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Pentatrap Quasiperiodic Tiling from Trig Geometry

Pentatrap Quasiperiodic Tiling from Trig Geometry
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2021-01-16 10:13
PolygonAngleTrigonometryTransformationGeometric Construction
This example demonstrates how to construct a visually ordered quasiperiodic tiling (a quasiperiodic tile pattern) using a small set of geometric “tile rules” built from trapezoid/isosceles-triangle relationships. The key math ideas include: geometric transformations that replicate a tile with consistent orientation, ratio-based constraints (integers/ratios and geometric-mean style relations as suggested by the original tags), and trigonometry—using sine-based parameters or angles—to determine the rotation/shape so that the resulting arrangement is ordered but not strictly periodic. From the GeoGebra construction summary, the build process uses Point to create the base and derived vertices (A, B, A′, B′, A″, etc.), Rotate to generate rotated copies that drive the tiling’s direction changes, Segment to define key edges, Polygon to assemble the tile boundaries, and Midpoint to enforce midpoint-controlled proportions and symmetry (which supports the isosceles/ratio constraints). The presence of many named points (C, D, E…; I, J, K, …) and grouped parameters (q1–q4, along with variables like a, c, d, b′, etc.) indicates a parameter-driven geometric construction. Suitable for high-school to early-university audiences and for classroom activities on transformation geometry, tilings, and trigonometric parameterization.

GeoGebra Commands & Parameters

  1. Point(Circle[A, 1]) → B
  2. Rotate(A, 108°, B) → A'
  3. Rotate(B, (-(72°)), A) → B'
  4. Rotate(A', 72°, B') → A''
  5. Segment(A, B) → f
  6. Midpoint(A'', B) → C
  7. Polygon(A, C, D, B') → q1, a, c, d, b'
  8. Point(Circle[E, Segment[A, C]]) → F
  9. Rotate(F + (Segment[C, D], 0), Angle[D - C] + Angle[F - E] - Angle[C - A], F) → G
  10. Rotate(G + (Segment[D, B'], 0), Angle[B' - D] + Angle[G - F] - Angle[D - C], G) → H
  11. Polygon(E, F, G, H) → q2, e, f_1, g, h
  12. Point(Circle[I, Segment[A, C]]) → J
  13. Rotate(J + (Segment[C, D], 0), Angle[D - C] + Angle[J - I] - Angle[C - A], J) → K
  14. Rotate(K + (Segment[D, B'], 0), Angle[B' - D] + Angle[K - J] - Angle[D - C], K) → L
  15. Polygon(I, J, K, L) → q3, i, j, k, l
  16. Point(Circle[M, Segment[A, C]]) → N
  17. Rotate(N + (Segment[C, D], 0), Angle[D - C] + Angle[N - M] - Angle[C - A], N) → O
  18. Rotate(O + (Segment[D, B'], 0), Angle[B' - D] + Angle[O - N] - Angle[D - C], O) → P
  19. Polygon(M, N, O, P) → q4, m, n, o, p
  20. Point(Circle[Q, Segment[A, C]]) → R
  21. Rotate(R + (Segment[C, D], 0), Angle[D - C] + Angle[R - Q] - Angle[C - A], R) → S
  22. Rotate(S + (Segment[D, B'], 0), Angle[B' - D] + Angle[S - R] - Angle[D - C], S) → T
  23. Polygon(Q, R, S, T) → q5, q, r, s, t
  24. Point(Circle[U, Segment[A, C]]) → V
  25. Rotate(V + (Segment[C, D], 0), Angle[D - C] + Angle[V - U] - Angle[C - A], V) → W
