From 8dd9d537f93cf8312b61d74b9942e0cabd8ad13f Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 03:03:56 -0400 Subject: [PATCH] =?UTF-8?q?coloring=5Fnested=5Ftire=5Fgraphs:=20redraw=20r?= =?UTF-8?q?ainbow=20orbit=20as=20edge=20colorings=20(=CF=83=20on=20spoke?= =?UTF-8?q?=20edges,=20not=20vertices)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Previous figures drew σ as VERTEX colors on a 6-cycle. This was misleading: σ = (1,2,3,2,3,1) is not a proper vertex 3-coloring of a hexagon (σ_5 = σ_0 = 1 at adjacent vertices), and the user correctly flagged this. σ is the coloring of the 6 *spoke edges* -- the G'-edges of G' that cross γ, equivalently the 6 edges of γ ⊂ G under the duality γ-edge ↔ crossing G'-edge. Adjacent γ-edges meet at γ-vertices, which are not G'-vertices, so σ has NO proper-coloring constraint on itself. Proper-edge-coloring constraints live on each tire's full annular cycle, which is longer than 6 (T_1's is 12, T_2's is 9), with γ-spokes interleaved among non-γ spokes; that's where the extendibility of σ is actually checked. Redrawn figures: - fig_rainbow_orbit.png: σ drawn as edge colors of γ (not vertex colors), all 6 orbit elements. - fig_rainbow_pattern.png: abstract pattern abcbca as edge labels, with explanatory text in the legend. - fig_rainbow_setup.png: shows γ between the two tires with each tire's full annular cycle (length 12 and 9), the interleaved non-γ dual vertices, the dashed G'-spoke edges crossing γ colored by σ, and T_1's antipodal chord in O_1. Co-Authored-By: Claude Opus 4.7 --- .../experiments/draw_rainbow_orbit.py | 402 ++++++++++-------- .../notes/fig_rainbow_orbit.png | Bin 90122 -> 111893 bytes .../notes/fig_rainbow_pattern.png | Bin 43448 -> 59442 bytes .../notes/fig_rainbow_setup.png | Bin 112159 -> 145047 bytes 4 files changed, 229 insertions(+), 173 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_rainbow_orbit.py b/papers/coloring_nested_tire_graphs/experiments/draw_rainbow_orbit.py index 07879f2..0c251ba 100644 --- a/papers/coloring_nested_tire_graphs/experiments/draw_rainbow_orbit.py +++ b/papers/coloring_nested_tire_graphs/experiments/draw_rainbow_orbit.py @@ -3,26 +3,29 @@ tire_fiber_step2.tex, Observation rainbow: k = 6, T_1 = (6, (0,3), SP) vs T_2 = (3, -, SR) |S_1 ∩ S_2| = 6 - The 6 elements are exactly {(a, b, c, b, c, a) : {a,b,c} = {1,2,3}} - --- the S_3 orbit of the pattern abstractly written abcbca. + The 6 elements are exactly {(a, b, c, b, c, a) : {a,b,c} = {1,2,3}}. -Produces three figures: +IMPORTANT INTERPRETATION: + σ ∈ {1,2,3}^6 is the coloring of the SIX G'-EDGES THAT CROSS γ, + equivalently a coloring of the 6 edges of the shared cycle γ ⊂ G. - fig_rainbow_pattern.png: - The abstract pattern abcbca on the shared 6-cycle, showing the - three antipodal/positional symmetry classes (positions {0,5}, - {1,3}, {2,4}). + σ is NOT a proper 3-coloring of a 6-cycle: adjacent entries CAN + coincide, because: + - T_1's annular cycle has length m_1 + |γ| = 6 + 6 = 12. + - T_2's annular cycle has length |γ| + k_2 = 6 + 3 = 9. + - σ records 6 "γ-side" spoke positions interleaved with + non-γ-side spoke positions on the longer cycle. - fig_rainbow_orbit.png: - All six concrete