a **dark-red** inner
apex with four spokes is a bite (its two paired annular edges). The label under
each diagram is the tooth word and the bite pairs (edge indices).
## Counts
There are **81** classes for `n = 9` (cf. `3:1, 4:1, 5:2, 6:6, 7:13, 8:36,
9:81` for `n = 3..9`, with the non-incidence stipulation in force). Breakdown
of the 81 classes:
| down teeth | classes | | bites | classes | | up teeth | classes |
|-----------:|--------:|---|------:|--------:|---|---------:|--------:|
| 0 | 1 | | 0 | 35 | | 3 | 23 |
| 2 | 3 | | 1 | 35 | | 4 | 29 |
| 3 | 7 | | 2 | 8 | | 5 | 18 |
| 4 | 18 | | 3 | 3 | | 6 | 7 |
| 5 | 29 | | | | | 7 | 3 |
| 6 | 23 | | | | | 9 | 1 |
46 of the 81 classes contain at least one bite. (Every singleton down tooth
must sit in a face holding `≥ 3` of them, so e.g. words with exactly one or two
down teeth only survive when those down teeth are paired into a bite — and now
only when the paired edges are non-incident, which is why the counts fall
sharply from the unrestricted `n = 9` total of 159.)
## Reproduce
```sh
# from this directory, using the repo .venv
../../../.venv/bin/python plot_full_medial_tire_n9.py # figure + index
python full_medial_tire_generator.py --min-n 9 --max-n 9 --dedup --show 5
```
`full_medial_tire_generator.py` is the generator (`generate(n, dedup=True)`
yields `FullMedialTireGraph` objects); `plot_full_medial_tire_n9.py` draws the
atlas.
papers/medial_tire_decompositions_of_plane_triangulations/experiments/full_medial_tire_n9_index.txt
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0 word=UUUUUUUUU up=9 down=0 bites=-
1 word=UUUUUUDUD up=7 down=2 bites=(6,8)
2 word=UUUUUUDDD up=6 down=3 bites=-
3 word=UUUUUDUUD up=7 down=2 bites=(5,8)
4 word=UUUUUDUDD up=6 down=3 bites=-
5 word=UUUUUDDDD up=5 down=4 bites=-
6 word=UUUUDUUUD up=7 down=2 bites=(4,8)
7 word=UUUUDUUDD up=6 down=3 bites=-
8 word=UUUUDUDUD up=6 down=3 bites=-
9 word=UUUUDUDDD up=5 down=4 bites=-
10 word=UUUUDDUDD up=5 down=4 bites=-
11 word=UUUUDDUDD up=5 down=4 bites=(4,8),(5,7)
12 word=UUUUDDDDD up=4 down=5 bites=-
13 word=UUUUDDDDD up=4 down=5 bites=(4,8)
14 word=UUUDUUUDD up=6 down=3 bites=-
15 word=UUUDUUDUD up=6 down=3 bites=-
16 word=UUUDUUDDD up=5 down=4 bites=-
17 word=UUUDUDUDD up=5 down=4 bites=-
18 word=UUUDUDUDD up=5 down=4 bites=(3,8),(5,7)
19 word=UUUDUDDUD up=5 down=4 bites=-
20 word=UUUDUDDUD up=5 down=4 bites=(3,5),(6,8)
21 word=UUUDUDDDD up=4 down=5 bites=-
22 word=UUUDUDDDD up=4 down=5 bites=(3,5)
23 word=UUUDUDDDD up=4 down=5 bites=(3,8)
24 word=UUUDDUUDD up=5 down=4 bites=-
25 word=UUUDDUUDD up=5 down=4 bites=(3,8),(4,7)
26 word=UUUDDUDDD up=4 down=5 bites=-
27 word=UUUDDUDDD up=4 down=5 bites=(4,6)
28 word=UUUDDUDDD up=4 down=5 bites=(3,8)
29 word=UUUDDDDDD up=3 down=6 bites=-
30 word=UUUDDDDDD up=3 down=6 bites=(3,8)
31 word=UUDUUDUUD up=6 down=3 bites=-
32 word=UUDUUDUDD up=5 down=4 bites=-
33 word=UUDUUDUDD up=5 down=4 bites=(2,8),(5,7)
34 word=UUDUUDDDD up=4 down=5 bites=-
35 word=UUDUUDDDD up=4 down=5 bites=(2,5)
36 word=UUDUDUUDD up=5 down=4 bites=-
37 word=UUDUDUUDD up=5 down=4 bites=(2,8),(4,7)
38 word=UUDUDUDUD up=5 down=4 bites=-
39 word=UUDUDUDUD up=5 down=4 bites=(2,4),(6,8)
40 word=UUDUDUDUD up=5 down=4 bites=(2,8),(4,6)
41 word=UUDUDUDDD up=4 down=5 bites=-
42 word=UUDUDUDDD up=4 down=5 bites=(4,6)
43 word=UUDUDUDDD up=4 down=5 bites=(2,4)
44 word=UUDUDUDDD up=4 down=5 bites=(2,8)
45 word=UUDUDDUDD up=4 down=5 bites=-
46 word=UUDUDDUDD up=4 down=5 bites=(5,7)
47 word=UUDUDDUDD up=4 down=5 bites=(2,4)
48 word=UUDUDDUDD up=4 down=5 bites=(2,8)
49 word=UUDUDDDUD up=4 down=5 bites=-
50 word=UUDUDDDUD up=4 down=5 bites=(6,8)
51 word=UUDUDDDUD up=4 down=5 bites=(2,8)
52 word=UUDUDDDDD up=3 down=6 bites=-
53 word=UUDUDDDDD up=3 down=6 bites=(2,4)
54 word=UUDUDDDDD up=3 down=6 bites=(2,8)
55 word=UUDDUUDDD up=4 down=5 bites=-
56 word=UUDDUUDDD up=4 down=5 bites=(3,6)
57 word=UUDDUDUDD up=4 down=5 bites=-
58 word=UUDDUDUDD up=4 down=5 bites=(5,7)
59 word=UUDDUDUDD up=4 down=5 bites=(2,8)
60 word=UUDDUDDDD up=3 down=6 bites=-
61 word=UUDDUDDDD up=3 down=6 bites=(3,5)
62 word=UUDDUDDDD up=3 down=6 bites=(2,8)
63 word=UUDDDUDDD up=3 down=6 bites=-
64 word=UUDDDUDDD up=3 down=6 bites=(4,6)
65 word=UUDDDUDDD up=3 down=6 bites=(2,8)
66 word=UUDDDUDDD up=3 down=6 bites=(2,8),(3,7),(4,6)
67 word=UDUDUDUDD up=4 down=5 bites=-
68 word=UDUDUDUDD up=4 down=5 bites=(5,7)
69 word=UDUDUDUDD up=4 down=5 bites=(3,5)
70 word=UDUDUDDDD up=3 down=6 bites=-
71 word=UDUDUDDDD up=3 down=6 bites=(3,5)
72 word=UDUDUDDDD up=3 down=6 bites=(1,3)
73 word=UDUDDUDDD up=3 down=6 bites=-
74 word=UDUDDUDDD up=3 down=6 bites=(4,6)
75 word=UDUDDUDDD up=3 down=6 bites=(1,3)
76 word=UDUDDUDDD up=3 down=6 bites=(1,8)
77 word=UDUDDUDDD up=3 down=6 bites=(1,8),(3,7),(4,6)
78 word=UDDUDDUDD up=3 down=6 bites=-
79 word=UDDUDDUDD up=3 down=6 bites=(5,7)
80 word=UDDUDDUDD up=3 down=6 bites=(1,8),(2,4),(5,7)
papers/medial_tire_decompositions_of_plane_triangulations/experiments/full_vs_reduced_medial_tire_findings.md
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# Full vs Reduced Medial Tire Findings
Question: do Definition 3.1 (full medial tire graph) and Definition 3.2
(reduced medial tire graph) differ?
## Experiment
Script:
```bash
python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py
```
The script compares two models.
- Ambient tread-face model: medial edges are contributed by annular
triangular faces of the tire tread inside the ambient triangulation.
- Standalone tire-with-boundary-faces model: the outer and inner
boundary walks are also treated as faces, as in the older drawing
script.
## Random Sweep
Command:
```bash
python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py
```
Result:
```text
ambient tread-face model
cases checked: 7200
cases where full != reduced: 0
removed-edge reasons: {}
standalone tire-with-boundary-faces model
cases checked: 7200
cases where full != reduced: 7200
removed-edge reasons: {'inner_boundary': 39600, 'outer_boundary': 39600}
first difference:
m=3 k=3 requested_chords=0 seed=0
path=IOOOII chords=[]
full_edges=24 reduced_edges=18
removed examples=[((0, 1), (0, 2)), ((0, 1), (1, 2)), ((0, 2), (1, 2)), ((3, 4), (3, 5)), ((3, 4), (4, 5))]
```
## Exhaustive Small Sweep
Command:
```bash
python3 papers/medial_tire_decompositions_of_plane_triangulations/experiments/compare_full_reduced_medial_tires.py --exhaustive --max-cycle 5 --max-chords 2
```
Result:
```text
exhaustive ambient tread-face model
cases checked: 5578
cases where full != reduced: 0
exhaustive standalone tire-with-boundary-faces model
cases checked: 5578
cases where full != reduced: 5578
first difference:
m=3 k=3 chords=() path=OOOIII
full_edges=24 reduced_edges=18
removed examples=[((0, 1), (0, 2)), ((0, 1), (1, 2)), ((0, 2), (1, 2)), ((3, 4), (3, 5)), ((3, 4), (4, 5))]
```
## Interpretation
For the intended ambient-triangulation definition, the experiments
support the suspicion that Definition 3.1 and Definition 3.2 coincide:
same-boundary medial edges do not arise from annular triangular tread
faces, and inner chords are not incident to tread triangles.
They differ only in the standalone tire-with-boundary-faces model,
where the artificial outer and inner boundary faces create medial edges
between consecutive boundary edges.
papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_branching_min_probe.py
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"""Smallest n admitting a BRANCHING tile (>=2 inner faces carrying interfaces),
both unrestricted and under the no-separating-triangle restriction.
A branching tile is a tree node with >=2 children: >=2 inner non-tooth faces each
holding singleton down teeth. Under "no separating triangle" we additionally forbid
any length-3 boundary (outer p=3, or an inner face with exactly 3 singletons), so
every inner face must have >=4 singletons and p>=4.
Purely structural (face singleton counts) -- no colouring enumeration.
Run: python3 kempe_branching_min_probe.py --min-n 9 --max-n 16
"""
from __future__ import annotations
import argparse
import sys
import time
from collections import defaultdict
from full_medial_tire_generator import generate, innermost_bite
def inner_face_sizes(g):
faces = defaultdict(int)
for e in g.singleton_down_edges:
faces[innermost_bite(e, g.bites)] += 1
return [c for c in faces.values() if c >= 3]
def run(args):
print("smallest n with branching tiles (>=2 inner faces holding singletons)\n")
print(f"{'n':>3} {'#tiles':>8} {'branching':>10} {'no-tri branch':>14} example (no-tri branching)")
print("-" * 78)
for n in range(args.min_n, args.max_n + 1):
t0 = time.time()
ntiles = nbr = nbr_notri = 0
example = None
for g in generate(n, min_up_teeth=3, dedup=True):
ntiles += 1
sizes = inner_face_sizes(g)
if len(sizes) >= 2:
nbr += 1
p = len(g.up_edges)
if p >= 4 and all(s >= 4 for s in sizes):
nbr_notri += 1
if example is None:
bites = ",".join(f"({i},{j})" for i, j in sorted(g.bites))
example = f"word={g.tooth_word} bites={bites} p={p} faces={sorted(sizes)}"
dt = time.time() - t0
print(f"{n:>3} {ntiles:>8} {nbr:>10} {nbr_notri:>14} {example or '-'} [{dt:.0f}s]")
sys.stdout.flush()
def main():
parser = argparse.ArgumentParser(description=__doc__)
parser.add_argument("--min-n", type=int, default=9)
parser.add_argument("--max-n", type=int, default=16)
run(parser.parse_args())
if __name__ == "__main__":
main()
papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences.py
+403
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"""Singleton-down-apex colour sequences of Kempe-balanced 3-colourings.
