MathPlotAI Logo MathPlotAI

Starting from a Dodecahedron to Create Islamic Geometric Patterns

Starting from a Dodecahedron to Create Islamic Geometric Patterns
Loading the math board and drawing, please wait…
2019-12-31 20:41
PolygonVectorSolid GeometryTransformationGeometric Construction
This example demonstrates how to construct intricate Islamic geometric patterns starting from a dodecahedron through transformations like rotation and reflection. Key mathematical concepts include symmetry, 3D vectors, polygon construction, intersection calculations involving spheres and segments, and angle measurements. The construction heavily utilizes commands such as Intersect, Point, Circle, Rotate, Polygon, Dodecahedron, Centroid, Midpoint, Sphere, Segment, CircleArc, Angle, Line, and Mirror as outlined in the construction summary. It is suitable for high school and advanced learners exploring solid geometry, vector transformations, and optimization design applications.

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. Midpoint(U, Z) → E_1
  16. Midpoint(Z, Q) → F_1
  17. Midpoint(Q, A_1) → G_1
  18. Midpoint(A_1, R) → H_1
  19. Midpoint(R, B_1) → I_1
  20. Midpoint(B_1, S) → J_1
  21. Midpoint(C_1, S) → K_1
  22. Midpoint(T, C_1) → L_1
  23. Midpoint(T, D_1) → M_1
  24. Midpoint(D_1, U) → N_1
  25. Segment(E_1, H_1) → k
  26. Segment(F_1, M_1) → l
  27. Segment(G_1, J_1) → m
  28. Segment(I_1, L_1) → n
  29. Segment(K_1, N_1) → p
  30. CircleArc(W, U, Q) → d
  31. Angle(U, W, Q) → HoekUQ
  32. Segment(W, Q) → r
  33. Segment(U, W) → q
  34. Intersect(Ray[W, E_1], d) → UQ,
  35. Intersect(Ray[W, F_1], d) → UUQ,
  36. CircleArc(W, Q, R) → e
  37. Intersect(Ray[W, G_1], e) → GR,
  38. Intersect(Ray[W, H_1], e) → RG,
  39. Line((0, 0, 0), (0, 0, 1)) → as
  40. Rotate(e, 72°, as) → e'
  41. Rotate(e, 144°, as) → e'_1
  42. Rotate(e, (-(144°)), as) → e'_2
  43. CircleArc(W, L, Q) → s
  44. CircleArc(W, P, U) → c_1
  45. CircleArc(W, K, L) → d_1
  46. CircleArc(W, P, K) → f_1
  47. CircleArc(W, O, T) → g_1
  48. CircleArc(W, O, J) → h_1
  49. CircleArc(W, J, P) → k_1
  50. CircleArc(W, N, S) → p_1
  51. CircleArc(W, N, I) → q_1
  52. CircleArc(W, I, O) → r_1
  53. CircleArc(W, I, E) → s_1
  54. CircleArc(W, H, N) → t_1
  55. CircleArc(W, M, H) → c_2
  56. CircleArc(W, M, R) → d_2
  57. CircleArc(W, J, F) → e_2
  58. CircleArc(W, K, B) → f_2
  59. CircleArc(W, L, G) → g_2
  60. CircleArc(W, G, C) → h_2
  61. CircleArc(W, G, M) → k_2
  62. CircleArc(W, H, D) → p_2
  63. CircleArc(W, D, E, xOyPlane) → r_2
  64. CircleArc(W, E, F) → s_2
  65. CircleArc(W, F, B) → t_2
  66. CircleArc(W, B, C) → c_3
  67. CircleArc(W, C, D) → d_3
  68. Rotate(GR, 144°, as) → GR'
  69. Rotate(GR, 72°, as) → GR'_1
  70. Rotate(GR, (-(144°)), as) → GR'_2
  71. Rotate(RG, 144°, as) → RG'
  72. Rotate(RG, (-(144°)), as) → RG'_1
  73. Rotate(RG, 72°, as) → RG'_2
  74. CircleArc(W, GR, RG'_2, xOyPlane) → t
  75. Rotate(t, 72°, as) → t'
  76. Rotate(t, 144°, as) → t'_1
  77. Rotate(t, (-(144°)), as) → t'_2
  78. CircleArc(W, UQ, RG) → e_1
  79. Mirror(ring, W) → ring'
  80. Centroid(zijvlakKLQUP) → O_1
  81. Angle(O_1, W, V) → hoekhoek
  82. Line(O_1, W) → asas
  83. Rotate(ring, 72°, asas) → ring'_1
  84. Rotate(ring'_1, 72°, asas) → l1
  85. Rotate(ring'_1, 72°, as) → l2
  86. Rotate(ring'_1, 144°, as) → l3
  87. Rotate(ring'_1, (-(72°)), as) → l4
  88. Rotate(ring'_1, (-(144°)), as) → l5
  89. Rotate(l1, (-(144°)), as) → l1'
  90. Rotate(l1, 144°, as) → l1'_1
  91. Rotate(l1, 72°, as) → l1'_2
  92. Rotate(l1, (-(72°)), as) → l1'_3
  93. Mirror(vlakring, W) → vlakring'
  94. Rotate(vlakring, 72°, asas) → vlakring'_1
  95. Rotate(vlakring'_1, 72°, asas) → l6
  96. Rotate(vlakring'_1, 72°, as) → l7
  97. Rotate(vlakring'_1, (-(72°)), as) → l8
  98. Rotate(vlakring'_1, (-(144°)), as) → l9
  99. Rotate(vlakring'_1, 144°, as) → l10
  100. Rotate(l6, 144°, as) → l6'
  101. Rotate(l6, (-(144°)), as) → l6'_1
  102. Rotate(l6, (-(72°)), as) → l6'_2
  103. Rotate(l6, 72°, as) → l6'_3

AI Drawing Prompt Reference

Start with a dodecahedron and generate intricate Islamic geometric patterns through rotations and reflections. First, create a circle, then rotate a point to form a pentagon, and construct a dodecahedron. Find the centroid of one of the dodecahedron's faces, use it as the center to construct multiple arcs and segments, and finally rotate these arcs to create a beautiful geometric pattern.
The prompt above was compiled from the original drawing commands and is for reference only.