colourings in the S_3 orbit, drawn as small - hexagons with positions colored. Highlights how each permutation - of {1,2,3} gives one element. + Proper-edge-coloring constraints live on each tire's full annular + cycle, not on σ directly. σ is simply the boundary data, like a + Birkhoff ring coloring. - fig_rainbow_setup.png: - The geometric setup: shared cycle gamma (the hexagon), T_1 - outside (with its antipodal chord), T_2 inside (the triangle), - spoke edges crossing gamma. Shows why the antipodal chord in T_1 - pins the pattern. +Produces: + fig_rainbow_orbit.png -- all 6 orbit elements as colored γ-edges + fig_rainbow_pattern.png -- abstract pattern (a,b,c,b,c,a) + fig_rainbow_setup.png -- geometric setup, showing why σ is on + γ's EDGES, with the longer annular + cycles indicated. """ import math import os @@ -30,17 +33,14 @@ import matplotlib.pyplot as plt import matplotlib.patches as patches -# Color palette: a 3-color set used consistently across all figures. COLORS = { - 1: '#1f77b4', # blue ("color 1") - 2: '#d62728', # red ("color 2") - 3: '#2ca02c', # green ("color 3") + 1: '#1f77b4', # blue + 2: '#d62728', # red + 3: '#2ca02c', # green } -NAMES = {1: '1', 2: '2', 3: '3'} def hex_positions(n=6, radius=1.0, angle_offset=math.pi / 2): - """Return positions of n evenly-spaced points on a circle of given radius.""" return [ (radius * math.cos(angle_offset - 2 * math.pi * i / n), radius * math.sin(angle_offset - 2 * math.pi * i / n)) @@ -48,79 +48,92 @@ def hex_positions(n=6, radius=1.0, angle_offset=math.pi / 2): ] -def draw_hex_with_pattern(ax, pattern, *, radius=1.0, label_size=10, - dot_size=20, ring_color='#888888'): - """Draw a 6-cycle with edges in ring_color and vertices colored by pattern - (length-6 list of colors). pattern[i] is the color of position i. - """ +def draw_gamma_with_edge_colors(ax, sigma, *, radius=1.0, edge_width=5.0, + vertex_size=8, vertex_color='#222', + show_indices=True): + """Draw the 6-cycle γ with edge i colored by sigma[i]. Vertices + are small black dots (they carry no color).""" pos = hex_positions(6, radius=radius) - # cycle edges + # Edges for i in range(6): x1, y1 = pos[i] x2, y2 = pos[(i + 1) % 6] - ax.plot([x1, x2], [y1, y2], color=ring_color, linewidth=1.6, zorder=1) - # vertices - for i, c in enumerate(pattern): + ax.plot([x1, x2], [y1, y2], color=COLORS[sigma[i]], + linewidth=edge_width, solid_capstyle='round', zorder=1) + # Edge index label, placed at midpoint, slightly inside + if show_indices: + mx, my = (x1 + x2) / 2, (y1 + y2) / 2 + # nudge toward origin + ix, iy = mx * 0.78, my * 0.78 + ax.annotate(f"$e_{i}$", (ix, iy), color='#444', + ha='center', va='center', fontsize=8, zorder=4) + # Vertices (small uncolored dots) + for i in range(6): x, y = pos[i] - ax.plot(x, y, 'o', color=COLORS[c], markersize=dot_size, zorder=2, - markeredgecolor='black', markeredgewidth=0.8) - # tiny label of position - lx, ly = 1.32 * x, 1.32 * y - ax.annotate(str(i), (lx, ly), color='#444', ha='center', va='center', - fontsize=label_size - 1, zorder=3) + ax.plot(x, y, 'o', color=vertex_color, markersize=vertex_size, + zorder=2) + if show_indices: + # vertex label outside + lx, ly = 1.28 * x, 1.28 * y + ax.annotate(f"$v_{i}$", (lx, ly), color='#555', + ha='center', va='center', fontsize=8, zorder=3) # --------------------------------------------------------------------------- -# Figure 1: the abstract pattern abcbca on a hexagon. +# Figure 1: abstract pattern abcbca, σ shown as EDGE colors. # --------------------------------------------------------------------------- def fig_pattern(outdir): - fig, ax = plt.subplots(figsize=(6.0, 5.4)) + fig, ax = plt.subplots(figsize=(7.0, 5.4)) pos = hex_positions(6, radius=1.0) - # Draw cycle edges - for i in range(6): - x1, y1 = pos[i] - x2, y2 = pos[(i + 1) % 6] - ax.plot([x1, x2], [y1, y2], color='#888888', linewidth=2.0, zorder=1) - - classes = {0: 'a', 1: 'b', 2: 'c', 3: 'b', 4: 'c', 5: 'a'} + # Edges with class letters as text labels at midpoint + classes = ['a', 'b', 'c', 'b', 'c', 'a'] class_colors = { 'a': '#bdbdbd', 'b': '#ffe082', 'c': '#a5d6a7', } + for i in range(6): + x1, y1 = pos[i] + x2, y2 = pos[(i + 1) % 6] + ax.plot([x1, x2], [y1, y2], color=class_colors[classes[i]], + linewidth=8.0, solid_capstyle='round', zorder=1) + mx, my = (x1 + x2) / 2, (y1 + y2) / 2 + ax.annotate(classes[i], (mx, my), color='black', ha='center', + va='center', fontsize=13, fontweight='bold', zorder=2) for i in range(6): x, y = pos[i] - cls = classes[i] - ax.plot(x, y, 'o', color=class_colors[cls], markersize=34, zorder=2, - markeredgecolor='black', markeredgewidth=1.0) - ax.annotate(cls, (x, y), color='black', ha='center', va='center', - fontsize=15, fontweight='bold', zorder=3) - lx, ly = 1.34 * x, 1.34 * y - ax.annotate(f'$v_{i}$', (lx, ly), color='#333', ha='center', - va='center', fontsize=11, zorder=3) + ax.plot(x, y, 'o', color='#222', markersize=11, zorder=3) + lx, ly = 1.25 * x, 1.25 * y + ax.annotate(f"$v_{i}$", (lx, ly), color='#444', ha='center', + va='center', fontsize=10, zorder=4) - # Legend showing the position-class assignments legend = ( - r"$\sigma = (a, b, c, b, c, a)$" + "\n" - + r"position classes:" + "\n" - + r" $a$: $\{v_0, v_5\}$" + "\n" - + r" $b$: $\{v_1, v_3\}$" + "\n" - + r" $c$: $\{v_2, v_4\}$" + r"$\sigma$ colors the 6 edges of $\gamma$" + "\n" + + r"(equivalently, the 6 dual edges of $G'$" + "\n" + + r" that cross $\gamma$)." + "\n\n" + + r"Pattern: $(e_0,\dots,e_5) = (a,b,c,b,c,a)$" + "\n" + + r" $a$ on $\{e_0, e_5\}$" + "\n" + + r" $b$ on $\{e_1, e_3\}$" + "\n" + + r" $c$ on $\{e_2, e_4\}$" + "\n\n" + + r"Adjacent edges *may* share a color:" + "\n" + + r" $e_5$ and $e_0$ both $= a$ here." + "\n" + + r" This is fine because proper-coloring" + "\n" + + r" constraints live on each tire's" + "\n" + + r" longer annular cycle, not on $\sigma$." ) - ax.text(1.85, 0.0, legend, fontsize=11, color='#222', - family='monospace', + ax.text(1.85, 0.0, legend, fontsize=9, color='#222', verticalalignment='center', bbox=dict(boxstyle='round,pad=0.5', facecolor='#f8f8f8', edgecolor='#bbb')) - ax.set_xlim(-1.6, 4.0) + ax.set_xlim(-1.6, 5.0) ax.set_ylim(-1.7, 1.7) ax.set_aspect('equal') ax.axis('off') - ax.set_title(r"Rainbow pattern $(a,b,c,b,c,a)$ on the shared cycle $\gamma$" - + " (length $k=6$)", + ax.set_title(r"Rainbow pattern $(a,b,c,b,c,a)$ as an edge coloring of " + + r"$\gamma$ ($|\gamma| = 