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>
This commit is contained in:
+312
@@ -0,0 +1,312 @@
|
||||
"""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()
|
||||
BIN
Binary file not shown.
BIN
Binary file not shown.
|
After Width: | Height: | Size: 469 KiB |
+69
@@ -0,0 +1,69 @@
|
||||
# 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
|
||||
|
||||
|
||||
|
||||
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.
|
||||
+81
@@ -0,0 +1,81 @@
|
||||
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)
|
||||
+139
@@ -0,0 +1,139 @@
|
||||
"""Draw every full medial tire graph with |A(T)| = 9.
|
||||
|
||||
Generates the dihedral-symmetry classes from ``full_medial_tire_generator`` and
|
||||
draws each one in the tooth style of the paper figures: the annular cycle A(T)
|
||||
as a thick ring of black vertices, up teeth pointing outward (blue apexes),
|
||||
singleton down teeth pointing inward (red apexes), and bites as a single inner
|
||||
apex (dark red) with four spokes to the two paired annular edges.
|
||||
|
||||
Output is a single grid figure (PNG + PDF) plus a text index of the classes.
|
||||
"""
|
||||
|
||||
from __future__ import annotations
|
||||
|
||||
import argparse
|
||||
import math
|
||||
import os
|
||||
|
||||
import matplotlib
|
||||
|
||||
matplotlib.use("Agg")
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
from full_medial_tire_generator import FullMedialTireGraph, generate
|
||||
|
||||
HERE = os.path.dirname(os.path.abspath(__file__))
|
||||
|
||||
|
||||
def vertex_xy(k: int, n: int, radius: float) -> tuple[float, float]:
|
||||
"""Annular vertex k, placed clockwise from the top of the unit circle."""
|
||||
angle = math.pi / 2 - 2 * math.pi * k / n
|
||||
return radius * math.cos(angle), radius * math.sin(angle)
|
||||
|
||||
|
||||
def edge_midpoint_angle(i: int, n: int) -> float:
|
||||
return math.pi / 2 - 2 * math.pi * (i + 0.5) / n
|
||||
|
||||
|
||||
def draw_graph(ax, g: FullMedialTireGraph) -> None:
|
||||
n = g.n
|
||||
ann = [vertex_xy(k, n, 1.0) for k in range(n)]
|
||||
matched = g.bite_edges
|
||||
|
||||
# annular cycle A(T)
|
||||
cyc_x = [p[0] for p in ann] + [ann[0][0]]
|
||||
cyc_y = [p[1] for p in ann] + [ann[0][1]]
|
||||
ax.plot(cyc_x, cyc_y, color="black", lw=1.6, zorder=2)
|
||||
|
||||
# up teeth (outer apexes) and singleton down teeth (inner apexes)
|
||||
for i, tooth in enumerate(g.tooth_word):
|
||||
if tooth == "U":
|
||||
r, color = 1.42, "#2b6cb0"
|
||||
elif i not in matched: # singleton down tooth
|
||||
r, color = 0.58, "#c53030"
|
||||
else:
|
||||
continue
|
||||
ang = edge_midpoint_angle(i, n)
|
||||
apex = (r * math.cos(ang), r * math.sin(ang))
|
||||
a0, a1 = ann[i], ann[(i + 1) % n]
|
||||
ax.plot([apex[0], a0[0]], [apex[1], a0[1]], color="#9a9a9a", lw=0.5, zorder=1)
|
||||
ax.plot([apex[0], a1[0]], [apex[1], a1[1]], color="#9a9a9a", lw=0.5, zorder=1)
|
||||
ax.scatter([apex[0]], [apex[1]], s=14, color=color, zorder=3)
|
||||
|
||||
# bites: one shared inner apex with four spokes
|
||||
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
|
||||
# pull slightly toward the centre so nested bites stay distinguishable
|
||||
apex = (cx * 0.82, cy * 0.82)
|
||||
for corner in corners:
|
||||
ax.plot([apex[0], corner[0]], [apex[1], corner[1]],
|
||||
color="#9a9a9a", lw=0.5, zorder=1)
|
||||
ax.scatter([apex[0]], [apex[1]], s=26, color="#7b1f1f",
|
||||
edgecolors="black", linewidths=0.4, zorder=4)
|
||||
|
||||
# annular vertices on top
|
||||
ax.scatter([p[0] for p in ann], [p[1] for p in ann],
|
||||
s=10, color="black", zorder=5)
|
||||
|
||||
bites = ",".join(f"{i}{j}" for i, j in sorted(g.bites)) or "-"
|
||||
ax.set_title(f"{g.tooth_word}\n{bites}", fontsize=5.5, pad=1.5)
|
||||
ax.set_xlim(-1.6, 1.6)
|
||||
ax.set_ylim(-1.6, 1.6)
|
||||
ax.set_aspect("equal")
|
||||
ax.axis("off")
|
||||
|
||||
|
||||
def run(args: argparse.Namespace) -> None:
|
||||
graphs = list(generate(args.n, min_up_teeth=args.min_up, dedup=True))
|
||||
print(f"n={args.n}: {len(graphs)} dihedral classes")
|
||||
|
||||
cols = args.cols
|
||||
rows = math.ceil(len(graphs) / cols)
|
||||
fig, axes = plt.subplots(rows, cols, figsize=(cols * 1.45, rows * 1.6))
|
||||
axes = axes.reshape(rows, cols)
|
||||
for idx in range(rows * cols):
|
||||
ax = axes[idx // cols][idx % cols]
