didericis
77093cb0b0
Add 21-vertex and 24-vertex examples showing recursive lopsidedness at d=2. Empirically confirm that the iterated algorithm (balanced switch when available, preprocess otherwise) drives every face to depth 0 on all tested configurations. Frame the remaining open question as identifying a strictly-decreasing monovariant under unbalanced preprocessing switches. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
251 lines
9.5 KiB
Python
251 lines
9.5 KiB
Python
"""Build a maximal-outerplanar L_k whose unique depth-2 face has NO
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balanced surface switch, then test whether preprocessing reaches one.
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Dual-tree blueprint:
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F (depth 2, degree 3)
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/|\\
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F1' F2' F3' each depth 1, degree 3
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/\\ /\\ /\\
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E_i G_i (per arm: E_i depth-0 ear, G_i depth-1 degree-3 node)
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/\\
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E'_i H_i (G_i's two non-F'_i children: both depth-0 ears)
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Inner-face count: 1 + 3 + 3 + 3 + 3 + 3 = 16. So polygon has n = 18.
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For each F'_i: non-F neighbours are E_i (depth 0) and G_i (depth 1).
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NOT balanced (G_i not depth 0). Hence F has no balanced surface switch.
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Concrete chord construction: place vertices 0..17 around the outer cycle.
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Allocate one "arm" of 6 outer-cycle vertices per F'_i, plus three vertices
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for F.
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Arm i (i = 0,1,2) covers outer-cycle positions [6i, 6i+5]; the F vertices
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are positions {0, 6, 12}.
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"""
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import os
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import math
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import networkx as nx
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import matplotlib.pyplot as plt
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from matplotlib.patches import Polygon
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OUT_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), os.pardir)
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n = 18
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POS = {i: (math.cos(math.radians(90 - i * 360 / n)),
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math.sin(math.radians(90 - i * 360 / n))) for i in range(n)}
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def outer_edges(num):
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return [(i, (i + 1) % num) for i in range(num)]
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def face_edges(f):
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return {frozenset((f[0], f[1])), frozenset((f[1], f[2])),
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frozenset((f[0], f[2]))}
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def compute_depths(faces, outer_edge_set):
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D = nx.Graph()
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D.add_nodes_from(range(len(faces)))
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for i, fi in enumerate(faces):
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for j, fj in enumerate(faces):
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if i < j and face_edges(fi) & face_edges(fj):
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D.add_edge(i, j)
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B = [i for i, f in enumerate(faces)
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if len(face_edges(f) & outer_edge_set) >= 1]
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if not B:
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return {i: float('inf') for i in range(len(faces))}, D
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depth = {i: min(nx.shortest_path_length(D, i, b) for b in B)
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for i in range(len(faces))}
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return depth, D
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# Apex vertices of F at outer positions 0, 6, 12.
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# Per arm i: outer positions p = [a, a+1, a+2, a+3, a+4, a+5] where a = 6i.
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# Reading positions: a = u_i, a+1 = e1, a+2 = m, a+3 = e2, a+4 = h, a+5 = v_i = u_{i+1}
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#
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# Inside the polygon arc (u_i ... u_{i+1}) we want to triangulate so that
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# the chord u_i--u_{i+1} corresponds to F'_i in the dual tree, with:
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# - F'_i adjacent to the apex chord
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# - non-apex side of F'_i forks into an ear E_i (depth 0) and the
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# degree-3 node G_i (depth 1)
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# - G_i further forks into two ears
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#
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# A clean way to realise this: pick F'_i = (u_i, m, u_{i+1}), then
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# E_i = (u_i, e1, m) -- needs chord u_i--m
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# G_i = (m, e2, ...) -- hmm, we need G_i to be degree-3 sharing one
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# edge with F'_i (chord m--something).
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#
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# Simpler: pick F'_i = (u_i, e2, u_{i+1}). Then F'_i has chords
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# u_i--e2 and e2--u_{i+1}.
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# Side u_i..e2 (covers e1, m): triangulate via chord u_i--m.
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# E_i = (u_i, e1, m) ear (outer edges u_i--e1, e1--m), depth 0.
