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Cubes Arranged Around a Torus

Cubes Arranged Around a Torus
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2020-07-20 20:31
SequenceSolid GeometryTransformationParametric EquationGraphing
① This example demonstrates how a series of cubes can be systematically placed along the surface of a torus, creating a decorative 3D arrangement that encircles the ring. ② Core mathematical concepts include parametric surfaces (torus), point generation along space curves, sequence and list operations, and rotational symmetry in solid geometry. ③ Construction highlights: using GeoGebra commands such as Join, Sequence, Surface, and Element, with objects t, b, a, n, O, g, k, m, liste3, Length, e, h, liste5, liste1, c, A, B, image1, C, D, image2, hauteur, l1, l2, Color, and A1 through A15, the torus is defined via parametric equations, and cube positions are generated sequentially on its surface, with color and height variables enhancing the visual effect. ④ Suitable for high school to university-level teaching in mathematics or computer graphics, this example helps students understand parametric surfaces, 3D modeling, and sequence generation, and can also serve as a creative geometric art project.

GeoGebra Commands & Parameters

  1. Join(Sequence[Sequence[(x(O) + ((a + 0.2 + ((b + 0.2) * cos(j + t))) * cos(s)), y(O) + ((a + 0.2 + ((b + 0.2) * cos(j + t))) * sin(s)), z(O) + ((b + 0.2) * sin(j + t))), s, 0, 6.28319, (2 * π / n)], j, 0, 6.28319, (2 * π / n)]) → liste3
  2. Sequence(Polygon[Element[liste3, i], Element[liste3, i + 1], Element[liste3, i + 1 + (2 * n) + e], Element[liste3, i + e + (2 * n)]], i, 1, Length[liste3], h) → liste5
  3. Sequence(Polygon[Element[liste3, i], Element[liste3, i + 1], Element[liste3, i + 1 + e + (2 * n)], Element[liste3, i + e + (2 * n)]], i, 1, Length[liste3], h) → liste1
  4. Surface(x(O) + ((a - 0.05 + ((b - 0.05) * cos(j))) * cos(s)), y(O) + ((a - 0.05 + ((b - 0.05) * cos(j))) * sin(s)), z(O) + ((b - 0.05) * sin(j)), j, 0, Length, s, 0, 6.28319) → c
  5. Sequence(Prism[Element[liste5, i], hauteur], i, 1, Length[liste5]) → l1
  6. Sequence(Prism[Element[liste5, i], hauteur], i, 1, Length[liste5]) → l2
  7. Element(l1, A1) → B1
  8. Element(l1, A2) → B2
  9. Element(l1, A3) → B3
  10. Element(l1, A4) → B4
  11. Element(l1, A5) → B5
  12. Element(l1, A6) → B6
  13. Element(l1, A7) → B7
  14. Element(l1, A8) → B8
  15. Element(l1, A9) → B9
  16. Element(l1, A10) → B10
  17. Element(l1, A11) → B11
  18. Element(l1, A12) → B12
  19. Element(l1, A13) → B13
  20. Element(l1, A14) → B14
  21. Element(l1, A15) → B15
  22. Element(l1, A16) → B16
  23. Element(l1, A17) → B17
  24. Element(l1, A18) → B18
  25. Element(l1, A19) → B19
  26. Element(l1, A20) → B20
  27. Element(l1, A21) → B21
  28. Element(l1, A22) → B22
  29. Element(l1, A23) → B23
  30. Element(l1, A24) → B24
  31. Element(l1, A25) → B25
  32. Element(l1, A26) → B26
  33. Element(l1, A27) → B27
  34. Element(l1, A28) → B28
  35. Element(l1, A29) → B29
  36. Element(l1, A30) → B30
  37. Element(l1, A31) → B31
  38. Element(l1, A32) → B32
  39. Element(l1, A33) → B33
  40. Element(l1, A34) → B34
  41. Element(l1, A35) → B35
  42. Element(l1, A36) → B36
  43. Element(l1, A37) → B37
  44. Element(l1, A38) → B38
  45. Element(l1, A39) → B39
  46. Element(l1, A40) → B40
  47. Element(l1, A41) → B41
  48. Element(l1, A42) → B42
  49. Element(l1, A43) → B43
  50. Element(l1, A44) → B44
  51. Element(l1, A45) → B45
  52. Element(l1, A46) → B46
  53. Element(l1, A47) → B47
  54. Element(l1, A48) → B48
  55. Element(l1, A49) → B49
  56. Element(l1, A50) → B50
  57. Element(l1, A51) → B51
  58. Element(l1, A52) → B52
  59. Element(l1, A53) → B53
  60. Element(l1, A54) → B54
  61. Element(l1, A55) → B55
  62. Element(l1, A56) → B56
  63. Element(l1, A57) → B57
  64. Element(l1, A58) → B58
  65. Element(l1, A59) → B59
  66. Element(l1, A60) → B60
  67. Element(l1, A61) → B61
  68. Element(l1, A62) → B62
  69. Element(l1, A63) → B63
  70. Element(l1, A64) → B64
  71. Element(l1, A65) → B65
  72. Element(l1, A66) → B66
  73. Element(l1, A67) → B67
  74. Element(l1, A68) → B68
  75. Element(l1, A69) → B69
  76. Element(l1, A70) → B70
  77. Element(l1, A71) → B71
  78. Element(l1, A72) → B72
  79. Element(l1, A73) → B73
  80. Element(l1, A74) → B74
  81. Element(l1, A75) → B75
  82. Element(l1, A76) → B76
  83. Element(l1, A77) → B77
  84. Element(l1, A78) → B78
  85. Element(l1, A79) → B79
  86. Element(l1, A80) → B80
  87. Element(l1, A81) → B81
  88. Element(l1, A82) → B82
  89. Element(l1, A83) → B83
  90. Element(l1, A84) → B84
  91. Element(l1, A85) → B85
  92. Element(l1, A86) → B86
  93. Element(l1, A87) → B87
  94. Element(l1, A88) → B88
  95. Element(l1, A89) → B89
  96. Element(l1, A90) → B90
  97. Element(l1, A91) → B91
  98. Element(l1, A92) → B92
  99. Element(l1, A93) → B93
  100. Element(l1, A94) → B94
  101. Element(l1, A95) → B95
  102. Element(l1, A96) → B96
  103. Element(l1, A97) → B97
  104. Element(l1, A98) → B98
  105. Element(l1, A99) → B99
  106. Element(l1, A100) → B100

AI Drawing Prompt Reference

Create a decorative 3D arrangement showing a series of cubes systematically placed along the surface of a torus, forming a ring around it. The torus is defined by parametric equations, and cubes are generated and positioned along its surface using sequences and lists, with rotational symmetry. Variable heights and colors of the cubes enhance the visual effect, resulting in a symmetric and layered artistic composition.
The prompt above was compiled from the original drawing commands and is for reference only.