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Girih Pattern Puzzle: Interactive Exploration of Symmetry and Geometric Transfor

Girih Pattern Puzzle: Interactive Exploration of Symmetry and Geometric Transfor
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2020-06-09 14:20
PolygonSequenceTransformationGeometric ConstructionLocus
In collections islamic geometric patterns
This GeoGebra example demonstrates a puzzle based on the traditional Islamic girih geometric pattern, constructed using commands such as Point, Polygon, Midpoint, Segment, Intersect, OrthogonalLine, Mirror, Centroid, Sequence, Rotate, Ray, and PolyLine. The core mathematical concepts involve geometric transformations (rotation and reflection), symmetry, polygon and centroid properties, and iterative pattern generation via sequences. The construction begins by defining base points A and B, creating the initial polygon veelhoek1, then progressively builds auxiliary lines, perpendiculars, and mirrored elements using Midpoint, Segment, Intersect, OrthogonalLine, Mirror, and Centroid. Finally, Sequence and Rotate generate repeated rotated motifs, while Ray and PolyLine complete the intricate tessellation. Numerous named objects (f through t, l1, veelhoek2, etc.) allow precise control over each geometric element. This example is suitable for upper secondary to early university students, helping them understand how transformations create symmetric designs, and serves as an excellent cross-disciplinary resource linking mathematics with art and cultural heritage.

GeoGebra Commands & Parameters

  1. Point(Circle[A, 1]) → B
  2. Polygon(A, B, 10) → veelhoek1, f, p, C, D, E, F, G, H, I, J, g, h, i, j, k, l, m, n
  3. Midpoint(f) → K
  4. Midpoint(i) → L
  5. Midpoint(g) → M
  6. Midpoint(h) → O
  7. Midpoint(p) → P
  8. Segment(K, L) → q
  9. Segment(O, P) → r
  10. Intersect(q, r) → N
  11. OrthogonalLine(K, f) → s
  12. Mirror(N, s) → N'
  13. Segment(N', K) → t
  14. Segment(K, N) → a
  15. Centroid(veelhoek1) → Q
  16. Sequence(Rotate[{a, t}, (nn * 36°), Q], nn, 1, 10) → l1
  17. Point(Circle[R, 1]) → S
  18. Polygon(R, S, 5) → veelhoek2, b, c, d, e, f_1, T, U, V
  19. Midpoint(b) → W
  20. Rotate(S, 54°, W) → S'
  21. Midpoint(c) → Z
  22. Ray(W, S') → g_1
  23. Rotate(S, (-(54°)), Z) → S'_1
  24. Ray(Z, S'_1) → h_1
  25. Intersect(g_1, h_1) → A_1
  26. Segment(W, A_1) → i_1
  27. Segment(A_1, Z) → j_1
  28. Centroid(veelhoek2) → B_1
  29. Sequence(Rotate[{i_1, j_1}, (nn * 72°), B_1], nn, 1, 5) → l2
  30. Point(Circle[C_1, 1]) → D_1
  31. Rotate(C_1, 144°, D_1) → E_1
  32. Rotate(D_1, 72°, C_1) → F_1
  33. Rotate(D_1, (-(72°)), E_1) → G_1
  34. Rotate(C_1, 72°, F_1) → H_1
  35. Polygon(F_1, C_1, D_1, E_1, G_1, H_1) → veelhoek3, k_1, c_1, d_1, e_1, l_1, m_1
  36. Midpoint(m_1) → I_1
  37. Midpoint(k_1) → J_1
  38. Midpoint(c_1) → K_1
  39. Midpoint(l_1) → L_1
  40. Midpoint(e_1) → M_1
  41. Midpoint(d_1) → N_1
  42. Segment(J_1, K_1) → n_1
  43. Rotate(J_1, (-(72°)), K_1) → O_1
  44. Rotate(K_1, (-(72°)), K_1) → P_1
  45. Segment(O_1, P_1) → p_1
  46. Segment(J_1, M_1) → q_1
  47. Intersect(p_1, q_1) → Q_1
  48. Segment(H_1, D_1) → r_1
  49. Mirror(Q_1, r_1) → R_1
  50. PolyLine(J_1, K_1, Q_1, I_1, J_1) → s_1
  51. Mirror(J_1, r_1) → S_1
  52. Mirror(K_1, r_1) → T_1
  53. Mirror(Q_1, r_1) → U_1
  54. Mirror(I_1, r_1) → V_1
  55. Mirror(J_1, r_1) → W_1
  56. PolyLine(S_1, T_1, U_1, V_1, W_1) → t_1
  57. Point(Circle[Z_1, 1]) → A_2
  58. Rotate(Z_1, 144°, A_2) → B_2
  59. Rotate(A_2, (-(144°)), Z_1) → C_2
  60. Segment(B_2, C_2) → a_1
  61. Mirror(A_2, a_1) → D_2
  62. Mirror(Z_1, a_1) → E_2
  63. Polygon(B_2, A_2, Z_1, C_2, E_2, D_2) → veelhoek4, b_2, a_2, z_1, c_2, e_2, d_2
  64. Midpoint(b_2) → F_2
  65. Midpoint(d_2) → G_2
  66. Midpoint(a_2) → H_2
  67. Midpoint(z_1) → I_2
  68. Rotate(A_2, 54°, F_2) → J_2
  69. Ray(F_2, J_2) → b_1
  70. Rotate(A_2, (-(54°)), H_2) → K_2
  71. Ray(H_2, K_2) → f_2
  72. Intersect(b_1, f_2) → L_2
  73. Mirror(L_2, a_1) → M_2
  74. Midpoint(e_2) → N_2
  75. Segment(H_2, N_2) → g_2
  76. Mirror(L_2, g_2) → O_2
  77. Mirror(M_2, g_2) → P_2
  78. Midpoint(c_2) → Q_2
  79. PolyLine(F_2, L_2, F_2, H_2, O_2, I_2, Q_2, P_2, N_2, M_2, G_2) → h_2
  80. Point(Circle[R_2, 1]) → S_2
  81. Rotate(R_2, 72°, S_2) → T_2
  82. Rotate(S_2, (-(108°)), R_2) → U_2
  83. Polygon(S_2, R_2, U_2, T_2) → vierhoek1, s_2, r_2, u_2, t_2
  84. Midpoint(s_2) → V_2
  85. Midpoint(t_2) → W_2
  86. Midpoint(u_2) → Z_2
  87. Midpoint(r_2) → A_3
  88. Rotate(W_2, (-(72°)), V_2) → B_3
  89. Ray(V_2, B_3) → i_2
  90. Segment(R_2, T_2) → j_2
  91. Mirror(B_3, j_2) → C_3
  92. Ray(A_3, C_3) → k_2
  93. Intersect(i_2, j_2) → D_3
  94. Segment(S_2, U_2) → l_2
  95. Mirror(D_3, l_2) → E_3
  96. PolyLine(V_2, D_3, A_3, Z_2, E_3, W_2, V_2) → m_2

AI Drawing Prompt Reference

Draw a puzzle based on the traditional Islamic Girih geometric pattern, showcasing symmetry and geometric transformations. Start by defining two base points and creating a decagon, then use midpoints, segments, intersections, perpendiculars, reflections, and centroids to construct auxiliary lines. Generate repeated rotated motifs using sequences and rotations, and complete the intricate tessellation with rays and polylines. The design should illustrate how geometric transformations create symmetric patterns.
The prompt above was compiled from the original drawing commands and is for reference only.