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Icosahedron from Golden Rectangles

Icosahedron from Golden Rectangles
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2017-04-20 03:16
Solid GeometryTransformationGeometric Construction
This example demonstrates the construction of an icosahedron using three golden rectangles through rotation, mirroring, and polygon creation. Key mathematical concepts include the golden ratio, solid geometry, and geometric transformations. The construction utilizes GeoGebra commands like Rotate, Mirror, and Polygon, along with objects such as k, p, A, A'1, A', A''. It is suitable for high school students and above, as well as for mathematical competitions or enrichment activities.

GeoGebra Commands & Parameters

  1. Rotate(A, (-(90°)), yAxis) → A'_1
  2. Mirror(A, xOyPlane) → A'
  3. Rotate(A', (-(90°)), yAxis) → A''
  4. Rotate(B, (-(90°)), yAxis) → B'_1
  5. Mirror(B, xOyPlane) → B'
  6. Rotate(B', (-(90°)), yAxis) → B''
  7. Polygon(A'_1, B'_1, B'', A'') → poly2, a'_1, b'_1, b'', a''
  8. Polygon(A, B, B', A') → poly1, a, b, b', a'
  9. Rotate(poly1, (-(((k * 90))°)), yAxis) → poly1'
  10. Rotate(poly2, (((k - 1) * 90))°, zAxis) → poly2'
  11. Rotate(A'', 90°, zAxis) → A'''
  12. Rotate(B'', 90°, zAxis) → B'''
  13. Rotate(B'_1, 90°, zAxis) → C
  14. Rotate(A'_1, 90°, zAxis) → D
  15. Polygon(D, A''', B''', C) → poly3, d, a''', b''', c
  16. Rotate(poly3, (-((((k - 2) * 90))°)), yAxis) → poly3'
  17. Rotate(D, (-(90°)), yAxis) → D'
  18. Rotate(C, (-(90°)), yAxis) → C'
  19. Rotate(A''', (-(90°)), yAxis) → E
  20. Rotate(B''', (-(90°)), yAxis) → F
  21. Polygon(D', E, F, C') → poly4, d', e, f, c'
  22. Rotate(poly4, (((k - 3) * 90))°, zAxis) → poly4'
  23. Rotate(E, 90°, zAxis) → E'
  24. Rotate(F, 90°, zAxis) → F'
  25. Rotate(C', 90°, zAxis) → C''
  26. Rotate(D', 90°, zAxis) → D''
  27. Polygon(B, A, D) → poly5, d_1, b_1, a_1
  28. Polygon(B, A''', A) → poly6, a_2, b_2, a'''_1
  29. Polygon(A', B''', B') → poly7, b'_2, a'_2, b'''_1
  30. Polygon(A', C, B') → poly8, b'_3, a'_3, c_1
  31. Polygon(A, D'', C'') → poly9, c'', a_3, d''
  32. Polygon(D'', C'', A') → poly10, a'_4, d''_1, c''_1
  33. Polygon(F', B, E') → poly11, e', f', b_3
  34. Polygon(F', E', B') → poly12, b'_4, f'_1, e'_1
  35. Polygon(D, C, D'') → poly13, d''_2, d_2, c_2
  36. Polygon(D, E', C) → poly14, c_3, d_3, e'_2
  37. Polygon(A''', F', B''') → poly15, b'''_2, a'''_2, f'_2
  38. Polygon(A''', B''', C'') → poly16, c''_2, a'''_3, b'''_3
  39. Polygon(C'', F', E', D'') → poly17, c''_3, f'_3, e'_3, d''_3
  40. Polygon(A''', B, F') → poly18, f'_4, a'''_4, b_4
  41. Polygon(A, D, D'') → poly19, d''_4, a_4, d_4
  42. Polygon(B, D, E') → poly20, e'_4, b_5, d_5
  43. Polygon(A, A''', C'') → poly21, c''_4, a_5, a'''_5
  44. Polygon(C'', B''', A') → poly22, a'_5, c''_5, b'''_4
  45. Polygon(C, D'', A') → poly23, a'_6, c_4, d''_5
  46. Polygon(C, B', E') → poly24, e'_5, c_5, b'_5