|
||||
if idx < len(graphs):
|
||||
draw_graph(ax, graphs[idx])
|
||||
else:
|
||||
ax.axis("off")
|
||||
|
||||
fig.suptitle(
|
||||
f"All full medial tire graphs with $|A(T)|={args.n}$ "
|
||||
f"({len(graphs)} classes up to dihedral symmetry)",
|
||||
fontsize=12, y=0.997,
|
||||
)
|
||||
fig.tight_layout(rect=(0, 0, 1, 0.985))
|
||||
|
||||
png = os.path.join(HERE, f"full_medial_tire_n{args.n}.png")
|
||||
pdf = os.path.join(HERE, f"full_medial_tire_n{args.n}.pdf")
|
||||
fig.savefig(png, dpi=200)
|
||||
fig.savefig(pdf)
|
||||
print(f"wrote {png}")
|
||||
print(f"wrote {pdf}")
|
||||
|
||||
if args.index:
|
||||
index_path = os.path.join(HERE, f"full_medial_tire_n{args.n}_index.txt")
|
||||
with open(index_path, "w") as fh:
|
||||
for idx, g in enumerate(graphs):
|
||||
bites = ",".join(f"({i},{j})" for i, j in sorted(g.bites)) or "-"
|
||||
fh.write(
|
||||
f"{idx:3d} word={g.tooth_word} up={len(g.up_edges)} "
|
||||
f"down={len(g.down_edges)} bites={bites}\n"
|
||||
)
|
||||
print(f"wrote {index_path}")
|
||||
|
||||
|
||||
def main() -> None:
|
||||
parser = argparse.ArgumentParser(description=__doc__)
|
||||
parser.add_argument("--n", type=int, default=9)
|
||||
parser.add_argument("--min-up", type=int, default=3)
|
||||
parser.add_argument("--cols", type=int, default=12)
|
||||
parser.add_argument("--index", action="store_true", default=True)
|
||||
run(parser.parse_args())
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
main()
|
||||
@@ -11,23 +11,30 @@
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Medial tire pieces}}{2}{}\protected@file@percent }
|
||||
\newlabel{def:full-medial-tire}{{3.1}{2}}
|
||||
\newlabel{thm:annular-medial-colour-bound}{{3.3}{3}}
|
||||
\newlabel{def:boundary-medial-vertices}{{3.4}{3}}
|
||||
\newlabel{def:medial-restriction-relation}{{3.5}{3}}
|
||||
\newlabel{def:annular-teeth}{{3.4}{3}}
|
||||
\newlabel{rem:teeth-sharing}{{3.5}{3}}
|
||||
\newlabel{rem:up-teeth-count}{{3.6}{3}}
|
||||
\newlabel{def:bite}{{3.7}{4}}
|
||||
\newlabel{rem:bite-face-count}{{3.8}{4}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A full medial tire graph $\mathsf {M}(T)$ illustrating the tooth terminology. The thick cycle is the annular medial cycle $A(T)$, whose black vertices are the annular medial vertices. Each edge of $A(T)$ carries one tooth: up teeth (blue apexes, outer-boundary medial vertices) point into the outer region, and down teeth (red apexes, inner-boundary medial vertices) point into the inner region. The two down teeth meeting at the central shared apex (larger red vertex) form a bite; that shared apex splits the inner region into two faces, one with four down teeth on its boundary and one with none.}}{4}{}\protected@file@percent }
|
||||
\newlabel{fig:medial-teeth-example}{{1}{4}}
|
||||
\newlabel{def:boundary-medial-vertices}{{3.9}{4}}
|
||||
\citation{bauerfeld-nested-tire-decompositions}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Three six-face full medial tire graphs found by the boundary-state restriction search. Black vertices are annular medial vertices; blue vertices are outer boundary medial vertices and red vertices are inner boundary medial vertices. The word below each diagram records the outer/inner type of the six annular faces in cyclic order. Boundary states are identified only up to colour permutation, not by rotation or reflection of the boundary order.}}{5}{}\protected@file@percent }
|
||||
\newlabel{fig:medial-restriction-worst-cases}{{2}{5}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces A proper vertex $3$-colouring of the full medial graph of the first seven-vertex counterexample found by the experiment. The medial vertex labelled $ij$ corresponds to the edge $(i,j)$ of the triangulation. For the vertex-source decomposition at source $1$, the highlighted annular medial cycle has colour counts $(2,2,2)$, so it is not coloured with two colours except at at most one vertex.}}{5}{}\protected@file@percent }
|
||||
\newlabel{fig:medial-annular-cycle-counterexample}{{3}{5}}
|
||||
\newlabel{def:medial-restriction-relation}{{3.10}{5}}
|
||||
