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Reconstruct the Girih Pattern

Reconstruct the Girih Pattern
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2019-03-06 20:29
PolygonCoordinate Geometry
This example demonstrates the reconstruction of the Girih pattern. Key mathematical concepts include polygons, symmetry, and coordinate geometry. Using GeoGebra commands such as Polygon, Intersect, and PolyLine, complex geometric patterns are constructed. Suitable for high school geometry education and exploration.

GeoGebra Commands & Parameters

  1. Polygon(F_1, E_2, 5) → veelhoek12, d_2, e_2, f_3, g_3, h_3, F_2, G_2, H_2
  2. Polygon(Q, I, 5) → veelhoek4, b, c, d, e, f_1, R, S, T
  3. Polygon(F_5, T, G_5, H_5, I_5, J_5, K_5, F_2) → veelhoek27, d_6, t_8, e_6, f_7, g_7, h_7, i_7, j_7
  4. Intersect(g_7, h_7) → E_7
  5. Polygon(C, D, 5) → veelhoek1, f, g, h, i, j, E, F, G
  6. Polygon(H, I, 5) → veelhoek2, k, l, m, n, p, J, K, L
  7. Polygon(C, M, 5) → veelhoek3, q, r, s, t, a, N, O, P
  8. Polygon(U, V, 5) → veelhoek5, g_1, h_1, i_1, j_1, k_1, W, Z, A_1
  9. Polygon(B_1, V, 5) → veelhoek6, l_1, m_1, n_1, p_1, q_1, C_1, D_1, E_1
  10. Polygon(F_1, G_1, 5) → veelhoek7, r_1, s_1, t_1, a_1, b_1, H_1, I_1, J_1
  11. Polygon(K_1, L_1, 5) → veelhoek8, c_1, d_1, e_1, f_2, g_2, M_1, N_1, O_1
  12. Polygon(K_1, P_1, 5) → veelhoek9, h_2, i_2, j_2, k_2, l_2, Q_1, R_1, S_1
  13. Polygon(T_1, U_1, 5) → veelhoek10, m_2, n_2, p_2, q_2, r_2, V_1, W_1, Z_1
  14. Polygon(A_2, U_1, 5) → veelhoek11, s_2, t_2, a_2, b_2, c_2, B_2, C_2, D_2
  15. Polygon(I_2, J_2, 5) → veelhoek13, i_3, j_3, k_3, l_3, m_3, K_2, L_2, M_2
  16. Polygon(I_2, N_2, 5) → veelhoek14, n_3, p_3, q_3, r_3, s_3, O_2, P_2, Q_2
  17. Polygon(R_2, N_2, 5) → veelhoek15, t_3, a_3, b_3, c_3, d_3, S_2, T_2, U_2
  18. Polygon(V_2, W_2, 5) → veelhoek16, e_3, f_4, g_4, h_4, i_4, Z_2, A_3, B_3
  19. Polygon(C_3, W_2, 5) → veelhoek17, j_4, k_4, l_4, m_4, n_4, D_3, E_3, F_3
  20. Polygon(G_3, H_3, 5) → veelhoek18, p_4, q_4, r_4, s_4, t_4, I_3, J_3, K_3
  21. Polygon(L_3, M_3, 5) → veelhoek19, a_4, b_4, c_4, d_4, e_4, N_3, O_3, P_3
  22. Polygon(Q_3, M_3, 5) → veelhoek20, f_5, g_5, h_5, i_5, j_5, R_3, S_3, T_3
  23. Polygon(M_1, O, 5) → veelhoek22, q_5, r_5, s_5, t_5, a_5, Z_3, A_4, B_4
  24. Polygon(L_3, C_4, 5) → veelhoek23, b_5, c_5, d_5, e_5, f_6, D_4, E_4, F_4
  25. Polygon(D_4, G_4, 5) → veelhoek24, g_6, h_6, i_6, j_6, k_6, H_4, I_4, J_4
  26. Polygon(G_4, K_4, 5) → veelhoek25, l_6, m_6, n_6, p_6, q_6, L_4, M_4, N_4
  27. Polygon(H_1, P_2, 5) → veelhoek29, m_7, n_7, p_7, q_7, r_7, Z_4, A_5, B_5
  28. Polygon(O_3, E_1, 5) → veelhoek30, s_7, t_7, a_7, b_7, c_7, C_5, D_5, E_5
  29. Intersect(r_7, t_7) → O_4
  30. Intersect(p_7, b_7) → P_4
  31. Polygon(O_3, E_1, O_4, H_1, P_2, F_3, P_4, I_3) → veelhoek26, o_3, r_6, o_4, s_6, t_6, a_6, b_6, c_6