Inner-face counterpart of ``kempe_up_tooth_sequences.py``. There the role of
the distinguished valid face was the unique *outer* face, which carries every
up-tooth apex; here we play the same game on an *inner* non-tooth face of B(T)
(the root face, or a bite's inner-gap face), which carries the singleton
down-tooth apexes assigned to it.
For a fixed annular size ``n`` and a fixed count ``m`` we:
1. take every full medial tire graph M(T) with |A(T)| = n (one representative
per dihedral symmetry class) that has an inner non-tooth face F holding
exactly ``m`` singleton down-tooth apexes -- by Remark 3.8 every such face
holds 0 or >= 3, so m >= 3. (At n = 9 each M(T) has at most one inner face
bearing singletons, so a graph and its face coincide one-to-one.)
2. enumerate the Kempe-balanced (``valid``) proper 3-colourings of M(T) and
read off the colour sequence of F's singleton down-tooth apexes d_i in
increasing annular-edge order (their cyclic order along F's arc);
3. reduce each sequence modulo the six colour permutations -- NOT modulo the
dihedral symmetry of the cycle -- to a canonical sequence;
4. group the configurations by the *set* of canonical down-apex sequences
they realise, and report how many share each set.
The Kempe-balanced rule is global, so the same valid colourings are used as in
the up-tooth experiment; only the apex set we read changes. Note that the rule
forces, on every valid face, each colour pair to meet the counted apexes an even
number of times -- so the down-apex sequences obey the same equal-parity law as
the up-tooth sequences.
Run: python3 kempe_down_face_sequences.py --n 9 --m 3
"""
from __future__ import annotations
import argparse
import math
import os
from collections import defaultdict
from full_medial_tire_generator import (
FullMedialTireGraph,
generate,
innermost_bite,
)
from kempe_valid_colorings import classify_colorings
from kempe_up_tooth_sequences import (
PALETTE,
PALETTE_NAME,
_parity_partitions,
_positions,
canonical_sequence,
compact_coloring,
dihedral_reading_sequences,
seq_str,
)
HERE = os.path.dirname(os.path.abspath(__file__))
Coloring = dict[str, int]
Bite = tuple[int, int]
FaceKey = Bite | None # None = root face
def face_name(face: FaceKey) -> str:
return "root" if face is None else f"bite({face[0]},{face[1]})"
def inner_face_singletons(graph: FullMedialTireGraph) -> dict[FaceKey, list[int]]:
"""Map each inner non-tooth face to its singleton down-tooth edges (sorted)."""
perface: dict[FaceKey, list[int]] = defaultdict(list)
for e in graph.singleton_down_edges:
perface[innermost_bite(e, graph.bites)].append(e)
return {face: sorted(edges) for face, edges in perface.items()}
def config_id(idx: int) -> str:
return f"C{idx:02d}"
def describe_config(graph: FullMedialTireGraph, face: FaceKey, edges) -> str:
bites = ",".join(f"({i},{j})" for i, j in sorted(graph.bites)) or "-"
apexes = ",".join(f"d{e}" for e in edges)
return (f"word={graph.tooth_word} bites={bites} "
f"face={face_name(face)} apexes=[{apexes}]")
# ---------------------------------------------------------------------------
# Data collection.
# ---------------------------------------------------------------------------
class Config:
"""A single (M(T), inner face) specimen with m singleton down apexes."""
def __init__(self, graph: FullMedialTireGraph, face: FaceKey, edges: list[int]):
self.graph = graph
self.face = face
self.edges = edges # singleton down edges on the face
self.apexes = [f"d{e}" for e in edges] # their apex vertex names
def sequence(self, coloring: Coloring) -> tuple[int, ...]:
return tuple(coloring[v] for v in self.apexes)
class Experiment:
def __init__(self, n: int, m: int, dihedral: bool = True):
self.n = n
self.m = m
self.dihedral = dihedral # read sequences off the un-deduped census
self.configs: list[Config] = []
for g in generate(n, min_up_teeth=3, dedup=True):
for face, edges in inner_face_singletons(g).items():
if len(edges) == m:
self.configs.append(Config(g, face, edges))
# per config: list of (coloring, canonical down-apex sequence)
self.colorings: list[list[tuple[Coloring, tuple[int, ...]]]] = []
# per config: set of canonical sequences it realises
self.config_seq_sets: list[frozenset[tuple[int, ...]]] = []
# canonical sequence -> list of (config_idx, coloring)
self.by_sequence: dict[tuple[int, ...], list[tuple[int, Coloring]]] = defaultdict(list)
for cidx, cfg in enumerate(self.configs):
entries: list[tuple[Coloring, frozenset]] = []
seqs: set[tuple[int, ...]] = set()
for coloring, verdict in classify_colorings(cfg.graph, dedup_colors=True):
if not verdict.valid:
continue
if self.dihedral:
cseqs = dihedral_reading_sequences(cfg.graph.n, coloring,
cfg.edges, "d")
else:
cseqs = {canonical_sequence(cfg.sequence(coloring))}
entries.append((coloring, frozenset(cseqs)))
for cseq in cseqs:
seqs.add(cseq)
self.by_sequence[cseq].append((cidx, coloring))
self.colorings.append(entries)
self.config_seq_sets.append(frozenset(seqs))
def groups(self):
groups: dict[frozenset[tuple[int, ...]], list[int]] = defaultdict(list)
for cidx, sset in enumerate(self.config_seq_sets):
groups[sset].append(cidx)
return sorted(groups.items(), key=lambda kv: (-len(kv[1]), len(kv[0])))
def sequences(self) -> list[tuple[int, ...]]:
return sorted(self.by_sequence)
# ---------------------------------------------------------------------------
# Drawing (rings the singleton down apexes of the chosen inner face).
# ---------------------------------------------------------------------------
def _draw(ax, cfg: Config, coloring, title):
graph = cfg.graph
pos = _positions(graph)
for u, v in graph.edges():
ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]],
color="#bbbbbb", lw=0.5, zorder=1)
for k in range(graph.n):
a, b = f"a{k}", f"a{(k + 1) % graph.n}"
ax.plot([pos[a][0], pos[b][0]], [pos[a][1], pos[b][1]],
color="#666666", lw=1.0, zorder=2)
for v, (x, y) in pos.items():
is_bite = v.startswith("p")
ax.scatter([x], [y], s=34 if is_bite else 24, color=PALETTE[coloring[v]],
edgecolors="black", linewidths=0.5 if is_bite else 0.3, zorder=3)
# ring the singleton down apexes of this inner face
dx = [pos[v][0] for v in cfg.apexes]
dy = [pos[v][1] for v in cfg.apexes]
ax.scatter(dx, dy, s=120, facecolors="none", edgecolors="#222222",
linewidths=1.4, zorder=4)
ax.set_xlim(-1.65, 1.65)
ax.set_ylim(-1.85, 1.65)
ax.set_aspect("equal")
ax.axis("off")
ax.set_title(title, fontsize=6, pad=1.5)
def draw_sequence(exp: Experiment, seq, out_png, out_pdf):
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
entries = exp.by_sequence[seq]
cols = 10
rows = math.ceil(len(entries) / cols)
fig, axes = plt.subplots(rows, cols, figsize=(cols * 1.5, rows * 1.7), squeeze=False)
for idx in range(rows * cols):
ax = axes[idx // cols][idx % cols]
if idx < len(entries):
cidx, coloring = entries[idx]
cfg = exp.configs[cidx]
dseq = seq_str(cfg.sequence(coloring))
_draw(ax, cfg, coloring, f"{config_id(cidx)} d={dseq}")
else:
ax.axis("off")
fig.suptitle(
f"Kempe-balanced colourings with inner-face singleton-down-apex "
f"sequence {seq_str(seq)} (mod colour permutation)\n"
f"n={exp.n}, m={exp.m} down apexes on the face — {len(entries)} "
f"colourings on {len({c for c, _ in entries})} configs; "
f"black rings mark the face's down apexes",
fontsize=11, y=0.998,
)
fig.tight_layout(rect=(0, 0, 1, 0.96))
fig.savefig(out_png, dpi=170)
fig.savefig(out_pdf)
plt.close(fig)
print(f"wrote {out_png}")
# ---------------------------------------------------------------------------
# Markdown notes.
# ---------------------------------------------------------------------------
def write_sequence_note(exp: Experiment, seq, path, fig_name):
s = seq_str(seq)
by_config: dict[int, list[Coloring]] = defaultdict(list)
for cidx, coloring in exp.by_sequence[seq]:
by_config[cidx].append(coloring)
cm: dict[int, int] = {}
for c in seq:
cm[c] = cm.get(c, 0) + 1
counts = ", ".join(f"{v}×colour{k}" for k, v in sorted(cm.items()))
lines = []
lines.append(f"# Inner-face down-apex sequence `{s}`")
lines.append("")
lines.append(
f"Canonical colour sequence of the singleton down-tooth apexes on a "
f"single inner non-tooth face (read in cyclic order, reduced modulo the "
f"six colour permutations) for Kempe-balanced 3-colourings of M(T) with "
f"**n = {exp.n}**, **m = {exp.m} singleton down apexes on the face**. "
f"Sequences are read off the un-deduped census (every cyclic orientation "
f"of each colouring), so one colouring may realise several sequences -- "
f"see `../kempe_sequence_orientation_note.md`."
)
lines.append("")
lines.append(f"- Colour multiset: {counts}.")
lines.append(f"- Realised by **{len(by_config)}** of {len(exp.configs)} "
f"configs (M(T), inner face).")
lines.append(f"- **{len(exp.by_sequence[seq])}** Kempe-balanced colourings "
f"(mod colour permutation) realise it in some orientation.")
lines.append(f"- Figure: `{fig_name}` (black rings mark the face's down apexes).")
lines.append("")
lines.append("Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` "
"up-tooth apexes; `D[...]` singleton down apexes `d` and bite "
"apexes `p`. Colours 0/1/2 = "
+ ", ".join(f"{c}:{PALETTE_NAME[c]}" for c in (0, 1, 2)) + ".")
lines.append("")
for cidx in sorted(by_config):
cfg = exp.configs[cidx]
cols = by_config[cidx]
lines.append(f"## {config_id(cidx)} — {describe_config(cfg.graph, cfg.face, cfg.edges)}")
lines.append("")
lines.append(f"{len(cols)} colouring(s) with down-apex sequence `{s}`:")
lines.append("")
for coloring in cols:
raw = seq_str(cfg.sequence(coloring))
lines.append(f"- face apex colours (canonical-rep edge order) `{raw}`, "
f"realises `{s}` in some orientation · "
f"`{compact_coloring(cfg.graph, coloring)}`")
lines.append("")
with open(path, "w") as fh:
fh.write("\n".join(lines) + "\n")
print(f"wrote {path}")
def write_summary(exp: Experiment, path):
lines = []
lines.append(f"# Inner-face singleton-down-apex sequences of Kempe-balanced "
f"colourings (n={exp.n}, m={exp.m})")
lines.append("")
reading = ("the un-deduped census (every cyclic orientation of each "
"colouring)" if exp.dihedral
else "a single anchored representative per dihedral class")
lines.append(
f"Every full medial tire graph M(T) with |A(T)| = {exp.n} (one "
f"representative per dihedral class) that has an inner non-tooth face "
f"holding exactly {exp.m} singleton down-tooth apexes: "
f"**{len(exp.configs)} configs (M(T), inner face)**. For each we "
f"enumerate the Kempe-balanced (valid) proper 3-colourings (modulo "
f"colour permutation), read the down-apex colour sequence in cyclic "
f"order off {reading}, and reduce it modulo colour permutation (NOT "
f"dihedral symmetry). Reading off the census makes the recorded "
f"vocabulary orientation-honest; see `../kempe_sequence_orientation_note.md`."