6$)", fontsize=11) out = os.path.join(outdir, 'fig_rainbow_pattern.png') @@ -130,11 +143,9 @@ def fig_pattern(outdir): # --------------------------------------------------------------------------- -# Figure 2: the 6 concrete colourings in the S_3 orbit. +# Figure 2: the 6 orbit elements as edge colorings of γ. # --------------------------------------------------------------------------- def fig_orbit(outdir): - # The 6 permutations of (a, b, c) -> (1, 2, 3) instantiated into the - # pattern (a, b, c, b, c, a). perms = [ (1, 2, 3), (1, 3, 2), @@ -144,26 +155,28 @@ def fig_orbit(outdir): (3, 2, 1), ] - fig, axes = plt.subplots(2, 3, figsize=(10.5, 7.5)) + fig, axes = plt.subplots(2, 3, figsize=(11.0, 7.6)) for idx, (a, b, c) in enumerate(perms): ax = axes[idx // 3, idx % 3] - pattern = [a, b, c, b, c, a] - draw_hex_with_pattern(ax, pattern, dot_size=24) - ax.set_xlim(-1.7, 1.7) - ax.set_ylim(-1.7, 1.7) + sigma = (a, b, c, b, c, a) + draw_gamma_with_edge_colors(ax, sigma) + ax.set_xlim(-1.6, 1.6) + ax.set_ylim(-1.6, 1.6) ax.set_aspect('equal') ax.axis('off') - # title shows pattern explicitly title = (f"$(a,b,c) = ({a},{b},{c})$\n" - + r"$\sigma = ($" + ",".join(str(x) for x in pattern) + r"$)$") + + r"$\sigma = ($" + ",".join(str(x) for x in sigma) + r"$)$") ax.set_title(title, fontsize=10) - fig.suptitle(r"The 6-element rainbow orbit: $S_3$ acting on pattern " - + r"$(a,b,c,b,c,a)$, the entire $|S_1 \cap S_2| = 6$ in step-2" + fig.suptitle(r"The 6-element rainbow orbit, drawn as edge colorings of " + + r"the shared cycle $\gamma$" + "\n" + + r"$\sigma$ is the coloring of the 6 dual edges of $G'$ " + + r"crossing $\gamma$, equivalently the 6 edges of $\gamma$" + "\n" - + r"$T_1 = (m{=}6, \mathrm{chord}{=}(0,3), \mathrm{SP})$ vs. " - + r"$T_2 = (k{=}3, \mathrm{SR})$ on shared cycle of length $k{=}6$", + + r"Case: $T_1{=}(m{=}6,\, \mathrm{chord}{=}(0,3),\, " + + r"\mathrm{SP})$ vs. $T_2{=}(k{=}3,\, \mathrm{SR})$, " + + r"$|S_1 \cap S_2| = 6$", fontsize=11, y=1.00) plt.tight_layout() out = os.path.join(outdir, 'fig_rainbow_orbit.png') @@ -173,127 +186,170 @@ def fig_orbit(outdir): # --------------------------------------------------------------------------- -# Figure 3: geometric setup --- T_1 (hexagon + antipodal chord) outside gamma, -# T_2 (triangle) inside gamma, the 6 spoke edges crossing gamma. +# Figure 3: geometric setup. Shows γ between T_1 and T_2; σ as colored +# G'-edges across γ; each tire's annular cycle indicated with the correct +# number of vertices and interleaved non-γ spokes. # --------------------------------------------------------------------------- def fig_setup(outdir): - fig, ax = plt.subplots(figsize=(8.5, 7.5)) + fig, ax = plt.subplots(figsize=(11.5, 7.2)) - # Shared cycle gamma at radius R_gamma - R_gamma = 1.4 + # γ at radius R_gamma; we draw it as a hexagon. + R_gamma = 1.5 gamma_pos = hex_positions(6, radius=R_gamma) - # T_1 lives outside gamma. Its "B_in" is gamma; its "B_out" is a bigger - # hexagon at R_out, with an antipodal chord across T_1's B_in side. - # For visualization, draw T_1's annular region as an annulus around gamma. - R_out = 2.5 - R_in = 0.55 - T1_outer = hex_positions(6, radius=R_out) - - # Draw T_1 outer