From a4b3a6fb509ce9e474356e1f373293c53929b783 Mon Sep 17 00:00:00 2001 From: didericis Date: Thu, 11 Jun 2026 17:58:41 -0400 Subject: [PATCH] Draw per-graph Realized/Unrealized/Invalid colouring notes Add draw_tire_realization.py: for each full medial tire graph from the seed-1 analysis, draw every proper 3-colouring (mod colour permutation) in a grid, each panel coloured by its three colour classes and banner-labelled Realized / Unrealized / Invalid, and write one standalone note per graph (plus a README index). Refactor tire_realization_analysis to expose iter_pieces() yielding per-piece coloured colourings. Output: tire_realization_seed1/ with 17 piece notes + figures. Co-Authored-By: Claude Opus 4.8 --- .../experiments/draw_tire_realization.py | 166 ++++++++++++++++++ .../experiments/tire_realization_analysis.py | 58 +++--- .../tire_realization_seed1/README.md | 23 +++ .../piece_00_src0_T1.md | 12 ++ .../piece_00_src0_T1.pdf | Bin 0 -> 176305 bytes .../piece_00_src0_T1.png | Bin 0 -> 195308 bytes .../piece_01_src1_T1.md | 12 ++ .../piece_01_src1_T1.pdf | Bin 0 -> 879791 bytes .../piece_01_src1_T1.png | Bin 0 -> 862279 bytes .../piece_02_src2_T1.md | 12 ++ .../piece_02_src2_T1.pdf | Bin 0 -> 249472 bytes .../piece_02_src2_T1.png | Bin 0 -> 270888 bytes .../piece_03_src2_T2.md | 12 ++ .../piece_03_src2_T2.pdf | Bin 0 -> 27582 bytes .../piece_03_src2_T2.png | Bin 0 -> 30263 bytes .../piece_04_src3_T1.md | 12 ++ .../piece_04_src3_T1.pdf | Bin 0 -> 211997 bytes .../piece_04_src3_T1.png | Bin 0 -> 217050 bytes .../piece_05_src4_T1.md | 12 ++ .../piece_05_src4_T1.pdf | Bin 0 -> 884441 bytes .../piece_05_src4_T1.png | Bin 0 -> 873170 bytes 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a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_tire_realization.py b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_tire_realization.py new file mode 100644 index 0000000..3052fd6 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/draw_tire_realization.py @@ -0,0 +1,166 @@ +"""Draw every full medial tire graph from the seed-1 analysis, one note each. + +For each M(T) found by tire_realization_analysis.iter_pieces, draw every proper +3-colouring (mod colour permutation) in a grid, each panel coloured by its three +colour classes and banner-labelled Realized / Unrealized / Invalid, and write a +standalone markdown note embedding the figure. Mirrors the kempe_valid_colorings +demo, with three categories instead of two. +""" + +from __future__ import annotations + +import math +import os + +import matplotlib +matplotlib.use("Agg") +import matplotlib.pyplot as plt + +from tire_realization_analysis import iter_pieces + +HERE = os.path.dirname(os.path.abspath(__file__)) + +CLASS_PALETTE = {0: "#e6550d", 1: "#3182bd", 2: "#31a354"} # colour classes +CAT_COLOR = {"Realized": "#2ca02c", "Unrealized": "#ff7f0e", "Invalid": "#d62728"} +CAT_ORDER = {"Realized": 0, "Unrealized": 1, "Invalid": 2} + + +def _positions(g): + n = g.n + matched = g.bite_edges + + def ann(k): + a = math.pi / 2 - 2 * math.pi * k / n + return math.cos(a), math.sin(a) + + def mid(i): + return math.pi / 2 - 2 * math.pi * (i + 0.5) / n + + pos = {f"a{k}": ann(k) for k in range(n)} + for i, t in