)
lines.append("")
total = sum(len(c) for c in exp.colorings)
lines.append(f"- Total Kempe-balanced colourings (mod colour permutation): "
f"**{total}**.")
lines.append(f"- Distinct canonical down-apex sequences overall: "
f"**{len(exp.by_sequence)}**.")
lines.append("")
lines.append("## Distinct canonical down-apex sequences")
lines.append("")
lines.append("| sequence | colour multiset | #configs realising | #colourings |")
lines.append("|---|---|---|---|")
for seq in exp.sequences():
cm: dict[int, int] = {}
for c in seq:
cm[c] = cm.get(c, 0) + 1
cms = "+".join(str(v) for v in sorted(cm.values(), reverse=True))
ncfg = len({c for c, _ in exp.by_sequence[seq]})
lines.append(f"| `{seq_str(seq)}` | {cms} | {ncfg} | "
f"{len(exp.by_sequence[seq])} |")
lines.append("")
parity = "even" if exp.m % 2 == 0 else "odd"
allowed = sorted(
{"+".join(str(v) for v in sorted(p, reverse=True))
for p in _parity_partitions(exp.m)}
)
lines.append("Note: every realised sequence has its three colour-counts of "
"**equal parity** — exactly the Kempe-parity constraint on the "
"inner face (each colour pair meets its singleton down apexes an "
f"even number of times). With m = {exp.m} apexes (m is "
f"{'even' if exp.m % 2 == 0 else 'odd'}) every count must be "
f"**{parity}**, so the only admissible colour multisets are "
+ ", ".join(allowed) + ".")
lines.append("")
lines.append("## Step 4 — grouping configs by their set of unique down-apex "
"sequences")
lines.append("")
groups = exp.groups()
lines.append(f"The {len(exp.configs)} configs fall into **{len(groups)}** "
f"groups by the set of canonical down-apex sequences they "
f"realise:")
lines.append("")
lines.append("| #configs | set of down-apex sequences | config ids |")
lines.append("|---|---|---|")
for sset, cidxs in groups:
seqs = ", ".join(f"`{seq_str(s)}`" for s in sorted(sset))
ids = ", ".join(config_id(i) for i in cidxs)
lines.append(f"| {len(cidxs)} | {{ {seqs} }} | {ids} |")
lines.append("")
lines.append("## Config atlas (ids)")
lines.append("")
lines.append("| id | word / bites / face / apexes | #Kempe-balanced | "
"down-apex sequence set |")
lines.append("|---|---|---|---|")
for cidx, cfg in enumerate(exp.configs):
sset = exp.config_seq_sets[cidx]
seqs = ", ".join(f"`{seq_str(s)}`" for s in sorted(sset))
lines.append(f"| {config_id(cidx)} | "
f"{describe_config(cfg.graph, cfg.face, cfg.edges)} | "
f"{len(exp.colorings[cidx])} | {{ {seqs} }} |")
lines.append("")
lines.append("## Per-sequence notes")
lines.append("")
for seq in exp.sequences():
lines.append(f"- [`{seq_str(seq)}`](seq_{seq_str(seq)}.md) — "
f"figure `seq_{seq_str(seq)}.png`")
lines.append("")
with open(path, "w") as fh:
fh.write("\n".join(lines) + "\n")
print(f"wrote {path}")
# ---------------------------------------------------------------------------
# Driver.
# ---------------------------------------------------------------------------
def run(args):
exp = Experiment(args.n, args.m, dihedral=not args.anchored)
mode = "anchored" if args.anchored else "census (orientation-honest)"
print(f"reading: {mode}")
print(f"n={args.n}, m={args.m}: {len(exp.configs)} configs (M(T), inner face)")
print(f"distinct canonical down-apex sequences: {len(exp.by_sequence)}")
for seq in exp.sequences():
nc = len({c for c, _ in exp.by_sequence[seq]})
print(f" {seq_str(seq)}: {nc} configs, {len(exp.by_sequence[seq])} colourings")
print(f"groups by sequence-set: {len(exp.groups())}")
notes_dir = os.path.join(HERE, f"kempe_down_face_sequences_n{args.n}_m{args.m}")
os.makedirs(notes_dir, exist_ok=True)
write_summary(exp, os.path.join(notes_dir, "summary.md"))
for seq in exp.sequences():
s = seq_str(seq)
fig_name = f"seq_{s}.png"
write_sequence_note(exp, seq, os.path.join(notes_dir, f"seq_{s}.md"), fig_name)
if not args.no_figures:
draw_sequence(exp, seq,
os.path.join(notes_dir, fig_name),
os.path.join(notes_dir, f"seq_{s}.pdf"))
def main():
parser = argparse.ArgumentParser(description=__doc__)
parser.add_argument("--n", type=int, default=9)
parser.add_argument("--m", type=int, default=3,
help="singleton down apexes on the inner face (>=3)")
parser.add_argument("--no-figures", action="store_true")
parser.add_argument("--anchored", action="store_true",
help="read one anchored representative per class "
"(old behaviour) instead of the un-deduped census")
run(parser.parse_args())
if __name__ == "__main__":
main()
papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.md
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# Inner-face down-apex sequence `012`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 3 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 1×colour0, 1×colour1, 1×colour2.
- Realised by **26** of 26 configs (M(T), inner face).
- **241** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_012.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUUUUDDD bites=- face=root apexes=[d6,d7,d8]
22 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2] D[d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2] D[d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1] D[d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1] D[d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u4:0 u5:1] D[d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u4:0 u5:1] D[d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u4:0 u5:2] D[d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u4:0 u5:2] D[d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2] D[d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2] D[d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2 u4:1 u5:1] D[d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u4:1 u5:1] D[d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u4:0 u5:2] D[d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u4:0 u5:2] D[d6:1 d7:0 d8:2]`
## C01 — word=UUUUUDUDD bites=- face=root apexes=[d5,d7,d8]
11 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u4:2 u6:2] D[d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u6:1] D[d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u6:2] D[d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u6:2] D[d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u6:1] D[d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u4:0 u6:1] D[d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u4:0 u6:1] D[d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u4:0 u6:2] D[d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u4:2 u6:2] D[d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u4:1 u6:1] D[d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u4:0 u6:2] D[d5:2 d7:0 d8:1]`
## C02 — word=UUUUDUUDD bites=- face=root apexes=[d4,d7,d8]
17 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u5:2 u6:2] D[d4:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u5:1 u6:1] D[d4:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u6:0] D[d4:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u5:2 u6:2] D[d4:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u5:1 u6:1] D[d4:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u5:1 u6:0] D[d4:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u5:1 u6:0] D[d4:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u5:2 u6:0] D[d4:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u5:2 u6:2] D[d4:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0] D[d4:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u5:1 u6:1] D[d4:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0] D[d4:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u5:2 u6:0] D[d4:1 d7:0 d8:2]`
## C03 — word=UUUUDUDUD bites=- face=root apexes=[d4,d6,d8]
26 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:2 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:1 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:1 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u5:0 u7:2] D[d4:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:2 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:1 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:2 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u5:0 u7:1] D[d4:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:2 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u5:0 u7:1] D[d4:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:1 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1 u3:1 u5:0 u7:2] D[d4:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0] D[d4:2 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0] D[d4:1 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:1 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u3:0 u5:0 u7:2] D[d4:0 d6:1 d8:2]`
## C04 — word=UUUUDDDDD bites=(4,8) face=bite(4,8) apexes=[d5,d6,d7]
12 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d5:2 d6:1 d7:0 p4_8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d5:0 d6:1 d7:2 p4_8:2]`
- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d5:1 d6:2 d7:0 p4_8:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d5:0 d6:2 d7:1 p4_8:1]`
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d5:2 d6:1 d7:0 p4_8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d5:0 d6:1 d7:2 p4_8:2]`
- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d5:1 d6:2 d7:0 p4_8:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d5:0 d6:2 d7:1 p4_8:1]`
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d5:2 d6:1 d7:0 p4_8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:1 d7:2 p4_8:2]`
- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d5:1 d6:2 d7:0 p4_8:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:2 d7:1 p4_8:1]`
## C05 — word=UUUDUUUDD bites=- face=root apexes=[d3,d7,d8]
13 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2 u5:2 u6:2] D[d3:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u6:0] D[d3:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u4:1 u5:1 u6:2] D[d3:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u4:2 u5:2 u6:1] D[d3:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1 u5:1 u6:1] D[d3:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u6:0] D[d3:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u4:2 u5:1 u6:0] D[d3:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u2:1 u4:0 u5:1 u6:2] D[d3:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u4:1 u5:2 u6:0] D[d3:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012021021 U[u0:2 u1:0 u2:1 u4:0 u5:2 u6:1] D[d3:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2 u5:2 u6:2] D[d3:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2 u5:0 u6:0] D[d3:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u4:1 u5:1 u6:2] D[d3:2 d7:0 d8:1]`
## C06 — word=UUUDUUDUD bites=- face=root apexes=[d3,d6,d8]
20 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u7:0] D[d3:2 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u4:1 u5:2 u7:1] D[d3:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u4:1 u5:1 u7:2] D[d3:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u5:0 u7:2] D[d3:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u7:0] D[d3:1 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u4:2 u5:2 u7:1] D[d3:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u4:2 u5:1 u7:2] D[d3:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0] D[d3:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0] D[d3:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u5:0 u7:1] D[d3:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u4:2 u5:1 u7:0] D[d3:2 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u4:1 u5:2 u7:0] D[d3:1 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2 u5:0 u7:0] D[d3:2 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u4:1 u5:2 u7:1] D[d3:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u4:1 u5:1 u7:2] D[d3:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u4:0 u5:2 u7:0] D[d3:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u4:0 u5:2 u7:0] D[d3:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u4:0 u5:0 u7:2] D[d3:0 d6:1 d8:2]`
## C07 — word=UUUDUDDDD bites=(3,5) face=root apexes=[d6,d7,d8]
6 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d6:2 d7:0 d8:1 p3_5:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d6:1 d7:0 d8:2 p3_5:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d6:2 d7:0 d8:1 p3_5:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d6:1 d7:0 d8:2 p3_5:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d6:2 d7:0 d8:1 p3_5:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d6:1 d7:0 d8:2 p3_5:2]`
## C08 — word=UUUDUDDDD bites=(3,8) face=bite(3,8) apexes=[d5,d6,d7]
6 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d5:2 d6:1 d7:0 p3_8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d5:0 d6:1 d7:2 p3_8:2]`
- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d5:1 d6:2 d7:0 p3_8:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d5:0 d6:2 d7:1 p3_8:1]`