boundary (B_out for T_1) - for i in range(6): - x1, y1 = T1_outer[i] - x2, y2 = T1_outer[(i + 1) % 6] - ax.plot([x1, x2], [y1, y2], color='#444', linewidth=2.0, zorder=2) - - # Draw the annular region for T_1 (shaded) - outer_poly = patches.Polygon(T1_outer + gamma_pos[::-1], closed=True, - facecolor='#e3f2fd', edgecolor='none', - alpha=0.55, zorder=0) - ax.add_patch(outer_poly) - - # Draw spokes between gamma_pos[i] and T1_outer[i] (representing radial spokes - # in T_1's annular triangulation, schematically). + # T_1's annular cycle is a 12-cycle in G'. In Π_G it sits between + # γ and a larger outer boundary; vertices of T_1's annular cycle are + # DUAL vertices of T_1's annular faces. 6 of them are on the + # γ-side, 6 on the m_1 side. We draw them as 12 points spread + # outside γ. + R_T1_inner = 2.0 # γ-side of T_1's annular cycle vertices + R_T1_outer = 3.2 # m_1-side of T_1's annular cycle vertices + # Position T_1 dual vertices: alternate inner / outer as one goes around. + # In a balanced spoke-only triangulation, the 12 dual vertices alternate + # between γ-side (I-move) and outer-side (O-move). + T1_inner_pts = [] + T1_outer_pts = [] for i in range(6): + # γ-side dual vertex: just outside γ-edge i, halfway between v_i and v_{i+1} x1, y1 = gamma_pos[i] - x2, y2 = T1_outer[i] - ax.plot([x1, x2], [y1, y2], color='#a0a0a0', linewidth=0.9, - linestyle=':', zorder=2) + x2, y2 = gamma_pos[(i + 1) % 6] + mx, my = (x1 + x2) / 2, (y1 + y2) / 2 + r_m = math.hypot(mx, my) + ux, uy = mx / r_m, my / r_m + T1_inner_pts.append((R_T1_inner * ux, R_T1_inner * uy)) + # m_1-side dual vertex: just past v_i, outside. + r_v = math.hypot(x1, y1) + ux, uy = x1 / r_v, y1 / r_v + T1_outer_pts.append((R_T1_outer * ux, R_T1_outer * uy)) - # T_1's antipodal chord across its B_in (= gamma): - # this chord is in the O of T_1, conceptually living outside gamma in T_1's - # interior outerplanar O. For the diagram we represent it as a curved chord - # outside gamma connecting two opposite gamma vertices. - chord_p1 = gamma_pos[0] - chord_p2 = gamma_pos[3] - # Draw outside gamma using an arc above + # T_1 outer boundary B_out (length m_1 = 6) and the 12-cycle of + # annular face dual vertices: alternate T1_inner_pts and T1_outer_pts + # going around. Draw the 12-cycle edges as light gray. + T1_cycle = [] + for i in range(6): + T1_cycle.append(T1_inner_pts[i]) + T1_cycle.append(T1_outer_pts[(i + 1) % 6]) + for i in range(12): + x1, y1 = T1_cycle[i] + x2, y2 = T1_cycle[(i + 1) % 12] + ax.plot([x1, x2], [y1, y2], color='#90a4ae', linewidth=1.0, + linestyle=':', zorder=1) + for (x, y) in T1_inner_pts: + ax.plot(x, y, 'o', color='#607d8b', markersize=7, zorder=3) + for (x, y) in T1_outer_pts: + ax.plot(x, y, 'o', color='#90a4ae', markersize=5, zorder=3) + + # T_1's chord (in O_1, antipodal) -- draw outside γ as an arc. + chord_p1 = T1_outer_pts[0] + chord_p2 = T1_outer_pts[3] chord_arc = patches.FancyArrowPatch(chord_p1, chord_p2, - connectionstyle='arc3,rad=0.55', + connectionstyle='arc3,rad=0.45', arrowstyle='-', color='#9c27b0', - linewidth=2.2, linestyle=(0, (3, 2)), - zorder=4) + linewidth=1.8, linestyle=(0, (3, 2)), + zorder=2) ax.add_patch(chord_arc) - # T_2 lives