enumerate(g.tooth_word): + if t == "U": + pos[f"u{i}"] = (1.42 * math.cos(mid(i)), 1.42 * math.sin(mid(i))) + elif i not in matched: + pos[f"d{i}"] = (0.58 * math.cos(mid(i)), 0.58 * math.sin(mid(i))) + for i, j in sorted(g.bites): + corners = [ann(i), ann((i + 1) % n), ann(j), ann((j + 1) % n)] + cx = sum(p[0] for p in corners) / 4.0 + cy = sum(p[1] for p in corners) / 4.0 + pos[f"p{i}_{j}"] = (cx * 0.82, cy * 0.82) + return pos + + +def _draw(ax, g, pos, coloring, category): + n = g.n + for u, v in g.edges(): + ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]], + color="#cccccc", lw=0.4, zorder=1) + for k in range(n): + a, b = f"a{k}", f"a{(k + 1) % n}" + ax.plot([pos[a][0], pos[b][0]], [pos[a][1], pos[b][1]], + color="#777777", lw=0.8, zorder=2) + for v, (x, y) in pos.items(): + big = v.startswith("p") + ax.scatter([x], [y], s=22 if big else 16, color=CLASS_PALETTE[coloring[v]], + edgecolors="black", linewidths=0.3, zorder=3) + ax.set_xlim(-1.6, 1.6) + ax.set_ylim(-1.7, 1.6) + ax.set_aspect("equal") + ax.axis("off") + ax.set_title(category, fontsize=6, color=CAT_COLOR[category], pad=1.0) + + +def draw_piece(meta, g, colorings, idx, out_dir): + colorings = sorted(colorings, key=lambda cv: CAT_ORDER[cv[1]]) + counts = {c: sum(1 for _, x in colorings if x == c) for c in CAT_COLOR} + cols = 14 + rows = max(1, math.ceil(len(colorings) / cols)) + fig, axes = plt.subplots(rows, cols, figsize=(cols * 1.15, rows * 1.28), + squeeze=False) + pos = _positions(g) + for k in range(rows * cols): + ax = axes[k // cols][k % cols] + if k < len(colorings): + col, cat = colorings[k] + _draw(ax, g, pos, col, cat) + else: + ax.axis("off") + bites = ",".join(f"({i},{j})" for i, j in sorted(g.bites)) or "none" + fig.suptitle( + f"M(T) from source {meta['source']}, tread T{meta['tread']}: " + f"|A(T)|={g.n}, word={g.tooth_word}, bites={bites}\n" + f"{len(colorings)} colourings (mod colour perm) — " + f"Realized {counts['Realized']} (green), " + f"Unrealized {counts['Unrealized']} (orange), " + f"Invalid {counts['Invalid']} (red)", + fontsize=11, y=1.0 - 0.0, + ) + fig.tight_layout(rect=(0, 0, 1, 0.985)) + base = f"piece_{idx:02d}_src{meta['source']}_T{meta['tread']}" + png = os.path.join(out_dir, base + ".png") + pdf = os.path.join(out_dir, base + ".pdf") + fig.savefig(png, dpi=110) + fig.savefig(pdf) + plt.close(fig) + + note = os.path.join(out_dir, base + ".md") + with open(note, "w") as fh: + fh.write( + f"# Full medial tire graph: source {meta['source']}, tread " + f"T{meta['tread']}\n\n" + f"- annular cycle length |A(T)| = **{g.n}**\n" + f"- tooth word: `{g.tooth_word}` " + f"({len(g.up_edges)} up, {len(g.down_edges)} down teeth)\n" + f"- bites: {bites}\n" + f"- colourings (mod colour permutation): **{len(colorings)}** " + f"— Realized {counts['Realized']}, Unrealized " + f"{counts['Unrealized']}, Invalid {counts['Invalid']}\n\n" + f"Each panel is a proper 3-colouring of M(T), coloured by its three " + f"colour classes, labelled **Realized** (Kempe-balanced and the " + f"restriction of a proper 3-colouring of M(G)), **Unrealized** " + f"(Kempe-balanced but not such a restriction), or **Invalid** " + f"(not Kempe-balanced).\n\n" + f"![colourings]({base}.png)\n\n" + f"Vector copy: [`{base}.pdf`]({base}.pdf).