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d5:2 d6:1 d7:0 p3_8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d5:0 d6:1 d7:2 p3_8:2]`
## C09 — word=UUUDDUDDD bites=(4,6) face=root apexes=[d3,d7,d8]
3 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d7:0 d8:1 p4_6:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d7:0 d8:2 p4_6:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d7:0 d8:1 p4_6:2]`
## C10 — word=UUUDDUDDD bites=(3,8) face=bite(3,8) apexes=[d4,d6,d7]
3 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d4:2 d6:1 d7:0 p3_8:2]`
- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d4:1 d6:2 d7:0 p3_8:1]`
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d4:2 d6:1 d7:0 p3_8:2]`
## C11 — word=UUDUUDUUD bites=- face=root apexes=[d2,d5,d8]
18 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2 u6:1 u7:1] D[d2:2 d5:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2 u6:0 u7:0] D[d2:2 d5:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u4:1 u6:2 u7:1] D[d2:2 d5:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0 u6:1 u7:1] D[d2:2 d5:0 d8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:2 d5:0 d8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u4:2 u6:1 u7:2] D[d2:1 d5:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1 u6:2 u7:2] D[d2:1 d5:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1 u6:0 u7:0] D[d2:1 d5:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0 u6:2 u7:2] D[d2:1 d5:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:1 d5:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0 u6:2 u7:2] D[d2:1 d5:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0 u6:0 u7:0] D[d2:1 d5:0 d8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012021201 U[u0:2 u1:0 u3:1 u4:0 u6:1 u7:2] D[d2:1 d5:0 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u4:2 u6:2 u7:0] D[d2:0 d5:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u4:1 u6:1 u7:0] D[d2:0 d5:1 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u4:1 u6:2 u7:1] D[d2:0 d5:2 d8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u4:1 u6:1 u7:2] D[d2:0 d5:1 d8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u4:0 u6:2 u7:0] D[d2:0 d5:2 d8:1]`
## C12 — word=UUDUUDDDD bites=(2,5) face=root apexes=[d6,d7,d8]
10 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d6:2 d7:0 d8:1 p2_5:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d6:1 d7:0 d8:2 p2_5:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d6:2 d7:0 d8:1 p2_5:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d6:1 d7:0 d8:2 p2_5:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d6:2 d7:0 d8:1 p2_5:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d6:1 d7:0 d8:2 p2_5:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d6:2 d7:0 d8:1 p2_5:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d6:1 d7:0 d8:2 p2_5:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d6:2 d7:0 d8:1 p2_5:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d6:1 d7:0 d8:2 p2_5:1]`
## C13 — word=UUDUDUDDD bites=(4,6) face=root apexes=[d2,d7,d8]
5 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d7:0 d8:1 p4_6:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d7:0 d8:1 p4_6:0]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d7:0 d8:2 p4_6:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d7:0 d8:2 p4_6:0]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d7:0 d8:2 p4_6:0]`
## C14 — word=UUDUDUDDD bites=(2,4) face=root apexes=[d6,d7,d8]
6 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d6:2 d7:0 d8:1 p2_4:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d6:1 d7:0 d8:2 p2_4:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d6:2 d7:0 d8:1 p2_4:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d6:1 d7:0 d8:2 p2_4:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d6:2 d7:0 d8:1 p2_4:0]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d6:1 d7:0 d8:2 p2_4:0]`
## C15 — word=UUDUDUDDD bites=(2,8) face=bite(2,8) apexes=[d4,d6,d7]
5 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d4:2 d6:1 d7:0 p2_8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d4:0 d6:1 d7:2 p2_8:2]`
- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d4:1 d6:2 d7:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d4:0 d6:2 d7:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d4:0 d6:2 d7:1 p2_8:1]`
## C16 — word=UUDUDDUDD bites=(5,7) face=root apexes=[d2,d4,d8]
6 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d8:1 p5_7:0]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d8:2 p5_7:0]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d8:2 p5_7:0]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d8:1 p5_7:0]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d8:2 p5_7:0]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d8:2 p5_7:2]`
## C17 — word=UUDUDDUDD bites=(2,4) face=root apexes=[d5,d7,d8]
3 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d5:2 d7:0 d8:1 p2_4:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d5:1 d7:0 d8:2 p2_4:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d5:2 d7:0 d8:1 p2_4:0]`
## C18 — word=UUDUDDUDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d7]
5 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d4:1 d5:0 d7:2 p2_8:2]`
- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d4:1 d5:2 d7:0 p2_8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d4:2 d5:0 d7:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d4:2 d5:1 d7:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d4:2 d5:1 d7:0 p2_8:1]`
## C19 — word=UUDUDDDUD bites=(6,8) face=root apexes=[d2,d4,d5]
3 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 p6_8:2]`
- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 p6_8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 p6_8:2]`
## C20 — word=UUDUDDDUD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6]
10 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d4:2 d5:0 d6:1 p2_8:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d4:1 d5:0 d6:2 p2_8:2]`
- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d4:1 d5:2 d6:0 p2_8:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d4:0 d5:2 d6:1 p2_8:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d4:2 d5:0 d6:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d4:1 d5:0 d6:2 p2_8:1]`
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d4:2 d5:1 d6:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d4:0 d5:1 d6:2 p2_8:1]`
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d4:2 d5:1 d6:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d4:0 d5:1 d6:2 p2_8:1]`
## C21 — word=UUDDUUDDD bites=(3,6) face=root apexes=[d2,d7,d8]
9 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d7:0 d8:1 p3_6:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d7:0 d8:1 p3_6:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d7:0 d8:1 p3_6:0]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d7:0 d8:1 p3_6:0]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d7:0 d8:2 p3_6:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d7:0 d8:2 p3_6:1]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d7:0 d8:2 p3_6:0]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d7:0 d8:2 p3_6:0]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d7:0 d8:2 p3_6:1]`
## C22 — word=UUDDUDUDD bites=(5,7) face=root apexes=[d2,d3,d8]
5 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d8:1 p5_7:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d8:1 p5_7:0]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d8:2 p5_7:2]`
- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d8:2 p5_7:0]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d8:1 p5_7:0]`
## C23 — word=UUDDUDUDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d7]
8 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d3:2 d5:1 d7:0 p2_8:2]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d3:0 d5:1 d7:2 p2_8:2]`
- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d3:1 d5:2 d7:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d3:0 d5:2 d7:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d3:2 d5:1 d7:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:1 d7:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:0 d7:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d3:1 d5:2 d7:0 p2_8:1]`
## C24 — word=UDUDUDUDD bites=(5,7) face=root apexes=[d1,d3,d8]
6 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d8:1 p5_7:1]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d8:1 p5_7:0]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d8:1 p5_7:1]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d8:1 p5_7:1]`
- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d8:2 p5_7:2]`
- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d8:1 p5_7:0]`
## C25 — word=UDUDUDUDD bites=(3,5) face=root apexes=[d1,d7,d8]
3 colouring(s) with down-apex sequence `012`:
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d7:0 d8:1 p3_5:2]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d7:0 d8:1 p3_5:0]`
- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d7:0 d8:1 p3_5:1]`
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# Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=3)
Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 3 singleton down-tooth apexes: **26 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in cyclic order off the un-deduped census (every cyclic orientation of each colouring), and reduce it modulo colour permutation (NOT dihedral symmetry). Reading off the census makes the recorded vocabulary orientation-honest; see `../kempe_sequence_orientation_note.md`.
- Total Kempe-balanced colourings (mod colour permutation): **241**.
- Distinct canonical down-apex sequences overall: **1**.
## Distinct canonical down-apex sequences
| sequence | colour multiset | #configs realising | #colourings |
|---|---|---|---|
| `012` | 1+1+1 | 26 | 241 |
Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 3 apexes (m is odd) every count must be **odd**, so the only admissible colour multisets are 1+1+1.
## Step 4 — grouping configs by their set of unique down-apex sequences
The 26 configs fall into **1** groups by the set of canonical down-apex sequences they realise:
| #configs | set of down-apex sequences | config ids |
|---|---|---|
| 26 | { `012` } | C00, C01, C02, C03, C04, C05, C06, C07, C08, C09, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19, C20, C21, C22, C23, C24, C25 |
## Config atlas (ids)
| id | word / bites / face / apexes | #Kempe-balanced | down-apex sequence set |
|---|---|---|---|
| C00 | word=UUUUUUDDD bites=- face=root apexes=[d6,d7,d8] | 22 | { `012` } |
| C01 | word=UUUUUDUDD bites=- face=root apexes=[d5,d7,d8] | 11 | { `012` } |
| C02 | word=UUUUDUUDD bites=- face=root apexes=[d4,d7,d8] | 17 | { `012` } |
| C03 | word=UUUUDUDUD bites=- face=root apexes=[d4,d6,d8] | 26 | { `012` } |
| C04 | word=UUUUDDDDD bites=(4,8) face=bite(4,8) apexes=[d5,d6,d7] | 12 | { `012` } |
| C05 | word=UUUDUUUDD bites=- face=root apexes=[d3,d7,d8] | 13 | { `012` } |
| C06 | word=UUUDUUDUD bites=- face=root apexes=[d3,d6,d8] | 20 | { `012` } |
| C07 | word=UUUDUDDDD bites=(3,5) face=root apexes=[d6,d7,d8] | 6 | { `012` } |
| C08 | word=UUUDUDDDD bites=(3,8) face=bite(3,8) apexes=[d5,d6,d7] | 6 | { `012` } |
| C09 | word=UUUDDUDDD bites=(4,6) face=root apexes=[d3,d7,d8] | 3 | { `012` } |
| C10 | word=UUUDDUDDD bites=(3,8) face=bite(3,8) apexes=[d4,d6,d7] | 3 | { `012` } |
| C11 | word=UUDUUDUUD bites=- face=root apexes=[d2,d5,d8] | 18 | { `012` } |
| C12 | word=UUDUUDDDD bites=(2,5) face=root apexes=[d6,d7,d8] | 10 | { `012` } |
| C13 | word=UUDUDUDDD bites=(4,6) face=root apexes=[d2,d7,d8] | 5 | { `012` } |
| C14 | word=UUDUDUDDD bites=(2,4) face=root apexes=[d6,d7,d8] | 6 | { `012` } |
| C15 | word=UUDUDUDDD bites=(2,8) face=bite(2,8) apexes=[d4,d6,d7] | 5 | { `012` } |
| C16 | word=UUDUDDUDD bites=(5,7) face=root apexes=[d2,d4,d8] | 6 | { `012` } |
| C17 | word=UUDUDDUDD bites=(2,4) face=root apexes=[d5,d7,d8] | 3 | { `012` } |
| C18 | word=UUDUDDUDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d7] | 5 | { `012` } |
| C19 | word=UUDUDDDUD bites=(6,8) face=root apexes=[d2,d4,d5] | 3 | { `012` } |
| C20 | word=UUDUDDDUD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6] | 10 | { `012` } |
| C21 | word=UUDDUUDDD bites=(3,6) face=root apexes=[d2,d7,d8] | 9 | { `012` } |
| C22 | word=UUDDUDUDD bites=(5,7) face=root apexes=[d2,d3,d8] | 5 | { `012` } |
| C23 | word=UUDDUDUDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d7] | 8 | { `012` } |
| C24 | word=UDUDUDUDD bites=(5,7) face=root apexes=[d1,d3,d8] | 6 | { `012` } |
| C25 | word=UDUDUDUDD bites=(3,5) face=root apexes=[d1,d7,d8] | 3 | { `012` } |
## Per-sequence notes
- [`012`](seq_012.md) — figure `seq_012.png`
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# Inner-face down-apex sequence `0000`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 4×colour0.
