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Slicing a Sphere into 8 Equal Pieces

Slicing a Sphere into 8 Equal Pieces
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2019-12-24 10:06
VectorTrigonometrySolid GeometryCalculusGeometric Construction
This example demonstrates how to slice a sphere into 8 equal volume pieces using specific planes. Key math concepts include solid geometry, sphere surface area, volume calculation, and spatial geometric transformations. Construction highlights involve commands like Surface, Mirror, Dilate, Vector, and Translate in GeoGebra. Suitable for high school and above students learning solid geometry, spatial visualization, and geometric constructions.

GeoGebra Commands & Parameters

  1. Surface(((r * cos(u)) * sin(v)), ((r * sin(u)) * sin(v)), (r * cos(v)), u, 0, π / 2, v, 0, π / 2) → a
  2. Mirror(a, eq1) → a'
  3. Mirror(a', f) → a''
  4. Mirror(a, f) → a'_1
  5. Mirror(a, g) → a'_2
  6. Mirror(a', g) → a''_1
  7. Mirror(a'', g) → a'''
  8. Mirror(a'_1, g) → c
  9. Surface(((r * v) * cos(u)), 0, ((r * v) * sin(u)), u, 0, π / 2, v, 0, 1) → b
  10. Surface(((r * v) * cos(u)), 0, ((r * v) * sin(u)), u, π / 2, π, v, 0, 1) → d
  11. Surface(((r * v) * cos(u)), 0, ((r * v) * sin(u)), u, π, ((3 * π)) / 2, v, 0, 1) → e
  12. Surface(((r * v) * cos(u)), 0, ((r * v) * sin(u)), u, ((3 * π)) / 2, (2 * π), v, 0, 1) → h
  13. Surface(0, ((r * v) * cos(u)), ((r * v) * sin(u)), u, 0, π / 2, v, 0, 1) → i
  14. Surface(0, ((r * v) * cos(u)), ((r * v) * sin(u)), u, π / 2, π, v, 0, 1) → j
  15. Surface(0, ((r * v) * cos(u)), ((r * v) * sin(u)), u, π, ((3 * π)) / 2, v, 0, 1) → l
  16. Surface(0, ((r * v) * cos(u)), ((r * v) * sin(u)), u, ((3 * π)) / 2, (2 * π), v, 0, 1) → m
  17. Surface(((r * v) * cos(u)), ((r * v) * sin(u)), 0, u, 0, π / 2, v, 0, 1) → n
  18. Surface(((r * v) * cos(u)), ((r * v) * sin(u)), 0, u, π / 2, π, v, 0, 1) → o
  19. Surface(((r * v) * cos(u)), ((r * v) * sin(u)), 0, u, π, ((3 * π)) / 2, v, 0, 1) → p
  20. Surface(((r * v) * cos(u)), ((r * v) * sin(u)), 0, u, ((3 * π)) / 2, (2 * π), v, 0, 1) → q
  21. Dilate(A, If[0 < SplitApart ≤ 1, 1 - SplitApart, 0], B) → A'
  22. Vector(A, A') → u
  23. Translate(a, u) → a'_3
  24. Translate(b, u) → b'
  25. Translate(i, u) → i'
  26. Translate(n, u) → n'
  27. Mirror(A', A) → A''
  28. Vector(A, A'') → v
  29. Translate(a''', v) → k
  30. Mirror(n', eq2) → n''
  31. Mirror(i', eq2) → i''
  32. Mirror(b', eq2) → b''
  33. Mirror(a'_3, eq2) → t
  34. Translate(e, v) → e'
  35. Translate(l, v) → l'
  36. Translate(p, v) → p'
  37. Mirror(p', z = 0) → p''
  38. Mirror(l', z = 0) → l''
  39. Mirror(e', z = 0) → e''
  40. Mirror(k, z = 0) → k'
  41. Mirror(k', eq2) → k''
  42. Mirror(e'', eq2) → e'''
  43. Mirror(p'', eq2) → p'''
  44. Mirror(l'', eq2) → l'''
  45. Mirror(l''', z = 0) → w
  46. Mirror(e''', z = 0) → a_1
  47. Mirror(p''', z = 0) → b_1
  48. Mirror(k'', z = 0) → k'''
  49. Mirror(k''', s) → c_1
  50. Mirror(b_1, s) → d_1
  51. Mirror(a_1, s) → e_1
  52. Mirror(w, s) → w'
  53. Mirror(w', eq2) → w''
  54. Mirror(e_1, eq2) → f_1
  55. Mirror(d_1, eq2) → g_1
  56. Mirror(c_1, eq2) → h_1