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Organic Columns: Dynamic Geometric Prism Structure

Organic Columns: Dynamic Geometric Prism Structure
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2018-10-16 19:02
PolygonSolid GeometryTransformationParametric EquationSlider Animation
① This example demonstrates an organic columnar structure composed of multiple prisms, whose rotation, translation, and size are controlled by parameters, creating a dynamic, growth-like geometric form. ② Core mathematical concepts include solid geometry (prisms), translation and reflection transformations, conditional probability (possibly for random distribution control), perimeter calculation, and parametric curves. ③ Construction highlights: uses GeoGebra commands such as CurveCartesian for parametric curves, Polygon and Prism for building faces and solids, and Sphere for adding spherical elements; objects include parameters size, rot, si, co, a button (button1), and a slider (Speed) for interactivity; vertices (A, B, C, D, etc.), faces (faceABEH, etc.), and edges (edgeAH, etc.) are explicitly defined to construct the complete prism structure. ④ Suitable for high school or college-level solid geometry and dynamic geometry teaching, useful for exploring parametric modeling, transformation combinations, and spatial visualization, as well as a mathematical art creation example.

GeoGebra Commands & Parameters

  1. CurveCartesian((((co * size) * (1 - i)) * cos((1 / 2 * i))) - (((si * size) * (1 - i)) * sin((1 / 2 * i))), (((si * size) * (1 - i)) * cos((1 / 2 * i))) + (((co * size) * (1 - i)) * sin((1 / 2 * i))), i + (8 * π), i, (-8 * π), 0) → a
  2. CurveCartesian(((1 / 5 * (1 - i)) * cos((1 / 2 * i))), ((1 / 5 * (1 - i)) * sin((1 / 2 * i))), i + (8 * π), i, (-8 * π), 0) → b
  3. Polygon(A, B, C, D) → q1, a_1, b_1, c, d
  4. Prism(q1, H) → e, E, edgeBE, edgeCF, edgeDG, edgeEH, edgeEF, edgeFG, edgeGH, F, G, faceABEH, faceBCFE, faceCDGF, faceADGH, faceEFGH, edgeAH
  5. Sphere(a((0 * (-8 * π / n))) + (-5, 0, 0), sqrt(0.7)) → A0
  6. Sphere(a((-8 * π / n)) + (-5, 0, 0), sqrt((0.7 * 2))) → A1
  7. Sphere(a((2 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 3))) → A2
  8. Sphere(a((3 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 4))) → A3
  9. Sphere(a((4 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 5))) → A4
  10. Sphere(a((5 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 6))) → A5
  11. Sphere(a((6 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 7))) → A6
  12. Sphere(a((7 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 8))) → A7
  13. Sphere(a((8 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 9))) → A8
  14. Sphere(a((9 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 10))) → A9
  15. Sphere(a((10 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 11))) → A10
  16. Sphere(a((11 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 12))) → A11
  17. Sphere(a((12 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 13))) → A12
  18. Sphere(a((13 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 14))) → A13
  19. Sphere(a((14 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 15))) → A14
  20. Sphere(a((15 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 16))) → A15
  21. Sphere(a((16 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 17))) → A16
  22. Sphere(a((17 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 18))) → A17
  23. Sphere(a((18 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 19))) → A18
  24. Sphere(a((19 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 20))) → A19
  25. Sphere(a((20 * (-8 * π / n))) + (-5, 0, 0), sqrt((0.7 * 21))) → A20
  26. Sphere(a((0 * (-8 * π / n))) + (5, 5, 0), sqrt(0.7)) → B0
  27. Sphere(a((-8 * π / n)) + (5, 5, 0), sqrt((0.7 * 2))) → B1
  28. Sphere(a((2 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 3))) → B2
  29. Sphere(a((3 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 4))) → B3
  30. Sphere(a((4 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 5))) → B4
  31. Sphere(a((5 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 6))) → B5
  32. Sphere(a((6 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 7))) → B6
  33. Sphere(a((7 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 8))) → B7
  34. Sphere(a((8 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 9))) → B8
  35. Sphere(a((9 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 10))) → B9
  36. Sphere(a((10 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 11))) → B10
  37. Sphere(a((11 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 12))) → B11
  38. Sphere(a((12 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 13))) → B12
  39. Sphere(a((13 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 14))) → B13
  40. Sphere(a((14 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 15))) → B14
  41. Sphere(a((15 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 16))) → B15
  42. Sphere(a((16 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 17))) → B16
  43. Sphere(a((17 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 18))) → B17
  44. Sphere(a((18 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 19))) → B18
  45. Sphere(a((19 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 20))) → B19
  46. Sphere(a((20 * (-8 * π / n))) + (5, 5, 0), sqrt((0.7 * 21))) → B20
  47. Sphere(a((0 * (-8 * π / n))) + (5, -5, 0), sqrt(0.7)) → C0
  48. Sphere(a((-8 * π / n)) + (5, -5, 0), sqrt((0.7 * 2))) → C1
  49. Sphere(a((2 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 3))) → C2
  50. Sphere(a((3 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 4))) → C3
  51. Sphere(a((4 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 5))) → C4
  52. Sphere(a((5 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 6))) → C5
  53. Sphere(a((6 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 7))) → C6
  54. Sphere(a((7 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 8))) → C7
  55. Sphere(a((8 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 9))) → C8
  56. Sphere(a((9 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 10))) → C9
  57. Sphere(a((10 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 11))) → C10
  58. Sphere(a((11 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 12))) → C11
  59. Sphere(a((12 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 13))) → C12
  60. Sphere(a((13 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 14))) → C13
  61. Sphere(a((14 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 15))) → C14
  62. Sphere(a((15 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 16))) → C15
  63. Sphere(a((16 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 17))) → C16
  64. Sphere(a((17 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 18))) → C17
  65. Sphere(a((18 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 19))) → C18
  66. Sphere(a((19 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 20))) → C19
  67. Sphere(a((20 * (-8 * π / n))) + (5, -5, 0), sqrt((0.7 * 21))) → C20

AI Drawing Prompt Reference

Create an organic columnar structure with two intertwined three-dimensional spiral curves, where the main curve is controlled by rotation and scaling parameters to form a twisting, plant-like growth form. Along the path of the main curve, arrange a series of spheres with gradually increasing sizes from bottom to top, with two groups of spheres extending in different directions at both ends. In the center, a quadrilateral base rises vertically to form a prism structure with clearly visible faces. The entire shape presents a dynamic growth sensation and can be controlled via sliders for rotation and size changes.
The prompt above was compiled from the original drawing commands and is for reference only.