- Realised by **21** of 23 configs (M(T), inner face).
- **64** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_0000.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8]
10 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:1 d8:1]`
## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8]
5 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:1 d8:1]`
## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8]
5 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:1 d8:1]`
## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8]
7 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:2 d8:2]`
## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
2 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:1 d8:1]`
## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
2 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:2 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:1 d6:1 d8:1]`
## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
7 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:2 d8:2]`
## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7]
4 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:2 d7:2 p3_8:2]`
- face apex colours (canonical-rep edge order) `0000`, realises `0000` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:0 d7:0 p3_8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:1 d7:1 p3_8:1]`
- face apex colours (canonical-rep edge order) `0000`, realises `0000` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:0 d7:0 p3_8:1]`
## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
4 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010120101 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010210202 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012010202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:1 d8:1]`
## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
4 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:1 d8:1]`
## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
1 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u7:1] D[d2:1 d4:1 d6:1 d8:1]`
## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8]
2 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:2 d8:2 p2_4:1]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:1 d8:1 p2_4:1]`
## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7]
2 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:1 d7:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `0000`, realises `0000` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:0 d7:0 p2_8:1]`
## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8]
1 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:1 d8:1 p3_5:1]`
## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7]
1 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:1 d7:1 p2_8:1]`
## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8]
1 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:1 d8:1 p4_6:1]`
## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7]
1 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:1 d7:1 p2_8:1]`
## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8]
1 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:2 d8:2 p3_5:0]`
## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8]
2 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012120101 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:2 d8:2 p1_3:0]`
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012120202 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:1 d8:1 p1_3:0]`
## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8]
1 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:1 d8:1 p1_3:0]`
## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7]
1 colouring(s) with down-apex sequence `0000`:
- face apex colours (canonical-rep edge order) `0000`, realises `0000` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:0 d7:0 p1_8:2]`
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# Inner-face down-apex sequence `0011`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 2×colour0, 2×colour1.
- Realised by **23** of 23 configs (M(T), inner face).
- **181** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_0011.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8]
20 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8]
10 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8]
15 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8]
14 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
13 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:1 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:1 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]`
## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
15 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]`
- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u7:2] D[d3:0 d5:0 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]`
## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
17 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:2 d4:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]`
## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7]
8 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2200`, realises `0011` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:0 d7:0 p3_8:2]`
- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d4:2 d5:1 d6:1 d7:2 p3_8:2]`
- face apex colours (canonical-rep edge order) `0110`, realises `0011` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1] D[d4:0 d5:1 d6:1 d7:0 p3_8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:2 d7:2 p3_8:2]`
- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d4:1 d5:2 d6:2 d7:1 p3_8:1]`
- face apex colours (canonical-rep edge order) `1100`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:0 d7:0 p3_8:1]`
- face apex colours (canonical-rep edge order) `0220`, realises `0011` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1] D[d4:0 d5:2 d6:2 d7:0 p3_8:1]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:1 d7:1 p3_8:1]`
## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
13 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010102121 U[u0:2 u1:2 u3:2 u4:1 u6:0] D[d2:2 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010120102 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010201212 U[u0:2 u1:2 u3:1 u4:2 u6:0] D[d2:1 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010210201 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012010201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u4:0 u6:0] D[d2:1 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:1 d8:1]`
## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
11 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u5:2 u6:1] D[d2:2 d4:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021012 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u6:0] D[d2:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:1 d8:1]`
## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
10 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:2 d6:1 d8:1]`
- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=010102021 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=010120201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:2 d4:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=010201012 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:1 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=010210102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u3:2 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u7:2] D[d2:1 d4:1 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u7:0] D[d2:1 d4:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u7:1] D[d2:0 d4:0 d6:1 d8:1]`
## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8]
4 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:1 d8:1 p2_4:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012020121 U[u0:2 u1:0 u3:1] D[d5:2 d6:0 d7:0 d8:2 p2_4:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:2 d8:2 p2_4:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d5:1 d6:0 d7:0 d8:1 p2_4:1]`
## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7]
4 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d4:1 d5:2 d6:2 d7:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `1100`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:0 d7:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `0220`, realises `0011` in some orientation · `A=012021012 U[u0:2 u1:0 u3:1] D[d4:0 d5:2 d6:2 d7:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:1 d7:1 p2_8:1]`
## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8]
2 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:2 d8:2 p3_5:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d6:0 d7:0 d8:1 p3_5:1]`
## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7]
2 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d3:1 d5:2 d6:2 d7:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `1100`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:0 d7:0 p2_8:1]`
## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8]
3 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:2 d8:2 p4_6:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:2 d8:2 p4_6:1]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:1 d8:1 p4_6:1]`
## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7]
3 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:1 d7:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `2200`, realises `0011` in some orientation · `A=012010212 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:0 d7:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `1100`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:0 d7:0 p2_8:1]`
## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8]
2 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:1 d8:1 p3_5:0]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d6:0 d7:0 d8:2 p3_5:0]`
## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8]
4 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012120102 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:1 d8:1 p1_3:0]`
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012120121 U[u0:2 u2:0 u4:1] D[d5:2 d6:0 d7:0 d8:2 p1_3:0]`
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012120201 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:2 d8:2 p1_3:0]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012120212 U[u0:2 u2:0 u4:1] D[d5:1 d6:0 d7:0 d8:1 p1_3:0]`
## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8]
3 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d7:0 d8:2 p4_6:0]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:2 d8:2 p4_6:1]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:1 d8:1 p4_6:1]`
## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8]
2 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012120201 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:2 d8:2 p1_3:0]`
- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012120212 U[u0:2 u2:0 u5:1] D[d4:1 d6:0 d7:0 d8:1 p1_3:0]`
## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7]
3 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:2 d7:2 p1_8:2]`
- face apex colours (canonical-rep edge order) `1100`, realises `0011` in some orientation · `A=010202121 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:0 d7:0 p1_8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:2 d7:2 p1_8:2]`
## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8]
3 colouring(s) with down-apex sequence `0011`:
- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d8:1 p5_7:1]`
- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=010202121 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:1 d8:2 p5_7:0]`
- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d8:1 p5_7:1]`
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# Inner-face down-apex sequence `0101`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 2×colour0, 2×colour1.
- Realised by **12** of 23 configs (M(T), inner face).
- **53** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_0101.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8]
5 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:2 d7:0 d8:2]`
## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
9 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:0 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:0 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:0 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:0 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:2 d7:0 d8:2]`
## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
11 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:0 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:0 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:0 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:0 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=012012012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=012021021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:1 d6:0 d8:1]`
## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
3 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u5:1 u6:0] D[d3:0 d4:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u5:2 u6:0] D[d3:0 d4:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u5:1 u6:0] D[d3:0 d4:1 d7:0 d8:1]`
## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
5 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:2 d7:0 d8:2]`
## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
3 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u5:2 u6:1] D[d2:0 d4:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:0 d4:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u6:0] D[d2:0 d4:1 d7:0 d8:1]`
## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
8 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010120102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:2 d4:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=010210201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:1 d4:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=012010201 U[u0:2 u1:0 u3:2 u5:1 u7:2] D[d2:1 d4:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u7:0] D[d2:0 d4:1 d6:0 d8:1]`
## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8]
1 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d7:0 d8:2 p4_6:1]`
## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7]
1 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `2020`, realises `0101` in some orientation · `A=012012012 U[u0:2 u1:0 u5:1] D[d3:2 d4:0 d6:2 d7:0 p2_8:1]`
## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8]
3 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d7:0 d8:2 p4_6:0]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d7:0 d8:1 p4_6:0]`
- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d7:0 d8:2 p4_6:1]`
## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7]
1 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=010201201 U[u0:2 u2:1 u5:0] D[d3:1 d4:2 d6:1 d7:2 p1_8:2]`
## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8]
3 colouring(s) with down-apex sequence `0101`:
- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d8:1 p5_7:0]`
- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d8:1 p5_7:1]`
- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d8:1 p5_7:0]`
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# Inner-face down-apex sequence `0110`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 2×colour0, 2×colour1.
- Realised by **23** of 23 configs (M(T), inner face).
- **181** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_0110.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8]
20 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]`
## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8]
10 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]`
## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8]
15 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]`
## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8]
14 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]`
## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8]
13 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:1 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:1 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]`
## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8]
15 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]`
- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u7:2] D[d3:0 d5:0 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]`
## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8]
17 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:2 d4:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]`
## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7]
8 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2200`, realises `0110` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:0 d7:0 p3_8:2]`
- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d4:2 d5:1 d6:1 d7:2 p3_8:2]`
- face apex colours (canonical-rep edge order) `0110`, realises `0110` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1] D[d4:0 d5:1 d6:1 d7:0 p3_8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:2 d7:2 p3_8:2]`
- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d4:1 d5:2 d6:2 d7:1 p3_8:1]`
- face apex colours (canonical-rep edge order) `1100`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:0 d7:0 p3_8:1]`
- face apex colours (canonical-rep edge order) `0220`, realises `0110` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1] D[d4:0 d5:2 d6:2 d7:0 p3_8:1]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:1 d7:1 p3_8:1]`
## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8]
13 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010102121 U[u0:2 u1:2 u3:2 u4:1 u6:0] D[d2:2 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010120102 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010201212 U[u0:2 u1:2 u3:1 u4:2 u6:0] D[d2:1 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010210201 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012010201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u4:0 u6:0] D[d2:1 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:1 d8:1]`
## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8]
11 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u5:2 u6:1] D[d2:2 d4:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021012 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u6:0] D[d2:1 d4:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:1 d8:1]`
## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8]
10 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:2 d6:1 d8:1]`
- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=010102021 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=010120201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:2 d4:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=010201012 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:1 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=010210102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u3:2 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u7:2] D[d2:1 d4:1 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u7:0] D[d2:1 d4:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u7:1] D[d2:0 d4:0 d6:1 d8:1]`
## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8]
4 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:1 d8:1 p2_4:1]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012020121 U[u0:2 u1:0 u3:1] D[d5:2 d6:0 d7:0 d8:2 p2_4:1]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:2 d8:2 p2_4:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d5:1 d6:0 d7:0 d8:1 p2_4:1]`
## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7]
4 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d4:1 d5:2 d6:2 d7:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `1100`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:0 d7:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `0220`, realises `0110` in some orientation · `A=012021012 U[u0:2 u1:0 u3:1] D[d4:0 d5:2 d6:2 d7:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:1 d7:1 p2_8:1]`
## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8]
2 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:2 d8:2 p3_5:1]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d6:0 d7:0 d8:1 p3_5:1]`
## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7]
2 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d3:1 d5:2 d6:2 d7:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `1100`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:0 d7:0 p2_8:1]`
## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8]
3 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:2 d8:2 p4_6:1]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:2 d8:2 p4_6:1]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:1 d8:1 p4_6:1]`
## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7]
3 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:1 d7:1 p2_8:1]`
- face apex colours (canonical-rep edge order) `2200`, realises `0110` in some orientation · `A=012010212 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:0 d7:0 p2_8:1]`
- face apex colours (canonical-rep edge order) `1100`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:0 d7:0 p2_8:1]`
## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8]
2 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:1 d8:1 p3_5:0]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d6:0 d7:0 d8:2 p3_5:0]`
## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8]
4 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012120102 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:1 d8:1 p1_3:0]`
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012120121 U[u0:2 u2:0 u4:1] D[d5:2 d6:0 d7:0 d8:2 p1_3:0]`
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012120201 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:2 d8:2 p1_3:0]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012120212 U[u0:2 u2:0 u4:1] D[d5:1 d6:0 d7:0 d8:1 p1_3:0]`
## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8]
3 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d7:0 d8:2 p4_6:0]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:2 d8:2 p4_6:1]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:1 d8:1 p4_6:1]`
## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8]
2 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012120201 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:2 d8:2 p1_3:0]`
- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012120212 U[u0:2 u2:0 u5:1] D[d4:1 d6:0 d7:0 d8:1 p1_3:0]`
## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7]
3 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:2 d7:2 p1_8:2]`
- face apex colours (canonical-rep edge order) `1100`, realises `0110` in some orientation · `A=010202121 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:0 d7:0 p1_8:2]`
- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:2 d7:2 p1_8:2]`
## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8]
3 colouring(s) with down-apex sequence `0110`:
- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d8:1 p5_7:1]`
- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=010202121 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:1 d8:2 p5_7:0]`
- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d8:1 p5_7:1]`
papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.pdf
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papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/summary.md
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# Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=4)
Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 4 singleton down-tooth apexes: **23 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in cyclic order off the un-deduped census (every cyclic orientation of each colouring), and reduce it modulo colour permutation (NOT dihedral symmetry). Reading off the census makes the recorded vocabulary orientation-honest; see `../kempe_sequence_orientation_note.md`.