\citation{bauerfeld-nested-tire-decompositions}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Three six-face full medial tire graphs found by the boundary-state restriction search. Black vertices are annular medial vertices; blue vertices are outer boundary medial vertices and red vertices are inner boundary medial vertices. The word below each diagram records the outer/inner type of the six annular faces in cyclic order. Boundary states are identified only up to colour permutation, not by rotation or reflection of the boundary order.}}{4}{}\protected@file@percent }
|
||||
\newlabel{fig:medial-restriction-worst-cases}{{1}{4}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A proper vertex $3$-colouring of the full medial graph of the first seven-vertex counterexample found by the experiment. The medial vertex labelled $ij$ corresponds to the edge $(i,j)$ of the triangulation. For the vertex-source decomposition at source $1$, the highlighted annular medial cycle has colour counts $(2,2,2)$, so it is not coloured with two colours except at at most one vertex.}}{4}{}\protected@file@percent }
|
||||
\newlabel{fig:medial-annular-cycle-counterexample}{{2}{4}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{4}{}\protected@file@percent }
|
||||
\newlabel{cor:medial-tire-decomposition}{{4.1}{4}}
|
||||
\newlabel{def:compatible-family}{{4.2}{5}}
|
||||
\newlabel{prop:gluing-criterion}{{4.3}{5}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{5}{}\protected@file@percent }
|
||||
\newlabel{def:medial-boundary-state}{{5.1}{5}}
|
||||
\newlabel{conj:medial-chain-pigeonhole}{{5.2}{5}}
|
||||
\newlabel{conj:medial-route-fct}{{5.3}{6}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Kempe-cycle conservation across medial tires}}{6}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{6}{}\protected@file@percent }
|
||||
\newlabel{cor:medial-tire-decomposition}{{4.1}{6}}
|
||||
\newlabel{def:compatible-family}{{4.2}{6}}
|
||||
\newlabel{prop:gluing-criterion}{{4.3}{6}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{6}{}\protected@file@percent }
|
||||
\newlabel{def:medial-boundary-state}{{5.1}{6}}
|
||||
\newlabel{conj:medial-chain-pigeonhole}{{5.2}{7}}
|
||||
\newlabel{conj:medial-route-fct}{{5.3}{7}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Kempe-cycle conservation across medial tires}}{7}{}\protected@file@percent }
|
||||
\bibcite{bauerfeld-nested-tire-decompositions}{1}
|
||||
\bibcite{tait-original}{2}
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
@@ -35,5 +42,5 @@
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{29.38873pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{8}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{10}
|
||||
|
||||
@@ -1,6 +1,5 @@
|
||||
# Fdb version 3
|
||||
["pdflatex"] 1780957013 "/Users/didericis/Code/math-research/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex" "paper.pdf" "paper" 1780957014
|
||||
"/Users/didericis/Code/math-research/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex" 1780957012 31877 e1531fb82c355208b5b7c8e57cfb6e8a ""
|
||||
["pdflatex"] 1781194762 "paper.tex" "paper.pdf" "paper" 1781194763
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
|
||||
@@ -133,8 +132,8 @@
|
||||
"/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 ""
|
||||
"/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc ""
|
||||
"/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c ""
|
||||
"paper.aux" 1780957014 3096 87a6f727fdb250a8819be3b4fc862d4e "pdflatex"
|
||||
"paper.tex" 1780957012 31877 e1531fb82c355208b5b7c8e57cfb6e8a ""
|
||||
"paper.aux" 1781194763 4035 146533a306519cb8688d4e85db1d3f80 "pdflatex"
|
||||
"paper.tex" 1781194559 37363 b7b005cfaefc0e3a582f756b993a034e ""
|
||||
(generated)
|
||||
"paper.aux"
|
||||
"paper.log"
|
||||
|
||||
@@ -2,7 +2,7 @@ PWD /Users/didericis/Code/math-research/papers/medial_tire_decompositions_of_pla
|
||||
INPUT /usr/local/texlive/2022/texmf.cnf
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf
|
||||
INPUT /usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt
|
||||
INPUT /Users/didericis/Code/math-research/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex
|
||||
INPUT paper.tex
|
||||
OUTPUT paper.log
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
|
||||
@@ -1,12 +1,12 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 8 JUN 2026 18:16
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 11 JUN 2026 12:19
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
file:line:error style messages enabled.
