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Sphere Tiling

Sphere Tiling
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2018-12-07 07:14
TriangleCircleTrigonometrySolid GeometryGeometric Construction
This example demonstrates the tiling of a sphere with equilateral triangles. It involves core mathematical concepts such as spherical geometry, equilateral triangle construction, and angle calculations. GeoGebra commands used include Circle, Intersect, Sphere, Angle, PointIn, CircleArc, Sequence, and Mirror. It is suitable for high school and above students to understand spherical and spatial geometry constructions, as well as for demonstrating random experiments in probability and statistics.

GeoGebra Commands & Parameters

  1. Circle(O, 1, Line[O, (1, 1, sqrt(2))]) → c1
  2. Circle(O, 1, Line[O, (-1, -1, sqrt(2))]) → c4
  3. Circle(O, 1, Line[O, (-1, 1, sqrt(2))]) → c3
  4. Intersect(c3, c4, 1) → A
  5. Circle(O, 1, Line[O, (1, -1, sqrt(2))]) → c2
  6. Intersect(c1, c2, 1) → B
  7. Sphere(O, 1) → sp
  8. Intersect(c2, c3, 2) → C
  9. Angle(B, O, C) → BOC
  10. Angle(A, O, C) → AOC
  11. Intersect(c2, c4, 2) → D
  12. Angle(A, O, D) → AOD
  13. Angle(C, O, D) → COD
  14. Angle(A, O, E) → AOE
  15. Angle(D, O, E) → DOE
  16. Angle(A, O, B) → AOB
  17. PointIn(sp) → P
  18. Angle(A, O, P) → AOP
  19. CircleArc(O, A, P) → ac1
  20. CircleArc(O, B, P) → ac2
  21. CircleArc(O, C, P) → ac3
  22. Circle(O, 1, xAxis) → cx
  23. Circle(O, 1, yAxis) → cy
  24. Circle(O, 1, zAxis) → cz
  25. Sequence(Rotate[l1, (i * 90°), Line[O, C]], i, 0, 3) → l2
  26. Sequence(Rotate[l2, (i * 120°), Line[O, A]], i, 0, 3) → l3
  27. Mirror(l3, O) → l4
  28. Circle(Point[Segment[O, P], p], sqrt(1 - p^(2)), Line[O, P]) → c
  29. Sequence(Rotate[c, (i * 90°), Line[O, C]], i, 0, 3) → lc1
  30. Sequence(Rotate[lc1, (i * 120°), Line[O, A]], i, 0, 3) → lc2
  31. Mirror(lc2, O) → lc4