- Total Kempe-balanced colourings (mod colour permutation): **298**.
- Distinct canonical down-apex sequences overall: **4**.
## Distinct canonical down-apex sequences
| sequence | colour multiset | #configs realising | #colourings |
|---|---|---|---|
| `0000` | 4 | 21 | 64 |
| `0011` | 2+2 | 23 | 181 |
| `0101` | 2+2 | 12 | 53 |
| `0110` | 2+2 | 23 | 181 |
Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 4 apexes (m is even) every count must be **even**, so the only admissible colour multisets are 2+2, 4.
## Step 4 — grouping configs by their set of unique down-apex sequences
The 23 configs fall into **3** groups by the set of canonical down-apex sequences they realise:
| #configs | set of down-apex sequences | config ids |
|---|---|---|
| 11 | { `0000`, `0011`, `0110` } | C00, C01, C03, C07, C11, C12, C13, C14, C17, C18, C20 |
| 10 | { `0000`, `0011`, `0101`, `0110` } | C02, C04, C05, C06, C08, C09, C10, C15, C16, C21 |
| 2 | { `0011`, `0101`, `0110` } | C19, C22 |
## Config atlas (ids)
| id | word / bites / face / apexes | #Kempe-balanced | down-apex sequence set |
|---|---|---|---|
| C00 | word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8] | 30 | { `0000`, `0011`, `0110` } |
| C01 | word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8] | 15 | { `0000`, `0011`, `0110` } |
| C02 | word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8] | 25 | { `0000`, `0011`, `0101`, `0110` } |
| C03 | word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8] | 21 | { `0000`, `0011`, `0110` } |
| C04 | word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8] | 24 | { `0000`, `0011`, `0101`, `0110` } |
| C05 | word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8] | 28 | { `0000`, `0011`, `0101`, `0110` } |
| C06 | word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8] | 27 | { `0000`, `0011`, `0101`, `0110` } |
| C07 | word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7] | 12 | { `0000`, `0011`, `0110` } |
| C08 | word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8] | 22 | { `0000`, `0011`, `0101`, `0110` } |
| C09 | word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8] | 18 | { `0000`, `0011`, `0101`, `0110` } |
| C10 | word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8] | 19 | { `0000`, `0011`, `0101`, `0110` } |
| C11 | word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8] | 6 | { `0000`, `0011`, `0110` } |
| C12 | word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7] | 6 | { `0000`, `0011`, `0110` } |
| C13 | word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } |
| C14 | word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7] | 3 | { `0000`, `0011`, `0110` } |
| C15 | word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8] | 5 | { `0000`, `0011`, `0101`, `0110` } |
| C16 | word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7] | 5 | { `0000`, `0011`, `0101`, `0110` } |
| C17 | word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } |
| C18 | word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8] | 6 | { `0000`, `0011`, `0110` } |
| C19 | word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8] | 6 | { `0011`, `0101`, `0110` } |
| C20 | word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } |
| C21 | word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7] | 5 | { `0000`, `0011`, `0101`, `0110` } |
| C22 | word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8] | 6 | { `0011`, `0101`, `0110` } |
## Per-sequence notes
- [`0000`](seq_0000.md) — figure `seq_0000.png`
- [`0011`](seq_0011.md) — figure `seq_0011.png`
- [`0101`](seq_0101.md) — figure `seq_0101.png`
- [`0110`](seq_0110.md) — figure `seq_0110.png`
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# Inner-face down-apex sequence `00012`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
- Realised by **10** of 10 configs (M(T), inner face).
- **161** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_00012.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
30 colouring(s) with down-apex sequence `00012`:
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
15 colouring(s) with down-apex sequence `00012`:
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
9 colouring(s) with down-apex sequence `00012`:
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `00012` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
25 colouring(s) with down-apex sequence `00012`:
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
16 colouring(s) with down-apex sequence `00012`:
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `00012` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
8 colouring(s) with down-apex sequence `00012`:
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21022`, realises `00012` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12011`, realises `00012` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]`
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
12 colouring(s) with down-apex sequence `00012`:
- face apex colours (canonical-rep edge order) `22012`, realises `00012` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `21022`, realises `00012` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `12011`, realises `00012` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]`
- face apex colours (canonical-rep edge order) `11021`, realises `00012` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `02221`, realises `00012` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `01112`, realises `00012` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `01112`, realises `00012` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `00012`, realises `00012` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]`
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
23 colouring(s) with down-apex sequence `00012`:
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `00012` in some orientation · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]`
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
12 colouring(s) with down-apex sequence `00012`:
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12011`, realises `00012` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
11 colouring(s) with down-apex sequence `00012`:
- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21022`, realises `00012` in some orientation · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `21022`, realises `00012` in some orientation · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02111`, realises `00012` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]`
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# Inner-face down-apex sequence `00102`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
- Realised by **7** of 10 configs (M(T), inner face).
- **68** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_00102.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
12 colouring(s) with down-apex sequence `00102`:
- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `00102` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02122`, realises `00102` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `00102` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `00102` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
8 colouring(s) with down-apex sequence `00102`:
- face apex colours (canonical-rep edge order) `02122`, realises `00102` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `00102` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]`
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
14 colouring(s) with down-apex sequence `00102`:
- face apex colours (canonical-rep edge order) `21101`, realises `00102` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21011`, realises `00102` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `21202`, realises `00102` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `21101`, realises `00102` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12022`, realises `00102` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]`
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
16 colouring(s) with down-apex sequence `00102`:
- face apex colours (canonical-rep edge order) `21202`, realises `00102` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `21101`, realises `00102` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `20221`, realises `00102` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `20212`, realises `00102` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `10121`, realises `00102` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10112`, realises `00102` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `10121`, realises `00102` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10112`, realises `00102` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `02212`, realises `00102` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `01121`, realises `00102` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]`
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
4 colouring(s) with down-apex sequence `00102`:
- face apex colours (canonical-rep edge order) `02122`, realises `00102` in some orientation · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `00102` in some orientation · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]`
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
6 colouring(s) with down-apex sequence `00102`:
- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12022`, realises `00102` in some orientation · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]`
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
8 colouring(s) with down-apex sequence `00102`:
- face apex colours (canonical-rep edge order) `21202`, realises `00102` in some orientation · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `21011`, realises `00102` in some orientation · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `21101`, realises `00102` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21011`, realises `00102` in some orientation · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02122`, realises `00102` in some orientation · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]`
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# Inner-face down-apex sequence `00120`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
- Realised by **10** of 10 configs (M(T), inner face).
- **161** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_00120.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
30 colouring(s) with down-apex sequence `00120`:
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
15 colouring(s) with down-apex sequence `00120`:
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
9 colouring(s) with down-apex sequence `00120`:
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `00120` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
25 colouring(s) with down-apex sequence `00120`:
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
16 colouring(s) with down-apex sequence `00120`:
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `00120` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
8 colouring(s) with down-apex sequence `00120`:
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21022`, realises `00120` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12011`, realises `00120` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]`
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
12 colouring(s) with down-apex sequence `00120`:
- face apex colours (canonical-rep edge order) `22012`, realises `00120` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `21022`, realises `00120` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `12011`, realises `00120` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]`
- face apex colours (canonical-rep edge order) `11021`, realises `00120` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `02221`, realises `00120` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `01112`, realises `00120` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `01112`, realises `00120` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `00012`, realises `00120` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]`
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
23 colouring(s) with down-apex sequence `00120`:
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `00120` in some orientation · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]`
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
12 colouring(s) with down-apex sequence `00120`:
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12011`, realises `00120` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
11 colouring(s) with down-apex sequence `00120`:
- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21022`, realises `00120` in some orientation · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `21022`, realises `00120` in some orientation · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02111`, realises `00120` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]`
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# Inner-face down-apex sequence `01002`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
- Realised by **7** of 10 configs (M(T), inner face).
- **68** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_01002.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
12 colouring(s) with down-apex sequence `01002`:
- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01002` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02122`, realises `01002` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01002` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01002` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
8 colouring(s) with down-apex sequence `01002`:
- face apex colours (canonical-rep edge order) `02122`, realises `01002` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01002` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]`
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
14 colouring(s) with down-apex sequence `01002`:
- face apex colours (canonical-rep edge order) `21101`, realises `01002` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21011`, realises `01002` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `21202`, realises `01002` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `21101`, realises `01002` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12022`, realises `01002` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]`
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
16 colouring(s) with down-apex sequence `01002`:
- face apex colours (canonical-rep edge order) `21202`, realises `01002` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `21101`, realises `01002` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `20221`, realises `01002` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `20212`, realises `01002` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `10121`, realises `01002` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10112`, realises `01002` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `10121`, realises `01002` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10112`, realises `01002` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `02212`, realises `01002` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `01121`, realises `01002` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]`
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
4 colouring(s) with down-apex sequence `01002`:
- face apex colours (canonical-rep edge order) `02122`, realises `01002` in some orientation · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01002` in some orientation · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]`
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
6 colouring(s) with down-apex sequence `01002`:
- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12022`, realises `01002` in some orientation · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]`
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
8 colouring(s) with down-apex sequence `01002`:
- face apex colours (canonical-rep edge order) `21202`, realises `01002` in some orientation · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `21011`, realises `01002` in some orientation · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `21101`, realises `01002` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21011`, realises `01002` in some orientation · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02122`, realises `01002` in some orientation · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]`
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# Inner-face down-apex sequence `01020`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
- Realised by **7** of 10 configs (M(T), inner face).