|
||||
%&-line parsing enabled.
|
||||
**/Users/didericis/Code/math-research/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex
|
||||
(/Users/didericis/Code/math-research/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex
|
||||
**paper.tex
|
||||
(./paper.tex
|
||||
LaTeX2e <2021-11-15> patch level 1
|
||||
L3 programming layer <2022-02-24> (/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
L3 programming layer <2022-02-24>
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
Document Class: amsart 2020/05/29 v2.20.6
|
||||
\linespacing=\dimen138
|
||||
\normalparindent=\dimen139
|
||||
@@ -18,14 +18,17 @@ Package: amsmath 2021/10/15 v2.17l AMS math features
|
||||
For additional information on amsmath, use the `?' option.
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
Package: amstext 2021/08/26 v2.01 AMS text
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
File: amsgen.sty 1999/11/30 v2.0 generic functions
|
||||
\@emptytoks=\toks16
|
||||
\ex@=\dimen140
|
||||
)) (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
))
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
Package: amsbsy 1999/11/29 v1.2d Bold Symbols
|
||||
\pmbraise@=\dimen141
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
Package: amsopn 2021/08/26 v2.02 operator names
|
||||
)
|
||||
\inf@bad=\count185
|
||||
@@ -66,10 +69,13 @@ LaTeX Font Info: Redeclaring font encoding OMS on input line 744.
|
||||
LaTeX Info: Redefining \[ on input line 2938.
|
||||
LaTeX Info: Redefining \] on input line 2939.
|
||||
)
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 397.
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 397
|
||||
.
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
|
||||
Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
|
||||
\symAMSa=\mathgroup4
|
||||
\symAMSb=\mathgroup5
|
||||
@@ -100,42 +106,63 @@ LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
|
||||
\thm@postskip=\skip55
|
||||
\thm@headsep=\skip56
|
||||
\dth@everypar=\toks26
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
|
||||
Package: amssymb 2013/01/14 v3.01 AMS font symbols
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
|
||||
Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
|
||||
Package: keyval 2014/10/28 v1.15 key=value parser (DPC)
|
||||
\KV@toks@=\toks27
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
|
||||
Package: graphics 2021/03/04 v1.4d Standard LaTeX Graphics (DPC,SPQR)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
|
||||
Package: trig 2021/08/11 v1.11 sin cos tan (DPC)
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
|
||||
File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration
|
||||
)
|
||||
Package graphics Info: Driver file: pdftex.def on input line 107.
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
|
||||
File: pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex
|
||||
))
|
||||
\Gin@req@height=\dimen150
|
||||
\Gin@req@width=\dimen151
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.te
|
||||
x
|
||||
\pgfutil@everybye=\toks28
|
||||
\pgfutil@tempdima=\dimen152
|
||||
\pgfutil@tempdimb=\dimen153
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-li
|
||||
sts.tex))
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def
|
||||
\pgfutil@abb=\box53
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex)
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex)
|
||||
Package: pgfrcs 2021/05/15 v3.1.9a (3.1.9a)
|
||||
))
|
||||
Package: pgf 2021/05/15 v3.1.9a (3.1.9a)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
|
||||
Package: pgfsys 2021/05/15 v3.1.9a (3.1.9a)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
|
||||
\pgfkeys@pathtoks=\toks29
|
||||
\pgfkeys@temptoks=\toks30
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.c
|
||||
ode.tex
|
||||
\pgfkeys@tmptoks=\toks31
|
||||
))
|
||||
\pgf@x=\dimen154
|
||||
@@ -158,23 +185,33 @@ Package: pgfsys 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\t@pgf@tokb=\toks33
|
||||
\t@pgf@tokc=\toks34
|
||||
\pgf@sys@id@count=\count276
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg
|
||||
File: pgf.cfg 2021/05/15 v3.1.9a (3.1.9a)
|
||||
)
|
||||
Driver file for pgf: pgfsys-pdftex.def
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.d
|
||||
ef
|
||||
File: pgfsys-pdftex.def 2021/05/15 v3.1.9a (3.1.9a)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-p
|
||||
df.def
|
||||
File: pgfsys-common-pdf.def 2021/05/15 v3.1.9a (3.1.9a)
|
||||
))) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
|
||||
)))
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.
|
||||
code.tex
|
||||
File: pgfsyssoftpath.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgfsyssoftpath@smallbuffer@items=\count277
|
||||
\pgfsyssoftpath@bigbuffer@items=\count278
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.
|
||||
code.tex
|
||||
File: pgfsysprotocol.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
)) (/usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
|
||||
Package: xcolor 2021/10/31 v2.13 LaTeX color extensions (UK)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
|
||||
File: color.cfg 2016/01/02 v1.6 sample color configuration
|
||||
)
|
||||
Package xcolor Info: Driver file: pdftex.def on input line 227.
|
||||
@@ -187,18 +224,43 @@ Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1372.
|
||||
Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1373.
|
||||
Package xcolor Info: Model `Gray' substituted by `gray' on input line 1374.
|
||||
Package xcolor Info: Model `wave' substituted by `hsb' on input line 1375.