  32. Polygon(R_1, Q_4, R_4, S_4, C_2, L, T_4, E) → veelhoek28, k_7, l_7, d_7, e_7, f_8, l_8, g_8, e_8
  33. Polygon(U_4, A_2, 5) → veelhoek31, h_8, i_8, j_8, k_8, m_8, V_4, W_4, L_5
  34. Polygon(Q_5, B_3, R_5, S_5, T_5, U_5, V_5, W_5) → veelhoek33, s_8, a_8, b_8, c_8, d_8, u_5, v_5, w_5
  35. PolyLine(M_5, C_6, F_5, H_6, E_2, I_6, J_1, J_6, C_1, K_6, U, L_6, M, D_6, G, E_6, J, F_6, Q, G_6) → k_9
  36. Segment(N_4, M_6) → l_9
  37. Segment(N_4, M_4) → m_9
  38. Segment(M_4, N_6) → p_9
  39. Segment(N_6, M_6) → q_9
  40. Segment(M_4, O_6) → r_9
  41. Segment(O_6, K_3) → s_9
  42. Segment(K_3, P_6) → t_9
  43. Segment(P_6, D_3) → a_9
  44. Segment(D_3, Q_6) → b_9
  45. Segment(Q_6, V_2) → c_9
  46. Segment(V_2, R_6) → d_9
  47. Segment(R_6, Q_5) → e_9
  48. Segment(O_4, C_5) → f_{10}
  49. Segment(C_5, D_5) → g_{10}
  50. Segment(D_5, P_4) → h_{10}
  51. Segment(O_4, B_5) → i_{10}
  52. Segment(A_5, P_4) → j_{10}
  53. Segment(B_5, A_5) → k_{10}
  54. Intersect(i_1, j_1) → V_3
  55. Polygon(U_3, W_3, 5) → veelhoek21, k_5, l_5, m_5, n_5, p_5, S_6, T_6, U_6
  56. Polygon(Z_5, A_6, 5) → veelhoek32, n_8, p_8, q_8, r_8, f_9, B_6, V_6, W_6
  57. Intersect(r_5, s_5) → Z_6
  58. Segment(Z_6, A_7) → g_9
  59. Intersect(k_5, p_5) → B_7
  60. Segment(A_7, B_7) → h_9
  61. Point(k_9) → C_7
  62. Point(k_9) → D_7
  63. Segment(C_7, D_7) → i_9
  64. Intersect(f_3, g_3) → F_7
  65. Intersect(c_3, d_3) → H_7
  66. Intersect(g_4, h_4) → I_7
  67. Segment(H_7, I_7) → l_{10}
  68. Intersect(b_8, c_8) → J_7
  69. Polygon(U_7, E_7, 5) → veelhoek34, j_9, n_9, n_{10}, p_{10}, q_{10}, K_7, L_7, M_7
  70. Segment(I_7, J_7) → m_{10}
  71. Intersect(g_9, h_9) → N_7
  72. Intersect(r_5, s_5) → O_7
  73. Intersect(r, s) → Q_7
  74. Intersect(j_1, k_1) → R_7
  75. Intersect(k_5, p_5) → T_7
  76. Polygon(N_7, O_7, P_7, Q_7, C_7, R_7, S_7, T_7) → veelhoek35, r_{10}, o_7, s_{10}, t_{10}, a_{10}, b_{10}, c_{10}, d_{10}
  77. Polygon(V_7, W_7, Z_7, A_8, B_8, C_8, D_8, E_8) → veelhoek36, v_7, w_7, z_7, e_{10}, f_{11}, g_{11}, h_{11}, i_{11}
  78. PolyLine(V_8, U_8, H_8, G_8, F_8, I_8) → j_{11}
  79. PolyLine(W_8, Z_8, I_9, L_9, K_9, J_9) → k_{11}
  80. PolyLine(H_8, T_8, R_8, J_8, S_8, G_8) → l_{11}
  81. PolyLine(R_8, U_9, B_9, Q_8, K_8, J_8) → m_{11}
  82. PolyLine(U_9, A_9, C_9, E_9, D_9, B_9) → n_{11}
  83. PolyLine(D_9, P_8, M_8, L_8, Q_8, B_9) → p_{11}
  84. PolyLine(C_9, G_9, Q_9, R_9, F_9, E_9) → q_{11}
  85. PolyLine(A_9, H_9, O_9, P_9, G_9, C_9) → r_{11}
  86. PolyLine(H_9, M_9, I_9, L_9, N_9, O_9) → s_{11}
  87. Segment(I_8, V_8) → t_{11}
  88. Segment(W_8, J_9) → a_{11}
  89. Segment(U_8, Z_8) → b_{11}
  90. Segment(F_9, S_9) → c_{11}
  91. Segment(R_9, T_9) → d_{11}
  92. Segment(M_8, N_8) → e_{11}
  93. Segment(P_8, O_8) → f_{12}
  94. Segment(V_9, J_9) → g_{12}