  26. Rotate(W + (Segment[D, B'], 0), Angle[B' - D] + Angle[W - V] - Angle[D - C], W) → Z
  27. Polygon(U, V, W, Z) → q6, u, v, w, z_1
  28. Point(Circle[A_1, Segment[A, C]]) → B_1
  29. Rotate(B_1 + (Segment[C, D], 0), Angle[D - C] + Angle[B_1 - A_1] - Angle[C - A], B_1) → C_1
  30. Rotate(C_1 + (Segment[D, B'], 0), Angle[B' - D] + Angle[C_1 - B_1] - Angle[D - C], C_1) → D_1
  31. Polygon(A_1, B_1, C_1, D_1) → q7, a_1, b_1, c_1, d_1
  32. Point(Circle[E_1, Segment[A, C]]) → F_1
  33. Rotate(F_1 + (Segment[C, D], 0), Angle[D - C] + Angle[F_1 - E_1] - Angle[C - A], F_1) → G_1
  34. Rotate(G_1 + (Segment[D, B'], 0), Angle[B' - D] + Angle[G_1 - F_1] - Angle[D - C], G_1) → H_1
  35. Polygon(E_1, F_1, G_1, H_1) → q8, e_1, b, g_1, h_1
  36. Point(Circle[I_1, Segment[A, C]]) → J_1
  37. Rotate(J_1 + (Segment[C, D], 0), Angle[D - C] + Angle[J_1 - I_1] - Angle[C - A], J_1) → K_1
  38. Rotate(K_1 + (Segment[D, B'], 0), Angle[B' - D] + Angle[K_1 - J_1] - Angle[D - C], K_1) → L_1
  39. Polygon(I_1, J_1, K_1, L_1) → q9, i_1, j_1, k_1, l_1
  40. Point(Circle[M_1, Segment[A, C]]) → N_1
  41. Rotate(N_1 + (Segment[C, D], 0), Angle[D - C] + Angle[N_1 - M_1] - Angle[C - A], N_1) → O_1
  42. Rotate(O_1 + (Segment[D, B'], 0), Angle[B' - D] + Angle[O_1 - N_1] - Angle[D - C], O_1) → P_1
  43. Polygon(M_1, N_1, O_1, P_1) → q10, m_1, n_1, o_1, p_1
  44. Point(Circle[Q_1, Segment[A, C]]) → R_1
  45. Rotate(R_1 + (Segment[C, D], 0), Angle[D - C] + Angle[R_1 - Q_1] - Angle[C - A], R_1) → S_1
  46. Rotate(S_1 + (Segment[D, B'], 0), Angle[B' - D] + Angle[S_1 - R_1] - Angle[D - C], S_1) → T_1
  47. Polygon(Q_1, R_1, S_1, T_1) → q11, q_1, r_1, s_1, t_1
  48. Point(Circle[U_1, Segment[A, C]]) → V_1
  49. Rotate(V_1 + (Segment[C, D], 0), Angle[D - C] + Angle[V_1 - U_1] - Angle[C - A], V_1) → W_1
  50. Rotate(W_1 + (Segment[D, B'], 0), Angle[B' - D] + Angle[W_1 - V_1] - Angle[D - C], W_1) → Z_1
  51. Polygon(U_1, V_1, W_1, Z_1) → q12, u_1, v_1, w_1, f_2
  52. Point(Circle[A_2, Segment[A, C]]) → B_2
  53. Rotate(B_2 + (Segment[C, D], 0), Angle[D - C] + Angle[B_2 - A_2] - Angle[C - A], B_2) → C_2
  54. Rotate(C_2 + (Segment[D, B'], 0), Angle[B' - D] + Angle[C_2 - B_2] - Angle[D - C], C_2) → D_2
  55. Polygon(A_2, B_2, C_2, D_2) → q13, a_2, b_2, c_2, d_2
  56. Point(Circle[E_2, Segment[A, C]]) → F_2
  57. Rotate(F_2 + (Segment[C, D], 0), Angle[D - C] + Angle[F_2 - E_2] - Angle[C - A], F_2) → G_2
  58. Rotate(G_2 + (Segment[D, B'], 0), Angle[B' - D] + Angle[G_2 - F_2] - Angle[D - C], G_2) → H_2
  59. Polygon(E_2, F_2, G_2, H_2) → q14, e_2, g_2, h_2, i_2
  60. Point(Circle[I_2, Segment[A, C]]) → J_2
  61. Rotate(J_2 + (Segment[C, D], 0), Angle[D - C] + Angle[J_2 - I_2] - Angle[C - A], J_2) → K_2
  62. Rotate(K_2 + (Segment[D, B'], 0), Angle[B' - D] + Angle[K_2 - J_2] - Angle[D - C], K_2) → L_2
  63. Polygon(I_2, J_2, K_2, L_2) → q15, j_2, k_2, l_2, m_2
  64. Point(Circle[M_2, Segment[A, C]]) → N_2
  65. Rotate(N_2 + (Segment[C, D], 0), Angle[D - C] + Angle[N_2 - M_2] - Angle[C - A], N_2) → O_2