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
|
||||
Package: pgfcore 2021/05/15 v3.1.9a (3.1.9a)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex
|
||||
\pgfmath@dimen=\dimen164
|
||||
\pgfmath@count=\count279
|
||||
\pgfmath@box=\box54
|
||||
\pgfmath@toks=\toks35
|
||||
\pgfmath@stack@operand=\toks36
|
||||
\pgfmath@stack@operation=\toks37
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex))) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.
|
||||
tex
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic
|
||||
.code.tex)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigo
|
||||
nometric.code.tex)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.rando
|
||||
m.code.tex)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.compa
|
||||
rison.code.tex)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.
|
||||
code.tex)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round
|
||||
.code.tex)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.
|
||||
code.tex)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integ
|
||||
erarithmetics.code.tex)))
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex
|
||||
\c@pgfmathroundto@lastzeros=\count280
|
||||
)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex
|
||||
)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.co
|
||||
de.tex
|
||||
File: pgfcorepoints.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgf@picminx=\dimen165
|
||||
\pgf@picmaxx=\dimen166
|
||||
@@ -214,76 +276,127 @@ File: pgfcorepoints.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgf@yy=\dimen176
|
||||
\pgf@zx=\dimen177
|
||||
\pgf@zy=\dimen178
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconst
|
||||
ruct.code.tex
|
||||
File: pgfcorepathconstruct.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgf@path@lastx=\dimen179
|
||||
\pgf@path@lasty=\dimen180
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage
|
||||
.code.tex
|
||||
File: pgfcorepathusage.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgf@shorten@end@additional=\dimen181
|
||||
\pgf@shorten@start@additional=\dimen182
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.co
|
||||
de.tex
|
||||
File: pgfcorescopes.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgfpic=\box55
|
||||
\pgf@hbox=\box56
|
||||
\pgf@layerbox@main=\box57
|
||||
\pgf@picture@serial@count=\count281
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicst
|
||||
ate.code.tex
|
||||
File: pgfcoregraphicstate.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgflinewidth=\dimen183
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransform
|
||||
ations.code.tex
|
||||
File: pgfcoretransformations.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgf@pt@x=\dimen184
|
||||
\pgf@pt@y=\dimen185
|
||||
\pgf@pt@temp=\dimen186
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.cod
|
||||
e.tex
|
||||
File: pgfcorequick.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.c
|
||||
ode.tex
|
||||
File: pgfcoreobjects.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathproce
|
||||
ssing.code.tex
|
||||
File: pgfcorepathprocessing.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.co
|
||||
de.tex
|
||||
File: pgfcorearrows.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgfarrowsep=\dimen187
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.cod
|
||||
e.tex
|
||||
File: pgfcoreshade.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgf@max=\dimen188
|
||||
\pgf@sys@shading@range@num=\count282
|
||||
\pgf@shadingcount=\count283
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.cod
|
||||
e.tex
|
||||
File: pgfcoreimage.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.
|
||||
code.tex
|
||||
File: pgfcoreexternal.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgfexternal@startupbox=\box58
|
||||
)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex
|
||||
))
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.co
|
||||
de.tex
|
||||
File: pgfcorelayers.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretranspare
|
||||
ncy.code.tex
|
||||
File: pgfcoretransparency.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.
|
||||
code.tex
|
||||
File: pgfcorepatterns.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.
|
||||
tex
|
||||
File: pgfcorerdf.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
))) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex
|
||||
)))
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.cod
|
||||
e.tex
|
||||
File: pgfmoduleshapes.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgfnodeparttextbox=\box59
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.
