diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_rainbow_orbit.py b/papers/coloring_nested_tire_graphs/experiments/draw_rainbow_orbit.py
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index 0000000..07879f2
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+++ b/papers/coloring_nested_tire_graphs/experiments/draw_rainbow_orbit.py
@@ -0,0 +1,314 @@
+"""Visualise the 'rainbow orbit' intersection from
+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.
+
+Produces three figures:
+
+  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}).
+
+  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.
+
+  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.
+"""
+import math
+import os
+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")
+}
+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))
+        for i in range(n)
+    ]
+
+
+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.
+    """
+    pos = hex_positions(6, radius=radius)
+    # cycle 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):
+        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)
+
+
+# ---------------------------------------------------------------------------
+# Figure 1: the abstract pattern abcbca on a hexagon.
+# ---------------------------------------------------------------------------
+def fig_pattern(outdir):
+    fig, ax = plt.subplots(figsize=(6.0, 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'}
+    class_colors = {
+        'a': '#bdbdbd',
+        'b': '#ffe082',
+        'c': '#a5d6a7',
+    }
+
+    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)
+
+    # 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"  $c$: $\{v_2, v_4\}$"
+    )
+    ax.text(1.85, 0.0, legend, fontsize=11, color='#222',
+            family='monospace',
+            verticalalignment='center',
+            bbox=dict(boxstyle='round,pad=0.5', facecolor='#f8f8f8',
+                      edgecolor='#bbb'))
+
+    ax.set_xlim(-1.6, 4.0)
+    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 $k=6$)",
+                 fontsize=11)
+
+    out = os.path.join(outdir, 'fig_rainbow_pattern.png')
+    plt.savefig(out, dpi=160, bbox_inches='tight')
+    plt.close()
+    print(f"wrote {out}")
+
+
+# ---------------------------------------------------------------------------
+# Figure 2: the 6 concrete colourings in the S_3 orbit.
+# ---------------------------------------------------------------------------
+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),
+        (2, 1, 3),
+        (2, 3, 1),
+        (3, 1, 2),
+        (3, 2, 1),
+    ]
+
+    fig, axes = plt.subplots(2, 3, figsize=(10.5, 7.5))
+
+    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)
+        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"$)$")
+        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"
+                 + "\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 $k{=}6$",
+                 fontsize=11, y=1.00)
+    plt.tight_layout()
+    out = os.path.join(outdir, 'fig_rainbow_orbit.png')
+    plt.savefig(out, dpi=160, bbox_inches='tight')
+    plt.close()
+    print(f"wrote {out}")
+
+
+# ---------------------------------------------------------------------------
+# Figure 3: geometric setup --- T_1 (hexagon + antipodal chord) outside gamma,
+# T_2 (triangle) inside gamma, the 6 spoke edges crossing gamma.
+# ---------------------------------------------------------------------------
+def fig_setup(outdir):
+    fig, ax = plt.subplots(figsize=(8.5, 7.5))
+
+    # Shared cycle gamma at radius R_gamma
+    R_gamma = 1.4
+    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).
+    for i in range(6):
+        x1, y1 = gamma_pos[i]
+        x2, y2 = T1_outer[i]
+        ax.plot([x1, x2], [y1, y2], color='#a0a0a0', linewidth=0.9,
+                linestyle=':', zorder=2)
+
+    # 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
+    chord_arc = patches.FancyArrowPatch(chord_p1, chord_p2,
+                                        connectionstyle='arc3,rad=0.55',
+                                        arrowstyle='-', color='#9c27b0',
+                                        linewidth=2.2, linestyle=(0, (3, 2)),
+                                        zorder=4)
+    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)
+    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)
+
+    # Place vertices of gamma colored by one of the rainbow patterns: (1,2,3,2,3,1)
+    pattern = [1, 2, 3, 2, 3, 1]
+    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)
+
+    # 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',
+                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',
+                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),
+                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',
+                ha='left', va='center',
+                arrowprops=dict(arrowstyle='->', color='#9c27b0', lw=1.0))
+
+    ax.set_xlim(-3.6, 5.4)
+    ax.set_ylim(-2.8, 3.2)
+    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",
+                 fontsize=11)
+
+    out = os.path.join(outdir, 'fig_rainbow_setup.png')
+    plt.savefig(out, dpi=160, bbox_inches='tight')
+    plt.close()
+    print(f"wrote {out}")
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    notes = os.path.normpath(os.path.join(here, '..', 'notes'))
+    fig_pattern(notes)
+    fig_orbit(notes)
+    fig_setup(notes)
+
+
+if __name__ == '__main__':
+    main()
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