  66. Rotate(O_2 + (Segment[D, B'], 0), Angle[B' - D] + Angle[O_2 - N_2] - Angle[D - C], O_2) → P_2
  67. Polygon(M_2, N_2, O_2, P_2) → q16, n_2, p_2, o_2, q_2
  68. Point(Circle[Q_2, Segment[A, C]]) → R_2
  69. Rotate(R_2 + (Segment[C, D], 0), Angle[D - C] + Angle[R_2 - Q_2] - Angle[C - A], R_2) → S_2
  70. Rotate(S_2 + (Segment[D, B'], 0), Angle[B' - D] + Angle[S_2 - R_2] - Angle[D - C], S_2) → T_2
  71. Polygon(Q_2, R_2, S_2, T_2) → q17, r_2, s_2, t_2, f_3
  72. Point(Circle[U_2, Segment[A, C]]) → V_2
  73. Rotate(V_2 + (Segment[C, D], 0), Angle[D - C] + Angle[V_2 - U_2] - Angle[C - A], V_2) → W_2
  74. Rotate(W_2 + (Segment[D, B'], 0), Angle[B' - D] + Angle[W_2 - V_2] - Angle[D - C], W_2) → Z_2
  75. Polygon(U_2, V_2, W_2, Z_2) → q18, u_2, v_2, w_2, z_2
  76. Point(Circle[A_3, Segment[A, C]]) → B_3
  77. Rotate(B_3 + (Segment[C, D], 0), Angle[D - C] + Angle[B_3 - A_3] - Angle[C - A], B_3) → C_3
  78. Rotate(C_3 + (Segment[D, B'], 0), Angle[B' - D] + Angle[C_3 - B_3] - Angle[D - C], C_3) → D_3
  79. Polygon(A_3, B_3, C_3, D_3) → q19, a_3, b_3, c_3, d_3
  80. Point(Circle[E_3, Segment[A, C]]) → F_3
  81. Rotate(F_3 + (Segment[C, D], 0), Angle[D - C] + Angle[F_3 - E_3] - Angle[C - A], F_3) → G_3
  82. Rotate(G_3 + (Segment[D, B'], 0), Angle[B' - D] + Angle[G_3 - F_3] - Angle[D - C], G_3) → H_3
  83. Polygon(E_3, F_3, G_3, H_3) → q20, e_3, g_3, h_3, i_3
  84. Point(Circle[I_3, Segment[A, C]]) → J_3
  85. Rotate(J_3 + (Segment[C, D], 0), Angle[D - C] + Angle[J_3 - I_3] - Angle[C - A], J_3) → K_3
  86. Rotate(K_3 + (Segment[D, B'], 0), Angle[B' - D] + Angle[K_3 - J_3] - Angle[D - C], K_3) → L_3
  87. Polygon(I_3, J_3, K_3, L_3) → q21, j_3, k_3, l_3, m_3
  88. Point(Circle[M_3, Segment[A, C]]) → N_3
  89. Rotate(N_3 + (Segment[C, D], 0), Angle[D - C] + Angle[N_3 - M_3] - Angle[C - A], N_3) → O_3
  90. Rotate(O_3 + (Segment[D, B'], 0), Angle[B' - D] + Angle[O_3 - N_3] - Angle[D - C], O_3) → P_3
  91. Polygon(M_3, N_3, O_3, P_3) → q22, n_3, p_3, o_3, q_3
  92. Point(Circle[Q_3, Segment[A, C]]) → R_3
  93. Rotate(R_3 + (Segment[C, D], 0), Angle[D - C] + Angle[R_3 - Q_3] - Angle[C - A], R_3) → S_3
  94. Rotate(S_3 + (Segment[D, B'], 0), Angle[B' - D] + Angle[S_3 - R_3] - Angle[D - C], S_3) → T_3
  95. Polygon(Q_3, R_3, S_3, T_3) → q23, r_3, s_3, t_3, f_4
  96. Point(Circle[U_3, Segment[A, C]]) → V_3
  97. Rotate(V_3 + (Segment[C, D], 0), Angle[D - C] + Angle[V_3 - U_3] - Angle[C - A], V_3) → W_3
  98. Rotate(W_3 + (Segment[D, B'], 0), Angle[B' - D] + Angle[W_3 - V_3] - Angle[D - C], W_3) → Z_3
  99. Polygon(U_3, V_3, W_3, Z_3) → q24, u_3, v_3, w_3, z_3
  100. Point(Circle[A_4, Segment[A, C]]) → B_4
  101. Rotate(B_4 + (Segment[C, D], 0), Angle[D - C] + Angle[B_4 - A_4] - Angle[C - A], B_4) → C_4
  102. Rotate(C_4 + (Segment[D, B'], 0), Angle[B' - D] + Angle[C_4 - B_4] - Angle[D - C], C_4) → D_4
  103. Polygon(A_4, B_4, C_4, D_4) → q25, a_4, b_4, c_4, d_4
  104. Point(Circle[E_4, Segment[A, C]]) → F_4
  105. Rotate(F_4 + (Segment[C, D], 0), Angle[D - C] + Angle[F_4 - E_4] - Angle[C - A], F_4) → G_4
  106. Rotate(G_4 + (Segment[D, B'], 0), Angle[B' - D] + Angle[G_4 - F_4] - Angle[D - C], G_4) → H_4
