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Icosahedron: Golden Rectangle & Tensegrity

Icosahedron: Golden Rectangle & Tensegrity
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2017-05-11 05:17
VectorCoordinate GeometrySolid GeometryGeometric Construction
In collections Polyeder, whose coordinates are golden numbers
This example demonstrates the geometric relationship between an icosahedron and a golden rectangle, highlighting tensegrity structures. Key math concepts include the golden ratio, polygon construction, vector operations, and analytical geometry. GeoGebra commands and objects used are Intersect, Polynomial, Plane, Mirror, Polygon, Point, Cube, Rotate, Translate, and PointIn. Suitable for high school students and geometry enthusiasts exploring spatial geometry and aesthetic principles.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → Z
  2. Polynomial(Lp) → f
  3. Plane(zAxis, xAxis) → b
  4. Mirror(I, b) → I'
  5. Plane(xAxis, yAxis) → c
  6. Mirror(I', c) → I''
  7. Mirror(I, c) → I'_1
  8. Polygon(I'', I'_1, I, I') → rec1, i'', i'_1, i, i'
  9. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  10. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  11. Mirror(I'_1, c) → J
  12. Rotate(rec1, 90°, xAxis) → poly1'
  13. Rotate(poly1', -90°, zAxis) → rec2
  14. Rotate(rec1, (-(90°)), yAxis) → poly1'_1
  15. Rotate(poly1'_1, 90°, zAxis) → rec3
  16. Translate(rec1, Vector[(f(t) * v)]) → rec1'
  17. Translate(rec2, Vector[(f(t) * u)]) → rec2'
  18. Translate(rec3, Vector[(f(t) * w)]) → rec3'
  19. PointIn(rec1') → K
  20. PointIn(rec1') → L
  21. PointIn(rec2') → M
  22. Polygon(K, L, M) → poly1, m, k, l
  23. PointIn(rec2') → N
  24. PointIn(rec2') → O
  25. PointIn(rec2') → P
  26. PointIn(rec1') → Q
  27. PointIn(rec1') → R
  28. PointIn(rec3') → S
  29. PointIn(rec3') → T
  30. PointIn(rec3') → U
  31. PointIn(rec3') → V
  32. Polygon(L, K, N) → poly2, n, l_1, k_1
  33. Polygon(M, U, P) → poly3, p, m_1, u_1
  34. Polygon(M, V, P) → poly4, p_1, m_2, v_1
  35. Polygon(N, S, O) → poly5, o, n_1, s
  36. Polygon(Q, P, R) → poly6, r, q, p_2
  37. Polygon(Q, O, R) → poly7, r_1, q_1, o_1
  38. Polygon(K, S, U) → poly8, u_2, k_2, s_1
  39. Polygon(S, U, Q) → poly9, q_2, s_2, u_3
  40. Polygon(L, V, T) → poly10, t_1, l_2, v_2
  41. Polygon(N, T, O) → poly11, o_2, n_2, t_2
  42. Polygon(V, T, R) → poly12, r_2, v_3, t_3

AI Drawing Prompt Reference

Draw an icosahedron and demonstrate its relationship with a golden rectangle. Start by constructing a golden rectangle, then use mirror and translation operations to generate multiple rectangles and polygons. Finally, apply rotation and translation operations to create a series of polygons, showcasing the structure and aesthetic principles of the icosahedron.
The prompt above was compiled from the original drawing commands and is for reference only.