Plot the n=12 size-7 universal breaker tile

Add plot_breaker_tile.py and figure for word=UUUDUDUDUDUD bite=(3,11): structure
(7 up teeth = size-7 outer rim, bite (3,11), singleton downs d5,d7,d9) plus a
Kempe-balanced colouring. Reconfirms the outer rim realises 9/10 admissible size-7
necklaces, never 0001112 -- the lone tile that empties the size-7 universal at n=12.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/medial_tire_decompositions_of_plane_triangulations/experiments/plot_breaker_tile.py

@@ -0,0 +1,138 @@
+"""Plot the n=12 size-7 universal breaker tile: word=UUUDUDUDUDUD, bite=(3,11).
+
+Left panel: structure (7 up teeth = the size-7 outer rim, ringed; the bite (3,11);
+the singleton down teeth).  Right panel: a Kempe-balanced colouring, with the outer
+rim necklace it presents.  This tile's outer rim realises 9 of the 10 admissible
+size-7 necklaces but never 0001112, which is why it alone empties the size-7
+universal at n=12.
+"""
+
+from __future__ import annotations
+
+import math
+import os
+
+from full_medial_tire_generator import FullMedialTireGraph
+from kempe_valid_colorings import classify_colorings
+from kempe_up_tooth_sequences import PALETTE, _positions, canonical_sequence, seq_str
+from kempe_transfer_relation_probe import necklace
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+
+WORD = "UUUDUDUDUDUD"
+BITE = frozenset({(3, 11)})
+N = 12
+TYPE_COLOR = {"a": "#999999", "u": "#e6550d", "d": "#31a354", "p": "#756bb1"}
+
+
+def vtype(v):
+    return v[0]
+
+
+def draw_structure(ax, g):
+    pos = _positions(g)
+    for u, v in g.edges():
+        ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]],
+                color="#cccccc", lw=0.6, zorder=1)
+    for k in range(g.n):
+        a, b = f"a{k}", f"a{(k + 1) % g.n}"
+        ax.plot([pos[a][0], pos[b][0]], [pos[a][1], pos[b][1]],
+                color="#555555", lw=1.2, zorder=2)
+    # highlight the bite spokes
+    for i, j in g.bites:
+        apex = f"p{i}_{j}"
+        for e in (i, j):
+            for av in (f"a{e}", f"a{(e + 1) % g.n}"):
+                ax.plot([pos[apex][0], pos[av][0]], [pos[apex][1], pos[av][1]],
+                        color="#756bb1", lw=1.6, zorder=2.5)
+    for v, (x, y) in pos.items():
+        t = vtype(v)
+        s = 90 if t in ("u", "p") else 60
+        ax.scatter([x], [y], s=s, color=TYPE_COLOR[t], edgecolors="black",
+                   linewidths=0.6, zorder=3)
+    # ring the up-tooth apexes (the size-7 outer rim / boundary)
+    ux = [pos[f"u{i}"][0] for i in g.up_edges]
+    uy = [pos[f"u{i}"][1] for i in g.up_edges]
+    ax.scatter(ux, uy, s=240, facecolors="none", edgecolors="#d62728",
+               linewidths=1.8, zorder=4)
+    # label up apexes in boundary order 0..6
+    for k, i in enumerate(g.up_edges):
+        x, y = pos[f"u{i}"]
+        ax.annotate(str(k), (x, y), textcoords="offset points", xytext=(0, 9),
+                    ha="center", fontsize=7, color="#d62728")
+    ax.set_xlim(-1.7, 1.7)
+    ax.set_ylim(-1.9, 1.7)
+    ax.set_aspect("equal")
+    ax.axis("off")
+    ax.set_title("structure: 7 up teeth (red-ringed = size-7 outer rim)\n"
+                 "bite (3,11) in purple; singleton down teeth d5,d7,d9",
+                 fontsize=8)
+
+
+def draw_colored(ax, g, coloring):
+    pos = _positions(g)
+    for u, v in g.edges():
+        ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]],
+                color="#cccccc", lw=0.6, zorder=1)
+    for k in range(g.n):
+        a, b = f"a{k}", f"a{(k + 1) % g.n}"
+        ax.plot([pos[a][0], pos[b][0]], [pos[a][1], pos[b][1]],
+                color="#555555", lw=1.2, zorder=2)
+    for v, (x, y) in pos.items():
+        is_bite = v.startswith("p")
+        ax.scatter([x], [y], s=80 if is_bite else 55, color=PALETTE[coloring[v]],
+                   edgecolors="black", linewidths=0.5, zorder=3)
+    ux = [pos[f"u{i}"][0] for i in g.up_edges]
+    uy = [pos[f"u{i}"][1] for i in g.up_edges]
+    ax.scatter(ux, uy, s=240, facecolors="none", edgecolors="#222222",
+               linewidths=1.6, zorder=4)
+    raw = seq_str(tuple(coloring[f"u{i}"] for i in g.up_edges))
+    neck = seq_str(necklace(tuple(coloring[f"u{i}"] for i in g.up_edges)))
+    ax.set_xlim(-1.7, 1.7)
+    ax.set_ylim(-1.9, 1.7)
+    ax.set_aspect("equal")
+    ax.axis("off")
+    ax.set_title(f"a Kempe-balanced colouring\nouter rim = {raw}  →  necklace {neck}",
+                 fontsize=8)
+
+
+def main():
+    import matplotlib
+    matplotlib.use("Agg")
+    import matplotlib.pyplot as plt
+
+    g = FullMedialTireGraph(n=N, tooth_word=WORD, bites=BITE)
+
+    # confirm: outer rim realises 9/10 admissible necklaces, never 0001112
+    realized = set()
+    sample = None
+    for c, v in classify_colorings(g, dedup_colors=True):
+        if not v.valid:
+            continue
+        nk = necklace(tuple(c[f"u{i}"] for i in g.up_edges))
+        realized.add(nk)
+        if sample is None:
+            sample = c
+    target = necklace(tuple(map(int, "0001112")))
+    print(f"breaker {WORD} bite (3,11): outer rim realises {len(realized)} necklaces")
+    print(f"  contains 0001112? {target in realized}  (expect False)")
+
+    fig, axes = plt.subplots(1, 2, figsize=(10, 5.4))
+    draw_structure(axes[0], g)
+    draw_colored(axes[1], g, sample)
+    fig.suptitle(
+        "Size-7 universal breaker (n=12): word=UUUDUDUDUDUD, bite=(3,11)\n"
+        "outer rim realises 9 of 10 admissible size-7 necklaces — never 0001112",
+        fontsize=11, y=0.99,
+    )
+    fig.tight_layout(rect=(0, 0, 1, 0.93))
+    png = os.path.join(HERE, "breaker_tile_n12.png")
+    pdf = os.path.join(HERE, "breaker_tile_n12.pdf")
+    fig.savefig(png, dpi=200)
+    fig.savefig(pdf)
+    print(f"wrote {png}")
+    print(f"wrote {pdf}")
+
+
+if __name__ == "__main__":
+    main()