  107. Polygon(E_4, F_4, G_4, H_4) → q26, e_4, g_4, h_4, i_4
  108. Point(Circle[I_4, Segment[A_4, B_4]]) → J_4
  109. Rotate(J_4 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[J_4 - I_4] - Angle[B_4 - A_4], J_4) → K_4
  110. Rotate(K_4 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[K_4 - J_4] - Angle[C_4 - B_4], K_4) → L_4
  111. Polygon(I_4, J_4, K_4, L_4) → q27, j_4, k_4, l_4, m_4
  112. Point(Circle[M_4, Segment[A_4, B_4]]) → N_4
  113. Rotate(N_4 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[N_4 - M_4] - Angle[B_4 - A_4], N_4) → O_4
  114. Rotate(O_4 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[O_4 - N_4] - Angle[C_4 - B_4], O_4) → P_4
  115. Polygon(M_4, N_4, O_4, P_4) → q28, n_4, p_4, o_4, q_4
  116. Point(Circle[Q_4, Segment[A_4, B_4]]) → R_4
  117. Rotate(R_4 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[R_4 - Q_4] - Angle[B_4 - A_4], R_4) → S_4
  118. Rotate(S_4 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[S_4 - R_4] - Angle[C_4 - B_4], S_4) → T_4
  119. Polygon(Q_4, R_4, S_4, T_4) → q29, r_4, s_4, t_4, f_5
  120. Point(Circle[U_4, Segment[A_4, B_4]]) → V_4
  121. Rotate(V_4 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[V_4 - U_4] - Angle[B_4 - A_4], V_4) → W_4
  122. Rotate(W_4 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[W_4 - V_4] - Angle[C_4 - B_4], W_4) → Z_4
  123. Polygon(U_4, V_4, W_4, Z_4) → q30, u_4, v_4, w_4, z_4
  124. Point(Circle[A_5, Segment[A_4, B_4]]) → B_5
  125. Rotate(B_5 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[B_5 - A_5] - Angle[B_4 - A_4], B_5) → C_5
  126. Rotate(C_5 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[C_5 - B_5] - Angle[C_4 - B_4], C_5) → D_5
  127. Polygon(A_5, B_5, C_5, D_5) → q31, a_5, b_5, c_5, d_5
  128. Point(Circle[E_5, Segment[A_4, B_4]]) → F_5
  129. Rotate(F_5 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[F_5 - E_5] - Angle[B_4 - A_4], F_5) → G_5
  130. Rotate(G_5 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[G_5 - F_5] - Angle[C_4 - B_4], G_5) → H_5
  131. Polygon(E_5, F_5, G_5, H_5) → q32, e_5, g_5, h_5, i_5
  132. Point(Circle[I_5, Segment[A_4, B_4]]) → J_5
  133. Rotate(J_5 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[J_5 - I_5] - Angle[B_4 - A_4], J_5) → K_5
  134. Rotate(K_5 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[K_5 - J_5] - Angle[C_4 - B_4], K_5) → L_5
  135. Polygon(I_5, J_5, K_5, L_5) → q33, j_5, k_5, l_5, m_5
  136. Point(Circle[M_5, Segment[A_4, B_4]]) → N_5
  137. Rotate(N_5 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[N_5 - M_5] - Angle[B_4 - A_4], N_5) → O_5
  138. Rotate(O_5 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[O_5 - N_5] - Angle[C_4 - B_4], O_5) → P_5
  139. Polygon(M_5, N_5, O_5, P_5) → q34, n_5, p_5, o_5, q_5
  140. Point(Circle[Q_5, Segment[A_4, B_4]]) → R_5
  141. Rotate(R_5 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[R_5 - Q_5] - Angle[B_4 - A_4], R_5) → S_5
  142. Rotate(S_5 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[S_5 - R_5] - Angle[C_4 - B_4], S_5) → T_5