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# X_i = (u_i, m, e2): edges u_i--m, m--e2 (chord? outer? m--e2 outer
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# since m, e2 are adjacent on outer cycle), u_i--e2 chord.
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# 1 outer edge: m--e2. Depth 0.
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# Hmm X_i is depth 0, not depth 1 as required for G_i.
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# This doesn't yet build the 4-deep structure I want. Let me redo.
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#
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# Required structure per arm to get a depth-1 face G_i:
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# need G_i with no outer edges. So all three of G_i's edges are chords.
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# each chord shared with another inner face (so G_i has 3 dual-neighbors).
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# For G_i depth 1: its three neighbours must include at least one depth-0
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# face. With G_i having 3 chord edges, it has 3 dual-neighbors.
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#
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# To realise G_i with 3 chord edges, we need >=4 outer-cycle vertices
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# inside G_i's region.
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# Let me redesign with more vertices per arm: 7 per arm instead of 6.
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# Then n = 3*7 + 3 = 24? Or with apex shared, n = 3*7 = 21.
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# Use n = 21. Apex at positions 0, 7, 14. Per arm i (i=0,1,2):
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# positions a = 7i, a+1, a+2, a+3, a+4, a+5, a+6 where a+7 = next apex.
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# So outer-cycle vertices in arm i: [u_i = 7i, 7i+1, ..., 7i+6, u_{i+1} = 7(i+1)].
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# That's 7 strict-interior vertices plus the two endpoints.
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# Wait, 7 outer-cycle positions from u_i to u_{i+1} (inclusive of both).
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#
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# Let me use simpler indexing. Apex U_0=0, U_1=7, U_2=14 (n=21).
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# Arm 0 covers outer positions 0..7 (inclusive), with internal vertices 1..6.
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print('Recomputing with n=21 layout...')
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n = 21
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POS = {i: (math.cos(math.radians(90 - i * 360 / n)),
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math.sin(math.radians(90 - i * 360 / n))) for i in range(n)}
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OUTER_EDGES = outer_edges(n)
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outer_set = {frozenset(e) for e in OUTER_EDGES}
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# Apex vertices of F:
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U0, U1, U2 = 0, 7, 14
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# Edges of F (chords)
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F_chords = [(U0, U1), (U1, U2), (U0, U2)]
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def arm_chords(a, b):
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"""For an arm from apex a (=7i) to apex b (=7(i+1) mod 21), with
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internal outer vertices a+1, a+2, ..., a+6 (6 internal verts), produce:
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F'_i = (a, mid, b) for some mid
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then sub-triangulate the (a, ..., mid) side and (mid, ..., b) side.
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Pick mid = a+4 (middle, gives 3 vertices on each side).
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a-side (a, a+1, a+2, a+3, mid=a+4): 4 strict-interior, need triangulating
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Add chord a--(a+2) and chord a--(a+4) [already F'_i].
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Triangles: (a, a+1, a+2) ear; (a, a+2, a+3); (a, a+3, a+4).
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But (a, a+2, a+3) has outer edge (a+2,a+3); depth 0.
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(a, a+3, a+4) has outer edge (a+3, a+4) wait that's not necessarily outer.
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Actually a+3, a+4 ARE outer-adjacent. So (a, a+3, a+4) has outer
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edge (a+3, a+4); 1 outer edge, depth 0.
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Hmm need to engineer the G_i = depth-1 face.
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Try mid = a+3. Then a-side has 2 internal vertices (a+1, a+2);
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b-side has 3 (a+4, a+5, a+6).
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a-side triangulation: chord a--(a+2). Triangles:
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(a, a+1, a+2) ear; (a, a+2, a+3) = E_i?
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(a, a+2, a+3): outer edge (a+2, a+3); 1 outer; depth 0.
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So E_i = (a, a+2, a+3) depth 0 with chord a-(a+2) leading to
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ear (a, a+1, a+2).
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b-side (a+3, a+4, a+5, a+6, b): 4 internal (a+4, a+5, a+6) wait
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that's 3. We need a depth-1 G_i in here. Triangulate with chord
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(a+3)--(a+5): triangles (a+3, a+4, a+5) ear; (a+3, a+5, b) and
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(a+5, a+6, b).