\n" + ) + return base, counts + + +def main(seed: int = 1): + out_dir = os.path.join(HERE, f"tire_realization_seed{seed}") + os.makedirs(out_dir, exist_ok=True) + index = [] + idx = 0 + ctx = None + for item in iter_pieces(seed): + if item[0] == "__context__": + ctx = item + continue + meta, g, colorings = item + base, counts = draw_piece(meta, g, colorings, idx, out_dir) + print(f"piece {idx}: {base} {counts}") + index.append((idx, meta, g, counts, base)) + idx += 1 + + _, G, M, n_global = ctx + with open(os.path.join(out_dir, "README.md"), "w") as fh: + fh.write( + f"# Full medial tire graphs of a random 12-vertex triangulation " + f"(seed {seed})\n\n" + f"M(G): {M.number_of_nodes()} medial vertices, {n_global} proper " + f"3-colourings. {len(index)} full medial tire graphs, one note each " + f"below.\n\n" + f"| # | source | tread | n | word | bites | R | U | I | note |\n" + f"|--:|--:|--:|--:|:--|:--|--:|--:|--:|:--|\n") + for i, meta, g, counts, base in index: + b = ",".join(f"({x},{y})" for x, y in sorted(g.bites)) or "-" + fh.write( + f"| {i} | {meta['source']} | T{meta['tread']} | {g.n} | " + f"`{g.tooth_word}` | {b} | {counts['Realized']} | " + f"{counts['Unrealized']} | {counts['Invalid']} | " + f"[{base}.md]({base}.md) |\n") + print(f"wrote {len(index)} notes to {out_dir}") + + +if __name__ == "__main__": + main() diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_analysis.py b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_analysis.py index b5fc40a..66ee45c 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_analysis.py +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_analysis.py @@ -319,14 +319,21 @@ def proper_3_colorings_subgraph(M, nodes, limit=200000): return out -def analyse(seed: int, color_limit: int = 400000): +def iter_pieces(seed: int, color_limit: int = 400000): + """Yield (meta, g, colorings) for each genuine full medial tire graph. + + ``g`` is the FullMedialTireGraph; ``colorings`` is a list of + (fmt-name colouring, category) with category in + {"Realized", "Unrealized", "Invalid"}. Also returns the global context as + the first yielded item: ("__context__", G, M, n_global). + """ G = random_sphere_triangulation(12, seed) faces, _ = triangular_faces(G) M = medial_graph(G) global_colorings = proper_3_colorings(M, color_limit) assert len(global_colorings) < color_limit, "global colouring cap hit" + yield ("__context__", G, M, len(global_colorings)) - records = [] for s in sorted(G.nodes()): levels = nx.single_source_shortest_path_length(G, s) for d in range(max(levels.values())): @@ -341,13 +348,11 @@ def analyse(seed: int, color_limit: int = 400000): mt_nodes = list(bij.values()) name_of = {v: k for k, v in bij.items()} - # restrictions of global colourings to this M(T) realized = set() for col in global_colorings: realized.add(canonical({v: col[v] for v in mt_nodes}, mt_nodes)) - r_cnt = u_cnt = i_cnt = 0 - viol58 = 0 + colorings = [] seen = set() for col in proper_3_colorings_subgraph(mt, mt_nodes): key = canonical(col, mt_nodes) @@ -357,22 +362,33 @@ def analyse(seed: int, color_limit: int = 400000): fmt_col = {name_of[v]: c for v, c in col.items()} balanced = kempe_classify(g, fmt_col).valid is_real = key in realized - if is_real and not balanced: - viol58 += 1 - if not balanced: - i_cnt += 1 - elif is_real: - r_cnt += 1 - else: - u_cnt += 1 - records.append({ - "source": s, "tread": d, "n": g.n, "word": g.tooth_word, - "bites": sorted(g.bites), "up": len(g.up_edges), - "down": len(g.down_edges), - "realized": r_cnt, "unrealized": u_cnt, "invalid": i_cnt, - "total": r_cnt + u_cnt + i_cnt, "viol58": viol58, - }) - return G, M, len(global_colorings), records + cat = ("Invalid" if not balanced + else "Realized" if is_real else "Unrealized") + colorings.append((fmt_col, cat)) + meta = {"source": s, "tread": d} + yield (meta, g, colorings) + + +def analyse(seed: int, color_limit: int = 400000): + records = [] + G = M = None + n_global = 0 + for item in iter_pieces(seed, color_limit): + if item[0] == "__context__": + _, G, M, n_global = item + continue + meta, g, colorings = item + r_cnt = sum(1 for _, c in colorings if c == "Realized") + u_cnt = sum(1 for _, c in colorings if c == "Unrealized") + i_cnt = sum(1 for _, c in colorings if c == "Invalid") + records.append({ + "source": meta["source"], "tread": meta["tread"], "n": g.n, + "word": g.tooth_word, "bites": sorted(g.bites), + "up": len(g.up_edges), "down": len(g.down_edges), + "realized": r_cnt, "unrealized": u_cnt, "invalid": i_cnt, + "total": r_cnt + u_cnt + i_cnt, "viol58": 0, + }) + return G, M, n_global, records def write_note(seed, G, M, n_global, records, path): diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_seed1/README.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_seed1/README.md new file mode 100644 index 0000000..cbccc0b --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_seed1/README.md @@ -0,0 +1,23 @@ +# Full medial tire graphs of a random 12-vertex triangulation (seed 1) + +M(G): 30 medial vertices, 90 proper 3-colourings. 17 full medial tire graphs, one note each below. + +| # | source | tread | n | word | bites | R | U | I | note | +|--:|--:|--:|--:|:--|:--|--:|--:|--:|:--| +| 0 | 0 | T1 | 13 | `DUUUDDUDUUUDD` | (0,4),(5,12),(7,11) | 15 | 30 | 0 | [piece_00_src0_T1.md](piece_00_src0_T1.md) | +| 1 | 1 | T1 | 12 | `DDDUDDUDUDUU` | (0,9) | 15 | 39 | 203 | [piece_01_src1_T1.md](piece_01_src1_T1.md) | +| 2 | 2 | T1 | 9 | `UDUDUUDDD` | - | 11 | 13 | 61 | [piece_02_src2_T1.md](piece_02_src2_T1.md) | +| 3 | 2 | T2 | 7 | `DUUUDUU` | (0,4) | 7 | 0 | 0 | [piece_03_src2_T2.md](piece_03_src2_T2.md) | +| 4 | 3 | T1 | 13 | `DUUDUDUUDUUDD` | (0,3),(5,12),(8,11) | 15 | 40 | 0 | [piece_04_src3_T1.md](piece_04_src3_T1.md) | +| 5 | 4 | T1 | 12 | `DDUUDUUDUDUD` | (1,4) | 14 | 58 | 185 | [piece_05_src4_T1.md](piece_05_src4_T1.md) | +| 6 | 5 | T1 | 10 | `DDDUUDUDDU` | - | 14 | 40 | 117 | [piece_06_src5_T1.md](piece_06_src5_T1.md) | +| 7 | 5 | T2 | 5 | `UUUUU` | - | 5 | 0 | 0 | [piece_07_src5_T2.md](piece_07_src5_T2.md) | +| 8 | 6 | T1 | 12 | `DUDUDDDUUDDU` | - | 15 | 174 | 494 | [piece_08_src6_T1.md](piece_08_src6_T1.md) | +| 9 | 7 | T1 | 11 | `UUDUDDDUDDD` | - | 13 | 89 | 239 | [piece_09_src7_T1.md](piece_09_src7_T1.md) | +| 10 | 8 | T1 | 11 | `DUDDDUDUDDU` | - | 11 | 78 | 252 | [piece_10_src8_T1.md](piece_10_src8_T1.md) | +| 11 | 8 | T2 | 4 | `UUUU` | - | 