  143. Polygon(Q_5, R_5, S_5, T_5) → q35, r_5, s_5, t_5, f_6
  144. Point(Circle[U_5, Segment[A_4, B_4]]) → V_5
  145. Rotate(V_5 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[V_5 - U_5] - Angle[B_4 - A_4], V_5) → W_5
  146. Rotate(W_5 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[W_5 - V_5] - Angle[C_4 - B_4], W_5) → Z_5
  147. Polygon(U_5, V_5, W_5, Z_5) → q36, u_5, v_5, w_5, z_5
  148. Point(Circle[A_6, Segment[A_4, B_4]]) → B_6
  149. Rotate(B_6 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[B_6 - A_6] - Angle[B_4 - A_4], B_6) → C_6
  150. Rotate(C_6 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[C_6 - B_6] - Angle[C_4 - B_4], C_6) → D_6
  151. Polygon(A_6, B_6, C_6, D_6) → q37, a_6, b_6, c_6, d_6
  152. Point(Circle[E_6, Segment[A_4, B_4]]) → F_6
  153. Rotate(F_6 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[F_6 - E_6] - Angle[B_4 - A_4], F_6) → G_6
  154. Rotate(G_6 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[G_6 - F_6] - Angle[C_4 - B_4], G_6) → H_6
  155. Polygon(E_6, F_6, G_6, H_6) → q38, e_6, g_6, h_6, i_6
  156. Point(Circle[I_6, Segment[A_4, B_4]]) → J_6
  157. Rotate(J_6 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[J_6 - I_6] - Angle[B_4 - A_4], J_6) → K_6
  158. Rotate(K_6 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[K_6 - J_6] - Angle[C_4 - B_4], K_6) → L_6
  159. Polygon(I_6, J_6, K_6, L_6) → q39, j_6, k_6, l_6, m_6
  160. Point(Circle[M_6, Segment[A_4, B_4]]) → N_6
  161. Rotate(N_6 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[N_6 - M_6] - Angle[B_4 - A_4], N_6) → O_6
  162. Rotate(O_6 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[O_6 - N_6] - Angle[C_4 - B_4], O_6) → P_6
  163. Polygon(M_6, N_6, O_6, P_6) → q40, n_6, p_6, o_6, q_6
  164. Point(Circle[Q_6, Segment[A_4, B_4]]) → R_6
  165. Rotate(R_6 + (Segment[B_4, C_4], 0), Angle[C_4 - B_4] + Angle[R_6 - Q_6] - Angle[B_4 - A_4], R_6) → S_6
  166. Rotate(S_6 + (Segment[C_4, D_4], 0), Angle[D_4 - C_4] + Angle[S_6 - R_6] - Angle[C_4 - B_4], S_6) → T_6
  167. Polygon(Q_6, R_6, S_6, T_6) → q41, r_6, s_6, t_6, f_7
  168. Point(Circle[U_6, Segment[E_4, F_4]]) → V_6
  169. Rotate(V_6 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[V_6 - U_6] - Angle[F_4 - E_4], V_6) → W_6
  170. Rotate(W_6 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[W_6 - V_6] - Angle[G_4 - F_4], W_6) → Z_6
  171. Polygon(U_6, V_6, W_6, Z_6) → q42, u_6, v_6, w_6, z_6
  172. Point(Circle[A_7, Segment[E_4, F_4]]) → B_7
  173. Rotate(B_7 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[B_7 - A_7] - Angle[F_4 - E_4], B_7) → C_7
  174. Rotate(C_7 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[C_7 - B_7] - Angle[G_4 - F_4], C_7) → D_7
  175. Polygon(A_7, B_7, C_7, D_7) → q43, a_7, b_7, c_7, d_7
  176. Point(Circle[E_7, Segment[E_4, F_4]]) → F_7
  177. Rotate(F_7 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[F_7 - E_7] - Angle[F_4 - E_4], F_7) → G_7
