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>
71 lines
2.9 KiB
Python
71 lines
2.9 KiB
Python
"""Visualize the 24-vertex doubly-lopsided d=2 example and the
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preprocessing trajectory."""
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import sys, os
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sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
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import matplotlib.pyplot as plt
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from matplotlib.patches import Polygon
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from d2_recursive_lopsided import (
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POS, n, OUTER_EDGES, outer_set, compute_depths, apply_switch,
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FACES as FACES0, CHORDS as CHORDS0
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)
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OUT_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), os.pardir)
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def draw(ax, faces, chords, depth, title,
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highlight_edges=None, green_edges=None):
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palette = {0: '#86efac', 1: '#fde68a', 2: '#fca5a5'}
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edge_pal = {0: '#16a34a', 1: '#d97706', 2: '#dc2626'}
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for i, f in enumerate(faces):
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d = depth[i]
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poly = Polygon([POS[v] for v in f], closed=True,
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facecolor=palette.get(d, '#ddd'),
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edgecolor=edge_pal.get(d, '#333'),
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linewidth=1.2, alpha=0.65, zorder=0)
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ax.add_patch(poly)
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cx = sum(POS[v][0] for v in f) / 3
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cy = sum(POS[v][1] for v in f) / 3
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ax.text(cx, cy, str(d), ha='center', va='center', fontsize=9,
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color=edge_pal.get(d, '#333'), fontweight='bold')
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for (a, b) in OUTER_EDGES + chords:
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color = '#333'; lw = 1.1
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if highlight_edges and ((a, b) in highlight_edges or
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(b, a) in highlight_edges):
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color = '#dc2626'; lw = 2.8
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if green_edges and ((a, b) in green_edges or
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(b, a) in green_edges):
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color = '#16a34a'; lw = 2.8
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ax.plot([POS[a][0], POS[b][0]], [POS[a][1], POS[b][1]],
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color=color, linewidth=lw, zorder=1)
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for i, (x, y) in POS.items():
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ax.scatter([x], [y], s=200, c='#1f2937', edgecolors='black',
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linewidths=0.6, zorder=2)
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ax.text(x, y, str(i), ha='center', va='center',
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fontsize=7, color='white', fontweight='bold', zorder=3)
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ax.set_aspect('equal'); ax.axis('off')
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ax.set_xlim(-1.2, 1.2); ax.set_ylim(-1.2, 1.2)
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ax.set_title(title, fontsize=10)
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depth0 = compute_depths(FACES0)
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faces1, chords1 = apply_switch(FACES0, CHORDS0, (0, 8), (16, 2))
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depth1 = compute_depths(faces1)
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faces2, chords2 = apply_switch(faces1, chords1, (8, 2), (16, 4))
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depth2 = compute_depths(faces2)
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fig, axes = plt.subplots(1, 3, figsize=(18, 6.5))
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draw(axes[0], FACES0, CHORDS0, depth0,
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'Start: F=(0,8,16) depth 2, all arms doubly-lopsided',
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highlight_edges=[(0, 8)])
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draw(axes[1], faces1, chords1, depth1,
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'After preprocess 1: F=(2,8,16) still depth 2, still no balanced switch',
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green_edges=[(2, 16)], highlight_edges=[(8, 2)])
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draw(axes[2], faces2, chords2, depth2,
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'After preprocess 2: F=(4,8,16) admits balanced switch on (4,8)',
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green_edges=[(4, 16)], highlight_edges=[(4, 8)])
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fig.tight_layout()
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out = os.path.join(OUT_DIR, 'fig_d2_recursive.png')
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fig.savefig(out, dpi=170, bbox_inches='tight')
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plt.close(fig)
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print(f'wrote {out}')
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