|
||||
tex
|
||||
File: pgfmoduleplot.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version
|
||||
-0-65.sty
|
||||
Package: pgfcomp-version-0-65 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgf@nodesepstart=\dimen189
|
||||
\pgf@nodesepend=\dimen190
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version
|
||||
-1-18.sty
|
||||
Package: pgfcomp-version-1-18 2021/05/15 v3.1.9a (3.1.9a)
|
||||
)) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex)) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
|
||||
))
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex)
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex))
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
|
||||
Package: pgffor 2021/05/15 v3.1.9a (3.1.9a)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)
|
||||
\pgffor@iter=\dimen191
|
||||
\pgffor@skip=\dimen192
|
||||
\pgffor@stack=\toks38
|
||||
\pgffor@toks=\toks39
|
||||
)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
|
||||
))
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.cod
|
||||
e.tex
|
||||
Package: tikz 2021/05/15 v3.1.9a (3.1.9a)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothan
|
||||
dlers.code.tex
|
||||
File: pgflibraryplothandlers.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgf@plot@mark@count=\count284
|
||||
\pgfplotmarksize=\dimen193
|
||||
@@ -304,26 +417,34 @@ File: pgflibraryplothandlers.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\tikznumberofchildren=\count286
|
||||
\tikznumberofcurrentchild=\count287
|
||||
\tikz@fig@count=\count288
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.cod
|
||||
e.tex
|
||||
File: pgfmodulematrix.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgfmatrixcurrentrow=\count289
|
||||
\pgfmatrixcurrentcolumn=\count290
|
||||
\pgf@matrix@numberofcolumns=\count291
|
||||
)
|
||||
\tikz@expandcount=\count292
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/librarie
|
||||
s/tikzlibrarytopaths.code.tex
|
||||
File: tikzlibrarytopaths.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
))) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex
|
||||
)))
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/librarie
|
||||
s/tikzlibrarybackgrounds.code.tex
|
||||
File: tikzlibrarybackgrounds.code.tex 2021/05/15 v3.1.9a (3.1.9a)
|
||||
\pgf@layerbox@background=\box64
|
||||
\pgf@layerboxsaved@background=\box65
|
||||
)
|
||||
\c@theorem=\count293
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX)
|
||||
\l__color_backend_stack_int=\count294
|
||||
\l__pdf_internal_box=\box66
|
||||
) (./paper.aux)
|
||||
)
|
||||
(./paper.aux)
|
||||
\openout1 = `paper.aux'.
|
||||
|
||||
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 29.
|
||||
@@ -341,13 +462,17 @@ LaTeX Font Info: ... okay on input line 29.
|
||||
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 29.
|
||||
LaTeX Font Info: ... okay on input line 29.
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 29.
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
|
||||
)
|
||||
LaTeX Font Info: Trying to load font information for U+msb on input line 29.
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
|
||||
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
|
||||
File: umsb.fd 2013/01/14 v3.01 AMS symbols B
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
|
||||
[Loading MPS to PDF converter (version 2006.09.02).]
|
||||
\scratchcounter=\count295
|
||||
\scratchdimen=\dimen259
|
||||
@@ -362,30 +487,58 @@ File: umsb.fd 2013/01/14 v3.01 AMS symbols B
|
||||
\everyMPtoPDFconversion=\toks41
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
|
||||
Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf
|
||||
Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 485.
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
|
||||
File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live
|
||||
)) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [2]
|
||||
Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4
|
||||
85.
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
|
||||
File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
|
||||
e
|
||||
))
|
||||
[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
|
||||
[2] [3]
|
||||
Overfull \hbox (62.13657pt too wide) in paragraph at lines 363--372
|
||||
[][]
|
||||
[]
|
||||
|
||||
|
||||
LaTeX Warning: `h' float specifier changed to `ht'.
|
||||
|
||||
|
||||
LaTeX Warning: `h' float specifier changed to `ht'.
|
||||
|
||||
[3] [4] [5] [6] [7] [8] (./paper.aux) )
|
||||
[4] [5] [6] [7] [8] [9] [10] (./paper.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
14159 strings out of 478268
|
||||
280330 string characters out of 5846347
|
||||
592946 words of memory out of 5000000
|
||||
31987 multiletter control sequences out of 15000+600000
|
||||
14415 strings out of 478268
|
||||
283664 string characters out of 5846347
|
||||
609309 words of memory out of 5000000
|
||||
32244 multiletter control sequences out of 15000+600000
|
||||
477048 words of font info for 58 fonts, out of 8000000 for 9000
|
||||
1302 hyphenation exceptions out of 8191
|
||||
84i,8n,89p,676b,841s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmss10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
|
||||
Output written on paper.pdf (8 pages, 265030 bytes).
|
||||
84i,8n,89p,736b,838s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</usr/local/texlive/2022/texmf-dist
|
||||
/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/
|
||||
fonts/type1/public/amsfonts/cm/cmbx8.pfb></usr/local/texlive/2022/texmf-dist/fo
|
||||
nts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fo
|
||||
nts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fon
|
||||
ts/type1/public/amsfonts/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts
|
||||
/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/t
|
||||
ype1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
|
||||
e1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1
|
||||
/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pu
|
||||
blic/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publi
|
||||
c/amsfonts/cm/cmss10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public
|
||||
/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
|
||||
amsfonts/cm/cmsy5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
|
||||
sfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
|
||||
onts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
|
||||
ts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
|
||||
s/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/
|
||||
cm/cmtt8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/sy
|
||||
mbols/msam10.pfb>
|
||||
Output written on paper.pdf (10 pages, 272876 bytes).
|
||||
PDF statistics:
|
||||
129 PDF objects out of 1000 (max. 8388607)
|
||||
80 compressed objects within 1 object stream
|
||||
135 PDF objects out of 1000 (max. 8388607)
|
||||
84 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
13 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
Binary file not shown.