  178. Rotate(G_7 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[G_7 - F_7] - Angle[G_4 - F_4], G_7) → H_7
  179. Polygon(E_7, F_7, G_7, H_7) → q44, e_7, g_7, h_7, i_7
  180. Point(Circle[I_7, Segment[E_4, F_4]]) → J_7
  181. Rotate(J_7 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[J_7 - I_7] - Angle[F_4 - E_4], J_7) → K_7
  182. Rotate(K_7 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[K_7 - J_7] - Angle[G_4 - F_4], K_7) → L_7
  183. Polygon(I_7, J_7, K_7, L_7) → q45, j_7, k_7, l_7, m_7
  184. Point(Circle[M_7, Segment[E_4, F_4]]) → N_7
  185. Rotate(N_7 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[N_7 - M_7] - Angle[F_4 - E_4], N_7) → O_7
  186. Rotate(O_7 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[O_7 - N_7] - Angle[G_4 - F_4], O_7) → P_7
  187. Polygon(M_7, N_7, O_7, P_7) → q46, n_7, p_7, o_7, q_7
  188. Point(Circle[Q_7, Segment[E_4, F_4]]) → R_7
  189. Rotate(R_7 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[R_7 - Q_7] - Angle[F_4 - E_4], R_7) → S_7
  190. Rotate(S_7 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[S_7 - R_7] - Angle[G_4 - F_4], S_7) → T_7
  191. Polygon(Q_7, R_7, S_7, T_7) → q47, r_7, s_7, t_7, f_8
  192. Point(Circle[U_7, Segment[E_4, F_4]]) → V_7
  193. Rotate(V_7 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[V_7 - U_7] - Angle[F_4 - E_4], V_7) → W_7
  194. Rotate(W_7 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[W_7 - V_7] - Angle[G_4 - F_4], W_7) → Z_7
  195. Polygon(U_7, V_7, W_7, Z_7) → q48, u_7, v_7, w_7, z_7
  196. Point(Circle[A_8, Segment[E_4, F_4]]) → B_8
  197. Rotate(B_8 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[B_8 - A_8] - Angle[F_4 - E_4], B_8) → C_8
  198. Rotate(C_8 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[C_8 - B_8] - Angle[G_4 - F_4], C_8) → D_8
  199. Polygon(A_8, B_8, C_8, D_8) → q49, a_8, b_8, c_8, d_8
  200. Point(Circle[E_8, Segment[E_4, F_4]]) → F_8
  201. Rotate(F_8 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[F_8 - E_8] - Angle[F_4 - E_4], F_8) → G_8
  202. Rotate(G_8 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[G_8 - F_8] - Angle[G_4 - F_4], G_8) → H_8
  203. Polygon(E_8, F_8, G_8, H_8) → q50, e_8, g_8, h_8, i_8
  204. Point(Circle[I_8, Segment[E_4, F_4]]) → J_8
  205. Rotate(J_8 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[J_8 - I_8] - Angle[F_4 - E_4], J_8) → K_8
  206. Rotate(K_8 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[K_8 - J_8] - Angle[G_4 - F_4], K_8) → L_8
  207. Polygon(I_8, J_8, K_8, L_8) → q51, j_8, k_8, l_8, m_8
  208. Point(Circle[M_8, Segment[E_4, F_4]]) → N_8
  209. Rotate(N_8 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[N_8 - M_8] - Angle[F_4 - E_4], N_8) → O_8
  210. Rotate(O_8 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[O_8 - N_8] - Angle[G_4 - F_4], O_8) → P_8
  211. Polygon(M_8, N_8, O_8, P_8) → q52, n_8, p_8, o_8, q_8
  212. Point(Circle[Q_8, Segment[E_4, F_4]]) → R_8
  213. Rotate(R_8 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[R_8 - Q_8] - Angle[F_4 - E_4], R_8) → S_8
