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Islamic Geometric Patterns Construction

Islamic Geometric Patterns Construction
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2019-12-31 20:45
CirclePolygonSolid GeometryTransformationGeometric Construction
This example demonstrates the construction of Islamic geometric patterns. It involves core mathematical concepts such as symmetry, polygon construction, rotation, and reflection transformations. Key GeoGebra commands and objects used include Intersect, Point, Circle, Rotate, Polygon, Dodecahedron, Centroid, Midpoint, Sphere, CircleArc, Angle, Segment, Line, and Mirror. Suitable for high school and above students to understand geometric transformations and pattern design.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Circle(A, B) → c
  4. Rotate(B, 72°) → C
  5. Polygon(B, C, 5) → veelhoek1, f, g, h, i, j, D, E, F
  6. Dodecahedron(veelhoek1, true) → a, G, P, Q, R, S, T, U, zijvlakBFJPK, zijvlakBCGLK, zijvlakCDHMG, zijvlakDEINH, H, zijvlakEFJOI, zijvlakJOTUP, zijvlakKLQUP, zijvlakGLQRM, zijvlakHMRSN, zijvlakINSTO, zijvlakQRSTU, RibbeBK, RibbeFJ, RibbeJP, I, RibbeKP, RibbeKL, RibbeGL, RibbeCG, RibbeGM, RibbeHM, RibbeDH, RibbeHN, RibbeIN, RibbeEI, J, RibbeIO, RibbeJO, RibbeOT, RibbeTU, RibbePU, RibbeQU, RibbeLQ, RibbeQR, RibbeMR, RibbeRS, K, RibbeNS, RibbeST, L, M, N, O
  7. Centroid(zijvlakQRSTU) → V
  8. Midpoint(A, V) → W
  9. Sphere(W, B) → b
  10. Midpoint(RibbeQU) → Z
  11. Midpoint(RibbeQR) → A_1
  12. Midpoint(RibbeRS) → B_1
  13. Midpoint(RibbeST) → C_1
  14. Midpoint(RibbeTU) → D_1
  15. CircleArc(W, U, Q) → d
  16. Angle(U, W, Q) → HoekUQ
  17. Segment(W, Q) → r
  18. Segment(U, W) → q
  19. CircleArc(W, Q, R) → e
  20. Line((0, 0, 0), (0, 0, 1)) → as
  21. Rotate(e, 72°, as) → e'
  22. Rotate(e, 144°, as) → e'_1
  23. Rotate(e, (-(144°)), as) → e'_2
  24. CircleArc(W, L, Q) → s
  25. CircleArc(W, P, U) → c_1
  26. CircleArc(W, K, L) → d_1
  27. CircleArc(W, P, K) → f_1
  28. CircleArc(W, O, T) → g_1
  29. CircleArc(W, O, J) → h_1
  30. CircleArc(W, J, P) → k_1
  31. CircleArc(W, N, S) → p_1
  32. CircleArc(W, N, I) → q_1
  33. CircleArc(W, I, O) → r_1
  34. CircleArc(W, I, E) → s_1
  35. CircleArc(W, H, N) → t_1
  36. CircleArc(W, M, H) → c_2
  37. CircleArc(W, M, R) → d_2
  38. CircleArc(W, J, F) → e_2
  39. CircleArc(W, K, B) → f_2
  40. CircleArc(W, L, G) → g_2
  41. CircleArc(W, G, C) → h_2
  42. CircleArc(W, G, M) → k_2
  43. CircleArc(W, H, D) → p_2
  44. CircleArc(W, D, E, xOyPlane) → r_2
  45. CircleArc(W, E, F) → s_2
  46. CircleArc(W, F, B) → t_2
  47. CircleArc(W, B, C) → c_3
  48. CircleArc(W, C, D) → d_3
  49. Midpoint(RibbeQU) → E_1
  50. Rotate(Q, 54°, Line[A_1, A_1 + (0, 0, 1)]) → Q'
  51. Rotate(R, 54°, Line[A_1, A_1 + (0, 0, 1)]) → R'
  52. Segment(Q', R') → RibbeQR'
  53. Rotate(R, (-(54°)), Line[B_1, B_1 + (0, 0, 1)]) → R'_1
  54. Rotate(S, (-(54°)), Line[B_1, B_1 + (0, 0, 1)]) → S'
  55. Segment(R'_1, S') → RibbeRS'
  56. Intersect(RibbeQR', RibbeRS') → F_1
  57. Segment(A_1, F_1) → k
  58. Segment(F_1, B_1) → l
  59. Rotate(seg, 72°, as) → seg'
  60. Rotate(seg', 72°, as) → seg''
  61. Rotate(seg', 144°, as) → seg''_1
  62. Rotate(seg', (-(144°)), as) → seg''_2
  63. Rotate(seg', (-(72°)), as) → seg''_3
  64. Mirror(ring, W) → ring'
  65. Centroid(zijvlakGLQRM) → G_1
  66. Line(W, G_1) → asas
  67. Rotate(ring, 72°, asas) → ring'_1
  68. Rotate(ring, 144°, asas) → ring'_2
  69. Rotate(ring, (-(144°)), asas) → ring'_3
  70. Rotate(ring, (-(72°)), asas) → ring'_4
  71. Rotate(ring'_1, 72°, as) → l1
  72. Rotate(ring'_1, (-(72°)), as) → l2
  73. Rotate(l2, (-(72°)), as) → l2'
  74. Rotate(ring'_3, 72°, as) → l3
  75. Rotate(l3, 72°, as) → l3'
  76. Rotate(l3, 144°, as) → l3'_1
  77. Intersect(Ray[W, F_1], b) → H_1,
  78. Segment(H_1, W) → m
  79. Intersect(Ray[W, A_1], b) → I_1,
  80. CircleArc(W, I_1, H_1) → p
  81. Intersect(Ray[W, B_1], b) → J_1,
  82. CircleArc(W, H_1, J_1) → t
  83. Rotate(hoekje, 72°, as) → hoekje'
  84. Rotate(hoekje', 144°, as) → hoekje''
  85. Rotate(hoekje', (-(72°)), as) → hoekje''_1
  86. Rotate(hoekje', 72°, as) → hoekje''_2
  87. Rotate(hoekje', (-(144°)), as) → hoekje''_3
  88. Rotate(bolring, 72°, asas) → bolring'
  89. Rotate(bolring, (-(72°)), asas) → bolring'_1
  90. Rotate(bolring, (-(144°)), asas) → bolring'_2
  91. Rotate(bolring, 144°, asas) → bolring'_3
  92. Rotate(bolring, 144°, asas) → bolring'_4
  93. Rotate(bolring', 72°, as) → bolring''
  94. Rotate(bolring', (-(72°)), as) → bolring''_1
  95. Rotate(bolring'_1, 72°, as) → l4
  96. Rotate(bolring'_2, 72°, as) → l5
  97. Rotate(bolring'_2, 144°, as) → l6
  98. Rotate(l6, 72°, as) → l6'
  99. Mirror(bolring, W) → bolring'_5