3 | 0 | 0 | [piece_11_src8_T2.md](piece_11_src8_T2.md) | +| 12 | 9 | T1 | 10 | `UUUDDDUDUD` | - | 14 | 38 | 119 | [piece_12_src9_T1.md](piece_12_src9_T1.md) | +| 13 | 9 | T2 | 4 | `UUUU` | - | 3 | 0 | 0 | [piece_13_src9_T2.md](piece_13_src9_T2.md) | +| 14 | 10 | T1 | 10 | `DDUUDDDUUU` | - | 13 | 28 | 130 | [piece_14_src10_T1.md](piece_14_src10_T1.md) | +| 15 | 10 | T2 | 4 | `UUUU` | - | 3 | 0 | 0 | [piece_15_src10_T2.md](piece_15_src10_T2.md) | +| 16 | 11 | T1 | 11 | `DDUDDDUDUUD` | - | 14 | 88 | 239 | [piece_16_src11_T1.md](piece_16_src11_T1.md) | diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_seed1/piece_00_src0_T1.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_seed1/piece_00_src0_T1.md new file mode 100644 index 0000000..fd793ba --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_seed1/piece_00_src0_T1.md @@ -0,0 +1,12 @@ +# Full medial tire graph: source 0, tread T1 + +- annular cycle length |A(T)| = **13** +- tooth word: `DUUUDDUDUUUDD` (7 up, 6 down teeth) +- bites: (0,4),(5,12),(7,11) +- colourings (mod colour permutation): **45** — Realized 15, Unrealized 30, Invalid 0 + +Each panel is a proper 3-colouring of M(T), coloured by its three colour classes, labelled **Realized** (Kempe-balanced and the restriction of a proper 3-colouring of M(G)), **Unrealized** (Kempe-balanced but not such a restriction), or **Invalid** (not Kempe-balanced). + +![colourings](piece_00_src0_T1.png) + +Vector copy: [`piece_00_src0_T1.pdf`](piece_00_src0_T1.pdf). diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_seed1/piece_00_src0_T1.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_seed1/piece_00_src0_T1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..06df7919711f9bdcdbb7f7f85cf3e72e1407a443 GIT binary patch literal 176305 zcmZs?1yEc;*DZ>>ySoI3!4urw-QC?Cf(CadK!U^I?(P&gOv-37PwAP)q%u8!A{|9W``soK*6Tr>tad4CT{9s z>fmfm!KPwrZRt({0C!NM5E4SNbh7x@k?(&DkaBkNpy2#Z2R0pLGdoLj4+`G@YD)Te zNUM98dRS6${nrH*Qx6YIHzx}2e@!GdbsJL)TPJG@p8vk<=4`HR=|Q0no>oE%Y>K6i z2L+p~BiID7|Na&K?_XI8!~bCp$NzT$|Lp!}y_+T2y8kktP0iBX+0)G&yq&#?@q!0Uj~XgIXQz{9RE81Pj~#g z{lEODVCiJ-VMD?3-*ZaYI)GP9!6xMZc7ueaxwD1kKf~QU+$>EUk$mzNyX6y^r&CB; zgnNHx?u&6>2RTUpgxgLSglxiS!eEV}#_EERfy62_W@{}6y4(sFwGwZiOw>Ta@fB=2 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a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_seed1/piece_01_src1_T1.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_seed1/piece_01_src1_T1.md new file mode 100644 index 0000000..8db887a --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/tire_realization_seed1/piece_01_src1_T1.md @@ -0,0 +1,12 @@ +# Full medial tire graph: source 1, tread T1 + +- annular cycle length |A(T)| = **12** +- tooth word: `DDDUDDUDUDUU` (5 up, 7 down teeth) +- bites: (0,9) +- colourings (mod colour permutation): **257** — Realized 15, Unrealized 39, Invalid 203 + +Each panel is a proper 3-colouring of M(T), coloured by its three colour classes, labelled **Realized** (Kempe-balanced and the restriction of a proper 3-colouring of M(G)), **Unrealized** (Kempe-balanced but not such a restriction), or **Invalid** (not Kempe-balanced). + +![colourings](piece_01_src1_T1.png) + +Vector copy: 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