  214. Rotate(S_8 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[S_8 - R_8] - Angle[G_4 - F_4], S_8) → T_8
  215. Polygon(Q_8, R_8, S_8, T_8) → q53, r_8, s_8, t_8, f_9
  216. Point(Circle[U_8, Segment[E_4, F_4]]) → V_8
  217. Rotate(V_8 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[V_8 - U_8] - Angle[F_4 - E_4], V_8) → W_8
  218. Rotate(W_8 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[W_8 - V_8] - Angle[G_4 - F_4], W_8) → Z_8
  219. Polygon(U_8, V_8, W_8, Z_8) → q54, u_8, v_8, w_8, z_8
  220. Point(Circle[A_9, Segment[E_4, F_4]]) → B_9
  221. Rotate(B_9 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[B_9 - A_9] - Angle[F_4 - E_4], B_9) → C_9
  222. Rotate(C_9 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[C_9 - B_9] - Angle[G_4 - F_4], C_9) → D_9
  223. Polygon(A_9, B_9, C_9, D_9) → q55, a_9, b_9, c_9, d_9
  224. Point(Circle[E_9, Segment[E_4, F_4]]) → F_9
  225. Rotate(F_9 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[F_9 - E_9] - Angle[F_4 - E_4], F_9) → G_9
  226. Rotate(G_9 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[G_9 - F_9] - Angle[G_4 - F_4], G_9) → H_9
  227. Polygon(E_9, F_9, G_9, H_9) → q56, e_9, g_9, h_9, i_9
  228. Point(Circle[I_9, Segment[E_4, F_4]]) → J_9
  229. Rotate(J_9 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[J_9 - I_9] - Angle[F_4 - E_4], J_9) → K_9
  230. Rotate(K_9 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[K_9 - J_9] - Angle[G_4 - F_4], K_9) → L_9
  231. Polygon(I_9, J_9, K_9, L_9) → q57, j_9, k_9, l_9, m_9
  232. Point(Circle[M_9, Segment[E_4, F_4]]) → N_9
  233. Rotate(N_9 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[N_9 - M_9] - Angle[F_4 - E_4], N_9) → O_9
  234. Rotate(O_9 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[O_9 - N_9] - Angle[G_4 - F_4], O_9) → P_9
  235. Polygon(M_9, N_9, O_9, P_9) → q58, n_9, p_9, o_9, q_9
  236. Point(Circle[Q_9, Segment[E_4, F_4]]) → R_9
  237. Rotate(R_9 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[R_9 - Q_9] - Angle[F_4 - E_4], R_9) → S_9
  238. Rotate(S_9 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[S_9 - R_9] - Angle[G_4 - F_4], S_9) → T_9
  239. Polygon(Q_9, R_9, S_9, T_9) → q59, r_9, s_9, t_9, f_{10}
  240. Point(Circle[U_9, Segment[E_4, F_4]]) → V_9
  241. Rotate(V_9 + (Segment[F_4, G_4], 0), Angle[G_4 - F_4] + Angle[V_9 - U_9] - Angle[F_4 - E_4], V_9) → W_9
  242. Rotate(W_9 + (Segment[G_4, H_4], 0), Angle[H_4 - G_4] + Angle[W_9 - V_9] - Angle[G_4 - F_4], W_9) → Z_9
  243. Polygon(U_9, V_9, W_9, Z_9) → q60, u_9, v_9, w_9, z_9

AI Drawing Prompt Reference

Create a quasiperiodic tiling pattern using pentagons and isosceles triangles. Start by drawing a circle and marking its intersection with the center. Rotate and connect these points to form multiple similar geometric shapes. Use the midpoints of these shapes and rotate them to build a complex tiling structure. The pattern should exhibit an ordered yet non-periodic arrangement through rotations and proportional constraints.
The prompt above was compiled from the original drawing commands and is for reference only.