inside gamma. T_2's B_out = gamma (length 6), - # T_2's B_in = a triangle of length 3. - triangle_pos = hex_positions(3, radius=R_in, angle_offset=math.pi / 2) - for i in range(3): - x1, y1 = triangle_pos[i] - x2, y2 = triangle_pos[(i + 1) % 3] - ax.plot([x1, x2], [y1, y2], color='#444', linewidth=2.0, zorder=2) - - # Draw T_2's annular region (between gamma and triangle, shaded) - inner_poly = patches.Polygon(gamma_pos + triangle_pos[::-1], closed=True, - facecolor='#fff3e0', edgecolor='none', - alpha=0.55, zorder=0) - ax.add_patch(inner_poly) - - # Draw shared cycle gamma (on top of everything) + # T_2's annular cycle (length |γ| + k_2 = 6 + 3 = 9) inside γ. + # 6 γ-side dual vertices and 3 k_2-side dual vertices. + R_T2_inner = 0.45 # k_2-side + R_T2_outer = 1.05 # γ-side + T2_outer_pts = [] for i in range(6): x1, y1 = gamma_pos[i] x2, y2 = gamma_pos[(i + 1) % 6] - ax.plot([x1, x2], [y1, y2], color='#333', linewidth=3.0, zorder=3) + mx, my = (x1 + x2) / 2, (y1 + y2) / 2 + r_m = math.hypot(mx, my) + ux, uy = mx / r_m, my / r_m + T2_outer_pts.append((R_T2_outer * ux, R_T2_outer * uy)) + T2_inner_pts = [] + for i in range(3): + angle = math.pi / 2 + 2 * math.pi * i / 3 + T2_inner_pts.append((R_T2_inner * math.cos(angle), + R_T2_inner * math.sin(angle))) + # T_2's 9-cycle: 2 γ-spokes, 1 k_2-spoke, ... let's just draw the cycle + # cleanly without committing to an exact interleaving. + T2_cycle = [] + for i in range(6): + T2_cycle.append(T2_outer_pts[i]) + if i % 2 == 1: + T2_cycle.append(T2_inner_pts[i // 2]) + for i in range(9): + x1, y1 = T2_cycle[i] + x2, y2 = T2_cycle[(i + 1) % 9] + ax.plot([x1, x2], [y1, y2], color='#ffb74d', linewidth=1.0, + linestyle=':', zorder=1) + for (x, y) in T2_outer_pts: + ax.plot(x, y, 'o', color='#f57c00', markersize=7, zorder=3) + for (x, y) in T2_inner_pts: + ax.plot(x, y, 'o', color='#ffb74d', markersize=5, zorder=3) - # Place vertices of gamma colored by one of the rainbow patterns: (1,2,3,2,3,1) - pattern = [1, 2, 3, 2, 3, 1] + # Now draw γ with edges colored by σ = (1, 2, 3, 2, 3, 1). + sigma = (1, 2, 3, 2, 3, 1) + for i in range(6): + x1, y1 = gamma_pos[i] + x2, y2 = gamma_pos[(i + 1) % 6] + ax.plot([x1, x2], [y1, y2], color=COLORS[sigma[i]], linewidth=5.0, + solid_capstyle='round', zorder=4) + mx, my = (x1 + x2) / 2, (y1 + y2) / 2 + # Also draw the dual G'-edges crossing γ at midpoint -- short segments + # from T_2-side dual vertex to T_1-side dual vertex. + sx1, sy1 = T2_outer_pts[i] + sx2, sy2 = T1_inner_pts[i] + ax.plot([sx1, sx2], [sy1, sy2], color=COLORS[sigma[i]], + linewidth=2.5, linestyle=(0, (4, 2)), zorder=2) + # γ vertices for i in range(6): x, y = gamma_pos[i] - ax.plot(x, y, 'o', color=COLORS[pattern[i]], markersize=26, zorder=4, - markeredgecolor='black', markeredgewidth=1.0) - ax.annotate(f"$v_{i}$", (x * 1.0, y * 1.0), color='white', - ha='center', va='center', fontsize=10, fontweight='bold', - zorder=5) + ax.plot(x, y, 'o', color='#222', markersize=10, zorder=5) + lx, ly = R_gamma * 1.12 * x / math.hypot(x, y), R_gamma * 1.12 * y / math.hypot(x, y) + ax.annotate(f"$v_{i}$", (lx + 0.05 * x, ly + 0.05 * y), + color='#222', ha='center', va='center', fontsize=9, + fontweight='bold', zorder=6) - # Triangle