- **68** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_01020.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
12 colouring(s) with down-apex sequence `01020`:
- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01020` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02122`, realises `01020` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01020` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01020` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
8 colouring(s) with down-apex sequence `01020`:
- face apex colours (canonical-rep edge order) `02122`, realises `01020` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01020` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]`
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
14 colouring(s) with down-apex sequence `01020`:
- face apex colours (canonical-rep edge order) `21101`, realises `01020` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21011`, realises `01020` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `21202`, realises `01020` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `21101`, realises `01020` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12022`, realises `01020` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]`
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
16 colouring(s) with down-apex sequence `01020`:
- face apex colours (canonical-rep edge order) `21202`, realises `01020` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `21101`, realises `01020` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `20221`, realises `01020` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `20212`, realises `01020` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `10121`, realises `01020` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10112`, realises `01020` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `10121`, realises `01020` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10112`, realises `01020` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `02212`, realises `01020` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `01121`, realises `01020` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]`
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
4 colouring(s) with down-apex sequence `01020`:
- face apex colours (canonical-rep edge order) `02122`, realises `01020` in some orientation · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01020` in some orientation · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]`
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
6 colouring(s) with down-apex sequence `01020`:
- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12022`, realises `01020` in some orientation · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]`
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
8 colouring(s) with down-apex sequence `01020`:
- face apex colours (canonical-rep edge order) `21202`, realises `01020` in some orientation · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `21011`, realises `01020` in some orientation · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `21101`, realises `01020` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21011`, realises `01020` in some orientation · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02122`, realises `01020` in some orientation · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]`
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# Inner-face down-apex sequence `01112`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
- Realised by **10** of 10 configs (M(T), inner face).
- **161** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_01112.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
30 colouring(s) with down-apex sequence `01112`:
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
15 colouring(s) with down-apex sequence `01112`:
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
9 colouring(s) with down-apex sequence `01112`:
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `01112` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
25 colouring(s) with down-apex sequence `01112`:
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
16 colouring(s) with down-apex sequence `01112`:
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `01112` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
8 colouring(s) with down-apex sequence `01112`:
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21022`, realises `01112` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12011`, realises `01112` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]`
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
12 colouring(s) with down-apex sequence `01112`:
- face apex colours (canonical-rep edge order) `22012`, realises `01112` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `21022`, realises `01112` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `12011`, realises `01112` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]`
- face apex colours (canonical-rep edge order) `11021`, realises `01112` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `02221`, realises `01112` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `01112`, realises `01112` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `01112`, realises `01112` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `00012`, realises `01112` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]`
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
23 colouring(s) with down-apex sequence `01112`:
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `01112` in some orientation · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]`
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
12 colouring(s) with down-apex sequence `01112`:
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12011`, realises `01112` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
11 colouring(s) with down-apex sequence `01112`:
- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21022`, realises `01112` in some orientation · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `21022`, realises `01112` in some orientation · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02111`, realises `01112` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]`
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# Inner-face down-apex sequence `01121`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
- Realised by **7** of 10 configs (M(T), inner face).
- **68** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_01121.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
12 colouring(s) with down-apex sequence `01121`:
- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01121` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02122`, realises `01121` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01121` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01121` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
8 colouring(s) with down-apex sequence `01121`:
- face apex colours (canonical-rep edge order) `02122`, realises `01121` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01121` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]`
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
14 colouring(s) with down-apex sequence `01121`:
- face apex colours (canonical-rep edge order) `21101`, realises `01121` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21011`, realises `01121` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `21202`, realises `01121` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `21101`, realises `01121` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12022`, realises `01121` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]`
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
16 colouring(s) with down-apex sequence `01121`:
- face apex colours (canonical-rep edge order) `21202`, realises `01121` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `21101`, realises `01121` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `20221`, realises `01121` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `20212`, realises `01121` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `10121`, realises `01121` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10112`, realises `01121` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `10121`, realises `01121` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10112`, realises `01121` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `02212`, realises `01121` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `01121`, realises `01121` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]`
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
4 colouring(s) with down-apex sequence `01121`:
- face apex colours (canonical-rep edge order) `02122`, realises `01121` in some orientation · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01121` in some orientation · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]`
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
6 colouring(s) with down-apex sequence `01121`:
- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12022`, realises `01121` in some orientation · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]`
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
8 colouring(s) with down-apex sequence `01121`:
- face apex colours (canonical-rep edge order) `21202`, realises `01121` in some orientation · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `21011`, realises `01121` in some orientation · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `21101`, realises `01121` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21011`, realises `01121` in some orientation · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02122`, realises `01121` in some orientation · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]`
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# Inner-face down-apex sequence `01200`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 3×colour0, 1×colour1, 1×colour2.
- Realised by **10** of 10 configs (M(T), inner face).
- **161** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_01200.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
30 colouring(s) with down-apex sequence `01200`:
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
15 colouring(s) with down-apex sequence `01200`:
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
9 colouring(s) with down-apex sequence `01200`:
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `01200` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
25 colouring(s) with down-apex sequence `01200`:
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
16 colouring(s) with down-apex sequence `01200`:
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `01200` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
8 colouring(s) with down-apex sequence `01200`:
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21022`, realises `01200` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12011`, realises `01200` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]`
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
12 colouring(s) with down-apex sequence `01200`:
- face apex colours (canonical-rep edge order) `22012`, realises `01200` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `21022`, realises `01200` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `12011`, realises `01200` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]`
- face apex colours (canonical-rep edge order) `11021`, realises `01200` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `02221`, realises `01200` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `01112`, realises `01200` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `01112`, realises `01200` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `00012`, realises `01200` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]`
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
23 colouring(s) with down-apex sequence `01200`:
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `01200` in some orientation · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]`
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
12 colouring(s) with down-apex sequence `01200`:
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12011`, realises `01200` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
11 colouring(s) with down-apex sequence `01200`:
- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21022`, realises `01200` in some orientation · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `21022`, realises `01200` in some orientation · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02111`, realises `01200` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]`
papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.pdf
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# Inner-face down-apex sequence `01211`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 1×colour0, 3×colour1, 1×colour2.
- Realised by **7** of 10 configs (M(T), inner face).
- **68** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_01211.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
12 colouring(s) with down-apex sequence `01211`:
- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01211` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02122`, realises `01211` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01211` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01211` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]`
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
8 colouring(s) with down-apex sequence `01211`:
- face apex colours (canonical-rep edge order) `02122`, realises `01211` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01211` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]`
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
14 colouring(s) with down-apex sequence `01211`:
- face apex colours (canonical-rep edge order) `21101`, realises `01211` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21011`, realises `01211` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `21202`, realises `01211` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `21101`, realises `01211` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12022`, realises `01211` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]`
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
16 colouring(s) with down-apex sequence `01211`:
- face apex colours (canonical-rep edge order) `21202`, realises `01211` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `21101`, realises `01211` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `20221`, realises `01211` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `20212`, realises `01211` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `10121`, realises `01211` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10112`, realises `01211` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `10121`, realises `01211` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10112`, realises `01211` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `02212`, realises `01211` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `01121`, realises `01211` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]`
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
4 colouring(s) with down-apex sequence `01211`:
- face apex colours (canonical-rep edge order) `02122`, realises `01211` in some orientation · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00102`, realises `01211` in some orientation · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]`
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
6 colouring(s) with down-apex sequence `01211`:
- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12022`, realises `01211` in some orientation · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]`
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
8 colouring(s) with down-apex sequence `01211`:
- face apex colours (canonical-rep edge order) `21202`, realises `01211` in some orientation · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `21011`, realises `01211` in some orientation · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `21101`, realises `01211` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21011`, realises `01211` in some orientation · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02122`, realises `01211` in some orientation · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]`
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# Inner-face down-apex sequence `01222`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 1×colour0, 1×colour1, 3×colour2.
- Realised by **10** of 10 configs (M(T), inner face).
- **161** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_01222.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8]
30 colouring(s) with down-apex sequence `01222`:
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]`
## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8]
15 colouring(s) with down-apex sequence `01222`:
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]`
## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8]
9 colouring(s) with down-apex sequence `01222`:
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `01222` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]`
## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8]
25 colouring(s) with down-apex sequence `01222`:
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]`
## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8]
16 colouring(s) with down-apex sequence `01222`:
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `01222` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]`
## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8]
8 colouring(s) with down-apex sequence `01222`:
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21022`, realises `01222` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `12011`, realises `01222` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]`
## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8]
12 colouring(s) with down-apex sequence `01222`:
- face apex colours (canonical-rep edge order) `22012`, realises `01222` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `21022`, realises `01222` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]`
- face apex colours (canonical-rep edge order) `12011`, realises `01222` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]`
- face apex colours (canonical-rep edge order) `11021`, realises `01222` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]`
- face apex colours (canonical-rep edge order) `02221`, realises `01222` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]`
- face apex colours (canonical-rep edge order) `01112`, realises `01222` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]`
- face apex colours (canonical-rep edge order) `01112`, realises `01222` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]`
- face apex colours (canonical-rep edge order) `00012`, realises `01222` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]`
## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8]
23 colouring(s) with down-apex sequence `01222`:
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `02111`, realises `01222` in some orientation · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]`
## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8]
12 colouring(s) with down-apex sequence `01222`:
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `12011`, realises `01222` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]`
## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8]
11 colouring(s) with down-apex sequence `01222`:
- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `21022`, realises `01222` in some orientation · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `21022`, realises `01222` in some orientation · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `02111`, realises `01222` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]`
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# Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=5)
Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 5 singleton down-tooth apexes: **10 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in cyclic order off the un-deduped census (every cyclic orientation of each colouring), and reduce it modulo colour permutation (NOT dihedral symmetry). Reading off the census makes the recorded vocabulary orientation-honest; see `../kempe_sequence_orientation_note.md`.
- Total Kempe-balanced colourings (mod colour permutation): **229**.
- Distinct canonical down-apex sequences overall: **10**.
## Distinct canonical down-apex sequences
| sequence | colour multiset | #configs realising | #colourings |
|---|---|---|---|
| `00012` | 3+1+1 | 10 | 161 |
| `00102` | 3+1+1 | 7 | 68 |
| `00120` | 3+1+1 | 10 | 161 |
| `01002` | 3+1+1 | 7 | 68 |
| `01020` | 3+1+1 | 7 | 68 |
| `01112` | 3+1+1 | 10 | 161 |
| `01121` | 3+1+1 | 7 | 68 |
| `01200` | 3+1+1 | 10 | 161 |
| `01211` | 3+1+1 | 7 | 68 |
| `01222` | 3+1+1 | 10 | 161 |
Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 5 apexes (m is odd) every count must be **odd**, so the only admissible colour multisets are 3+1+1.
## Step 4 — grouping configs by their set of unique down-apex sequences
The 10 configs fall into **2** groups by the set of canonical down-apex sequences they realise:
| #configs | set of down-apex sequences | config ids |
|---|---|---|
| 7 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | C02, C04, C05, C06, C07, C08, C09 |
| 3 | { `00012`, `00120`, `01112`, `01200`, `01222` } | C00, C01, C03 |
## Config atlas (ids)
| id | word / bites / face / apexes | #Kempe-balanced | down-apex sequence set |
|---|---|---|---|
| C00 | word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8] | 30 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
| C01 | word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8] | 15 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
| C02 | word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] | 21 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
| C03 | word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8] | 25 | { `00012`, `00120`, `01112`, `01200`, `01222` } |
| C04 | word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] | 24 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
| C05 | word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] | 22 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
| C06 | word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] | 28 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
| C07 | word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] | 27 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
| C08 | word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] | 18 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
| C09 | word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] | 19 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } |
## Per-sequence notes
- [`00012`](seq_00012.md) — figure `seq_00012.png`
- [`00102`](seq_00102.md) — figure `seq_00102.png`
- [`00120`](seq_00120.md) — figure `seq_00120.png`
- [`01002`](seq_01002.md) — figure `seq_01002.png`
- [`01020`](seq_01020.md) — figure `seq_01020.png`
- [`01112`](seq_01112.md) — figure `seq_01112.png`
- [`01121`](seq_01121.md) — figure `seq_01121.png`
- [`01200`](seq_01200.md) — figure `seq_01200.png`
- [`01211`](seq_01211.md) — figure `seq_01211.png`
- [`01222`](seq_01222.md) — figure `seq_01222.png`
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# Inner-face down-apex sequence `000000`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 6×colour0.