@@ -195,8 +195,8 @@ Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire tread with
|
||||
non-degenerate boundaries and simple inner boundary $B_{\mathrm{in}}$.
|
||||
Let $A(T)$ be the subgraph of $\mathsf{M}(T)$ induced by the annular
|
||||
medial vertices. For a graph $H$, write $\operatorname{Col}_3(H)$ for
|
||||
the set of proper $3$-vertex-colourings of $H$. Then $A(T)$ is a cycle
|
||||
and
|
||||
the set of proper $3$-vertex-colourings of $H$. Then $A(T)$ is a
|
||||
cycle---the \emph{annular cycle} of $T$---and
|
||||
\[
|
||||
|\operatorname{Col}_3(\mathsf{M}(T))|
|
||||
\;\leq\; |\operatorname{Col}_3(A(T))|.
|
||||
@@ -234,6 +234,145 @@ same restriction to $A(T)$ are identical, so the restriction map is
|
||||
injective. The stated inequality follows.
|
||||
\end{proof}
|
||||
|
||||
\begin{definition}[Annular teeth]
|
||||
\label{def:annular-teeth}
|
||||
By Theorem~\ref{thm:annular-medial-colour-bound} the annular medial
|
||||
vertices induce the cycle $A(T)$ in $\mathsf{M}(T)$, the annular cycle, so
|
||||
the edges of $\mathsf{M}(T)$ joining two annular medial vertices are
|
||||
exactly the edges of $A(T)$. Each such edge lies in exactly one triangle ($3$-cycle) of
|
||||
$\mathsf{M}(T)$, and the third vertex of that triangle is necessarily
|
||||
non-annular, since $A(T)$ has no chords. We call this triangle an
|
||||
\emph{annular tooth} and its non-annular vertex the \emph{apex} of the
|
||||
tooth.
|
||||
|
||||
The cycle $A(T)$ separates the plane into two regions: the \emph{outer
|
||||
region}, which contains the outer-boundary medial vertices, and the
|
||||
\emph{inner region}, which contains the inner-boundary medial vertices.
|
||||
An annular tooth is an \emph{up tooth} if its apex lies in the outer
|
||||
region, and a \emph{down tooth} if its apex lies in the inner region.
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
\label{rem:teeth-sharing}
|
||||
The apexes of annular teeth satisfy two sharing bounds: no two up teeth
|
||||
share an apex, and at most two down teeth share an apex.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
\label{rem:up-teeth-count}
|
||||
The number of up teeth in $\mathsf{M}(T)$ is at least three.
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}[Bites]
|
||||
\label{def:bite}
|
||||
By Remark~\ref{rem:teeth-sharing} an apex is shared by at most two down
|
||||
teeth. When two down teeth share an apex, the pair is called a
|
||||
\emph{bite}, and their common apex is the \emph{apex of the bite}. A down
|
||||
tooth that belongs to a bite is a \emph{bite tooth}. We further require the
|
||||
two annular edges carrying the teeth of a bite to be \emph{non-incident}:
|
||||
they share no annular vertex of $A(T)$. Equivalently, the two bite teeth
|
||||
meet only at their common apex.
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
\label{rem:bite-face-count}
|
||||
Let $B(T)$ be the subgraph of $\mathsf{M}(T)$ consisting of $A(T)$
|
||||
together with all bite apexes (equivalently, $A(T)$ together with all
|
||||
bite teeth), drawn as a plane graph. For every interior face of $B(T)$
|
||||
that is not a bite tooth, the number of down teeth whose apex lies in
|
||||
the interior of that face is either $0$ or at least $3$. Such an apex is
|
||||
necessarily that of a singleton down tooth: every bite apex is a vertex
|
||||
of $B(T)$, so it lies on a face boundary rather than in any face
|
||||
interior.