vertices - for i, (x, y) in enumerate(triangle_pos): - ax.plot(x, y, 'o', color='#666', markersize=14, zorder=4, - markeredgecolor='black', markeredgewidth=0.6) - - # Annotate which regions are T_1 and T_2 (placed off to the side with arrows) - ax.annotate(r'$T_1$ annulus' + '\n' - + r'$m_1 = 6$,' + '\n' - + r'chord $(v_0, v_3) \in O_1$' + '\n' - + r'Steiner-poor', - xy=(R_out - 0.1, -0.5), xytext=(3.7, -1.4), - fontsize=10, color='#1565c0', + # Labels with arrows + ax.annotate(r'$T_1$ annular cycle, length $|γ|+m_1 = 12$' + '\n' + + r'γ-side dual vertices (dark): 6' + '\n' + + r'$m_1$-side dual vertices (light): 6', + xy=(R_T1_inner * 1.0, 0.0), xytext=(4.0, 1.0), + fontsize=9, color='#37474f', ha='left', va='center', - arrowprops=dict(arrowstyle='->', color='#1565c0', lw=1.0)) - ax.annotate(r'$T_2$ annulus' + '\n' - + r'$k_2 = 3$, no chord' + '\n' - + r'Steiner-rich', - xy=(R_in + 0.2, 0.0), xytext=(3.7, 0.8), - fontsize=10, color='#e65100', + arrowprops=dict(arrowstyle='->', color='#37474f', lw=0.8)) + ax.annotate(r'$T_2$ annular cycle, length $|γ|+k_2 = 9$' + '\n' + + r'γ-side dual vertices (orange): 6' + '\n' + + r'$k_2$-side dual vertices (light): 3', + xy=(R_T2_outer * 0.7, -0.7), xytext=(4.0, -1.0), + fontsize=9, color='#e65100', ha='left', va='center', - arrowprops=dict(arrowstyle='->', color='#e65100', lw=1.0)) - ax.annotate(r'shared cycle $\gamma$' + '\n' + r'(length $k = 6$)', - xy=(R_gamma * 0.9, R_gamma * 0.4), xytext=(3.7, 2.1), + arrowprops=dict(arrowstyle='->', color='#e65100', lw=0.8)) + ax.annotate(r'antipodal chord' + '\n' + r'$(v_0, v_3) \in O_1$', + xy=(0, R_T1_outer * 1.0 - 0.1), xytext=(-4.6, 2.4), + fontsize=9, color='#9c27b0', + ha='left', va='center', + arrowprops=dict(arrowstyle='->', color='#9c27b0', lw=0.8)) + ax.annotate(r'shared cycle $\gamma$' + '\n' + + r'$|γ|=6$, drawn with' + '\n' + + r'$\sigma=(1,2,3,2,3,1)$', + xy=(R_gamma * 0.7, 0.7), xytext=(-5.2, 0.0), fontsize=10, color='#222', ha='left', va='center', - arrowprops=dict(arrowstyle='->', color='#222', lw=1.0)) - ax.annotate(r'antipodal chord' + '\n' + r'$(v_0, v_3)$ in $O_1$', - xy=(-0.55, 1.7), xytext=(-3.2, 2.4), - fontsize=10, color='#9c27b0', + arrowprops=dict(arrowstyle='->', color='#222', lw=0.8)) + ax.annotate(r'$\sigma$ is the coloring of the 6 dual' + '\n' + + r'$G\,\!\!\!\!^{\,\prime}$-edges crossing $γ$' + '\n' + + r'(dashed, same colors as $γ$-edges)', + xy=(R_gamma * 1.05, -R_gamma * 0.05), xytext=(-5.2, -1.4), + fontsize=9, color='#333', ha='left', va='center', - arrowprops=dict(arrowstyle='->', color='#9c27b0', lw=1.0)) + arrowprops=dict(arrowstyle='->', color='#333', lw=0.8)) - ax.set_xlim(-3.6, 5.4) - ax.set_ylim(-2.8, 3.2) + ax.set_xlim(-5.5, 7.5) + ax.set_ylim(-3.8, 3.8) ax.set_aspect('equal') ax.axis('off') - ax.set_title(r"Geometric setup behind the rainbow orbit: $T_1$ outside $\gamma$," - + r" $T_2$ inside $\gamma$" + "\n" - + r"shown with the spoke configuration $\sigma = (1,2,3,2,3,1)$" - + r" from the orbit", + ax.set_title(r"Geometric setup: $\sigma = (1,2,3,2,3,1)$ is one of the " + + r"6 orbit elements. " + "\n" + + r"It colors the 6 $G'$-edges that cross $\gamma$. 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