- Realised by **4** of 7 configs (M(T), inner face).
- **5** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_000000.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
2 colouring(s) with down-apex sequence `000000`:
- face apex colours (canonical-rep edge order) `222222`, realises `000000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `111111`, realises `000000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:1 d8:1]`
## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
1 colouring(s) with down-apex sequence `000000`:
- face apex colours (canonical-rep edge order) `111111`, realises `000000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:1 d8:1]`
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
1 colouring(s) with down-apex sequence `000000`:
- face apex colours (canonical-rep edge order) `111111`, realises `000000` in some orientation · `A=012020202 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:1 d8:1]`
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
1 colouring(s) with down-apex sequence `000000`:
- face apex colours (canonical-rep edge order) `111111`, realises `000000` in some orientation · `A=012020202 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:1 d8:1]`
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# Inner-face down-apex sequence `000011`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 4×colour0, 2×colour1.
- Realised by **7** of 7 configs (M(T), inner face).
- **42** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_000011.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
12 colouring(s) with down-apex sequence `000011`:
- face apex colours (canonical-rep edge order) `222211`, realises `000011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `222002`, realises `000011` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `221122`, realises `000011` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `221111`, realises `000011` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `200222`, realises `000011` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `200002`, realises `000011` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `112222`, realises `000011` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `112211`, realises `000011` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `111122`, realises `000011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `111001`, realises `000011` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `100111`, realises `000011` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `100001`, realises `000011` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:0 d7:0 d8:1]`
## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
6 colouring(s) with down-apex sequence `000011`:
- face apex colours (canonical-rep edge order) `112222`, realises `000011` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `112211`, realises `000011` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `111122`, realises `000011` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `111001`, realises `000011` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `100111`, realises `000011` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `100001`, realises `000011` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:0 d7:0 d8:1]`
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
6 colouring(s) with down-apex sequence `000011`:
- face apex colours (canonical-rep edge order) `112222`, realises `000011` in some orientation · `A=012020101 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `112211`, realises `000011` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `111122`, realises `000011` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `111001`, realises `000011` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `002222`, realises `000011` in some orientation · `A=012120101 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `001111`, realises `000011` in some orientation · `A=012120202 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]`
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
4 colouring(s) with down-apex sequence `000011`:
- face apex colours (canonical-rep edge order) `122111`, realises `000011` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `111122`, realises `000011` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `111001`, realises `000011` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `001111`, realises `000011` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]`
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
4 colouring(s) with down-apex sequence `000011`:
- face apex colours (canonical-rep edge order) `200222`, realises `000011` in some orientation · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `200002`, realises `000011` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `002222`, realises `000011` in some orientation · `A=012120101 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `001111`, realises `000011` in some orientation · `A=012120202 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]`
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
4 colouring(s) with down-apex sequence `000011`:
- face apex colours (canonical-rep edge order) `211222`, realises `000011` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `200222`, realises `000011` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `200002`, realises `000011` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `001111`, realises `000011` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]`
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
6 colouring(s) with down-apex sequence `000011`:
- face apex colours (canonical-rep edge order) `221122`, realises `000011` in some orientation · `A=010120201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `221111`, realises `000011` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `220022`, realises `000011` in some orientation · `A=010121201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `001111`, realises `000011` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `000022`, realises `000011` in some orientation · `A=012121201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `000011`, realises `000011` in some orientation · `A=012121202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]`
papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.pdf
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# Inner-face down-apex sequence `000101`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 4×colour0, 2×colour1.
- Realised by **4** of 7 configs (M(T), inner face).
- **20** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_000101.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
2 colouring(s) with down-apex sequence `000101`:
- face apex colours (canonical-rep edge order) `020222`, realises `000101` in some orientation · `A=012102101 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `020002`, realises `000101` in some orientation · `A=012102121 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
6 colouring(s) with down-apex sequence `000101`:
- face apex colours (canonical-rep edge order) `020222`, realises `000101` in some orientation · `A=012012101 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `020002`, realises `000101` in some orientation · `A=012012121 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `010111`, realises `000101` in some orientation · `A=012021202 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `010001`, realises `000101` in some orientation · `A=012021212 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `020222`, realises `000101` in some orientation · `A=012102101 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `020002`, realises `000101` in some orientation · `A=012102121 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
6 colouring(s) with down-apex sequence `000101`:
- face apex colours (canonical-rep edge order) `212122`, realises `000101` in some orientation · `A=010201201 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `212111`, realises `000101` in some orientation · `A=010201202 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `020222`, realises `000101` in some orientation · `A=012012101 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `020002`, realises `000101` in some orientation · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `010111`, realises `000101` in some orientation · `A=012021202 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `010001`, realises `000101` in some orientation · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]`
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
6 colouring(s) with down-apex sequence `000101`:
- face apex colours (canonical-rep edge order) `220202`, realises `000101` in some orientation · `A=010121021 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `212122`, realises `000101` in some orientation · `A=010210201 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `212111`, realises `000101` in some orientation · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `011101`, realises `000101` in some orientation · `A=012020212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `010001`, realises `000101` in some orientation · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `000202`, realises `000101` in some orientation · `A=012121021 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]`
papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.pdf
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# Inner-face down-apex sequence `000110`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 4×colour0, 2×colour1.
- Realised by **7** of 7 configs (M(T), inner face).
- **42** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_000110.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8]
12 colouring(s) with down-apex sequence `000110`:
- face apex colours (canonical-rep edge order) `222211`, realises `000110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `222002`, realises `000110` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `221122`, realises `000110` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `221111`, realises `000110` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `200222`, realises `000110` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `200002`, realises `000110` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `112222`, realises `000110` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `112211`, realises `000110` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `111122`, realises `000110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `111001`, realises `000110` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `100111`, realises `000110` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `100001`, realises `000110` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:0 d7:0 d8:1]`
## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8]
6 colouring(s) with down-apex sequence `000110`:
- face apex colours (canonical-rep edge order) `112222`, realises `000110` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `112211`, realises `000110` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `111122`, realises `000110` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `111001`, realises `000110` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `100111`, realises `000110` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `100001`, realises `000110` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:0 d7:0 d8:1]`
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
6 colouring(s) with down-apex sequence `000110`:
- face apex colours (canonical-rep edge order) `112222`, realises `000110` in some orientation · `A=012020101 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `112211`, realises `000110` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `111122`, realises `000110` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `111001`, realises `000110` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `002222`, realises `000110` in some orientation · `A=012120101 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `001111`, realises `000110` in some orientation · `A=012120202 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]`
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
4 colouring(s) with down-apex sequence `000110`:
- face apex colours (canonical-rep edge order) `122111`, realises `000110` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `111122`, realises `000110` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `111001`, realises `000110` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `001111`, realises `000110` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]`
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
4 colouring(s) with down-apex sequence `000110`:
- face apex colours (canonical-rep edge order) `200222`, realises `000110` in some orientation · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `200002`, realises `000110` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `002222`, realises `000110` in some orientation · `A=012120101 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `001111`, realises `000110` in some orientation · `A=012120202 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]`
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
4 colouring(s) with down-apex sequence `000110`:
- face apex colours (canonical-rep edge order) `211222`, realises `000110` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `200222`, realises `000110` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `200002`, realises `000110` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `001111`, realises `000110` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]`
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
6 colouring(s) with down-apex sequence `000110`:
- face apex colours (canonical-rep edge order) `221122`, realises `000110` in some orientation · `A=010120201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `221111`, realises `000110` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `220022`, realises `000110` in some orientation · `A=010121201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `001111`, realises `000110` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `000022`, realises `000110` in some orientation · `A=012121201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `000011`, realises `000110` in some orientation · `A=012121202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]`
papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.pdf
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# Inner-face down-apex sequence `001001`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 4×colour0, 2×colour1.
- Realised by **4** of 7 configs (M(T), inner face).
- **8** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_001001.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
2 colouring(s) with down-apex sequence `001001`:
- face apex colours (canonical-rep edge order) `002002`, realises `001001` in some orientation · `A=012120121 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `001001`, realises `001001` in some orientation · `A=012120212 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:0 d7:0 d8:1]`
## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8]
2 colouring(s) with down-apex sequence `001001`:
- face apex colours (canonical-rep edge order) `122122`, realises `001001` in some orientation · `A=012010201 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `001001`, realises `001001` in some orientation · `A=012120212 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:0 d7:0 d8:1]`
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
2 colouring(s) with down-apex sequence `001001`:
- face apex colours (canonical-rep edge order) `002002`, realises `001001` in some orientation · `A=012120121 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `001001`, realises `001001` in some orientation · `A=012120212 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:0 d7:0 d8:1]`
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
2 colouring(s) with down-apex sequence `001001`:
- face apex colours (canonical-rep edge order) `211211`, realises `001001` in some orientation · `A=010202102 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `001001`, realises `001001` in some orientation · `A=012120212 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:0 d7:0 d8:1]`
papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.pdf
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# Inner-face down-apex sequence `001010`
Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`.
- Colour multiset: 4×colour0, 2×colour1.
- Realised by **4** of 7 configs (M(T), inner face).
- **20** Kempe-balanced colourings (mod colour permutation) realise it in some orientation.
- Figure: `seq_001010.png` (black rings mark the face's down apexes).
Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green.
## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8]
2 colouring(s) with down-apex sequence `001010`:
- face apex colours (canonical-rep edge order) `020222`, realises `001010` in some orientation · `A=012102101 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `020002`, realises `001010` in some orientation · `A=012102121 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8]
6 colouring(s) with down-apex sequence `001010`:
- face apex colours (canonical-rep edge order) `020222`, realises `001010` in some orientation · `A=012012101 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `020002`, realises `001010` in some orientation · `A=012012121 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `010111`, realises `001010` in some orientation · `A=012021202 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `010001`, realises `001010` in some orientation · `A=012021212 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `020222`, realises `001010` in some orientation · `A=012102101 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `020002`, realises `001010` in some orientation · `A=012102121 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]`
## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8]
6 colouring(s) with down-apex sequence `001010`:
- face apex colours (canonical-rep edge order) `212122`, realises `001010` in some orientation · `A=010201201 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `212111`, realises `001010` in some orientation · `A=010201202 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `020222`, realises `001010` in some orientation · `A=012012101 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `020002`, realises `001010` in some orientation · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `010111`, realises `001010` in some orientation · `A=012021202 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `010001`, realises `001010` in some orientation · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]`
## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8]
6 colouring(s) with down-apex sequence `001010`:
- face apex colours (canonical-rep edge order) `220202`, realises `001010` in some orientation · `A=010121021 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]`
- face apex colours (canonical-rep edge order) `212122`, realises `001010` in some orientation · `A=010210201 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]`
- face apex colours (canonical-rep edge order) `212111`, realises `001010` in some orientation · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]`
- face apex colours (canonical-rep edge order) `011101`, realises `001010` in some orientation · `A=012020212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:1 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `010001`, realises `001010` in some orientation · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]`
- face apex colours (canonical-rep edge order) `000202`, realises `001010` in some orientation · `A=012121021 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]`
papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.pdf
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