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(a+3, a+5, b): edges (a+3,a+5) chord, (a+5, b) chord, (a+3, b)
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which is F'_i edge. 0 outer edges. Depth >= 1.
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(a+5, a+6, b): edges (a+5, a+6) outer, (a+6, b) outer, (a+5, b)
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chord. 2 outer edges; ear; depth 0.
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So (a+3, a+5, b) has 0 outer edges, depth ?. Its neighbours:
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across (a+3, a+5): ear (a+3, a+4, a+5) depth 0
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across (a+5, b): ear (a+5, a+6, b) depth 0
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across (a+3, b): F'_i
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So depth = 1 (via two depth-0 neighbours).
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F'_i = (a, a+3, b). Edges (a, a+3) chord, (a+3, b) chord, (a, b)
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apex-chord (shared with F). 0 outer. Neighbours:
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across (a, a+3): E_i = (a, a+2, a+3) depth 0
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across (a+3, b): (a+3, a+5, b) depth 1 = G_i
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across (a, b): F depth ?
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depth(F'_i) = 1 + min(0, 1) = 1. ✓
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Non-(a,b) neighbours of F'_i: E_i depth 0 ✓ and G_i depth 1 ✗.
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LOPSIDED, hence unbalanced. ✓
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chords for arm a..b (= a + 7):
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(a, a+2), (a, a+3), (a+3, a+5), (a+3, b), (a+5, b)
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"""
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return [(a, a + 2), (a, a + 3),
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(a + 3, a + 5), (a + 3, b), (a + 5, b)]
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def arm_faces(a, b):
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return [
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(a, a + 1, a + 2), # ear of arm
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(a, a + 2, a + 3), # E_i: 1 outer edge -> depth 0
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(a + 3, a + 4, a + 5), # ear
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(a + 3, a + 5, b), # G_i: 0 outer edges
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(a + 5, a + 6, b), # ear
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(a, a + 3, b), # F'_i
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]
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CHORDS = list(F_chords)
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FACES = [(U0, U1, U2)] # F itself
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for (a, b) in [(0, 7), (7, 14), (14, 0)]:
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CHORDS.extend(arm_chords(a, b))
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FACES.extend(arm_faces(a, b))
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depth, D = compute_depths(FACES, outer_set)
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print(f'Total faces: {len(FACES)}')
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for i, f in enumerate(FACES):
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print(f' {f} -> depth {depth[i]}')
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print(f'B (depth-0 faces): {[FACES[i] for i in range(len(FACES)) if depth[i] == 0]}')
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# Identify the depth-2 face
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d2_faces = [i for i in range(len(FACES)) if depth[i] == 2]
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print(f'Depth-2 faces: {[FACES[i] for i in d2_faces]}')
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def check_balanced(F_idx, faces, depth_, outer_edge_set):
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"""Check if face F_idx admits a balanced surface switch on some edge."""
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F = faces[F_idx]
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fe = face_edges(F)
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for e in fe:
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if e in outer_edge_set:
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continue
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# Find the inner face sharing e with F
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candidates = [j for j in range(len(faces))
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if j != F_idx and e in face_edges(faces[j])]
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if not candidates:
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continue
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Fp_idx = candidates[0]
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if depth_[Fp_idx] != depth_[F_idx] - 1:
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continue
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# Found a depth-(d-1) neighbour F'. Check balancedness.
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Fp = faces[Fp_idx]
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fpe = face_edges(Fp)
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other_edges = [e2 for e2 in fpe if e2 != e]
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d = depth_[F_idx]
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ok = True
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for e2 in other_edges:
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if e2 in outer_edge_set:
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continue # outer-cycle edge is fine
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# find inner face across e2
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others = [j for j in range(len(faces))
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if j != Fp_idx and e2 in face_edges(faces[j])]
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if not others:
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ok = False
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break
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other_face = others[0]
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if depth_[other_face] != d - 2:
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ok = False
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break
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if ok:
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return True, F_idx, Fp_idx, e
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return False, None, None, None
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for F_idx in d2_faces:
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ok, _, fp, e = check_balanced(F_idx, FACES, depth, outer_set)
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print(f'F = {FACES[F_idx]}: balanced switch exists? {ok}')
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