|
||||
\end{remark}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\begin{tikzpicture}[scale=2.3,
|
||||
ann/.style={circle, fill=black, inner sep=1.1pt},
|
||||
upv/.style={circle, draw=blue!70!black, fill=blue!15, inner sep=1.5pt},
|
||||
downv/.style={circle, draw=red!70!black, fill=red!15, inner sep=1.5pt},
|
||||
bitev/.style={circle, draw=red!70!black, fill=red!35, inner sep=1.9pt},
|
||||
cyc/.style={black, line width=1.0pt},
|
||||
tth/.style={black!55, line width=0.45pt},
|
||||
lbl/.style={font=\scriptsize},
|
||||
lead/.style={black!55, line width=0.3pt}]
|
||||
|
||||
% annular medial vertices forming the cycle A(T)
|
||||
\foreach \k/\a in {0/105,1/75,2/45,3/15,4/-15,5/-45,6/-75,7/-105,8/-135,9/-165,10/165,11/135}
|
||||
\coordinate (v\k) at (\a:1);
|
||||
|
||||
% down-tooth apexes (inner region)
|
||||
\coordinate (d1) at (60:0.60);
|
||||
\coordinate (d2) at (30:0.60);
|
||||
\coordinate (d3) at (0:0.60);
|
||||
\coordinate (d4) at (-30:0.60);
|
||||
|
||||
% shared apex of the bite
|
||||
\coordinate (p) at (0,0);
|
||||
|
||||
% up-tooth apexes (outer region)
|
||||
\coordinate (u1) at (-60:1.35);
|
||||
\coordinate (u2) at (-120:1.35);
|
||||
\coordinate (u3) at (-150:1.35);
|
||||
\coordinate (u4) at (180:1.35);
|
||||
\coordinate (u5) at (150:1.35);
|
||||
\coordinate (u6) at (120:1.35);
|
||||
|
||||
% annular medial cycle
|
||||
\draw[cyc] (v0)--(v1)--(v2)--(v3)--(v4)--(v5)--(v6)--(v7)--(v8)--(v9)--(v10)--(v11)--cycle;
|
||||
|
||||
% down teeth into the right inner region
|
||||
\draw[tth] (d1)--(v1) (d1)--(v2);
|
||||
\draw[tth] (d2)--(v2) (d2)--(v3);
|
||||
\draw[tth] (d3)--(v3) (d3)--(v4);
|
||||
\draw[tth] (d4)--(v4) (d4)--(v5);
|
||||
|
||||
% up teeth
|
||||
\draw[tth] (u1)--(v5) (u1)--(v6);
|
||||
\draw[tth] (u2)--(v7) (u2)--(v8);
|
||||
\draw[tth] (u3)--(v8) (u3)--(v9);
|
||||
\draw[tth] (u4)--(v9) (u4)--(v10);
|
||||
\draw[tth] (u5)--(v10) (u5)--(v11);
|
||||
\draw[tth] (u6)--(v11) (u6)--(v0);
|
||||
|
||||
% the bite: two down teeth sharing the apex p
|
||||
\draw[tth] (p)--(v0) (p)--(v1) (p)--(v6) (p)--(v7);
|
||||
|
||||
% vertices
|
||||
\foreach \k in {0,...,11} \node[ann] at (v\k) {};
|
||||
\foreach \u in {u1,u2,u3,u4,u5,u6} \node[upv] at (\u) {};
|
||||
\foreach \dd in {d1,d2,d3,d4} \node[downv] at (\dd) {};
|
||||
\node[bitev] at (p) {};
|
||||
|
||||
% annotations
|
||||
\node[lbl, anchor=west] at (42:1.78) (Ldt) {down tooth};
|
||||
\draw[lead] (Ldt.west) -- (30:0.74);
|
||||
|
||||
\node[lbl, anchor=east] at (150:1.86) (Lut) {up tooth};
|
||||
\draw[lead] (Lut.east) -- (150:1.22);
|
||||
|
||||
\node[lbl] at (90:1.62) (Lbite) {bite};
|
||||
\draw[lead] (Lbite.south) -- (0,0.08);
|
||||
|
||||
\node[lbl, anchor=west] at (-12:1.78) (L4) {region with 4 down teeth};
|
||||
\draw[lead] (L4.west) -- (-9:0.80);
|
||||
|
||||
\node[lbl, anchor=east] at (192:1.86) (L0) {region with 0 down teeth};
|
||||
\draw[lead] (L0.east) -- (180:0.45);
|
||||
\end{tikzpicture}
|
||||
\caption{A full medial tire graph $\mathsf{M}(T)$ illustrating the tooth
|
||||
terminology. The thick cycle is the annular medial cycle $A(T)$, whose
|
||||
black vertices are the annular medial vertices. Each edge of $A(T)$
|
||||
carries one tooth: up teeth (blue apexes, outer-boundary medial vertices)
|
||||
point into the outer region, and down teeth (red apexes, inner-boundary
|
||||
medial vertices) point into the inner region. The two down teeth meeting
|
||||
at the central shared apex (larger red vertex) form a bite; that shared
|
||||
apex splits the inner region into two faces, one with four down teeth on
|
||||
its boundary and one with none.}
|
||||
\label{fig:medial-teeth-example}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\begin{tikzpicture}[scale=0.82,
|
||||
|
||||
Reference in New Issue
Block a user