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Density of Sphere Packing in a Cube

Density of Sphere Packing in a Cube
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2017-05-12 22:18
Solid GeometryGeometric Construction
In collections surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1
This example demonstrates the density of sphere packing in a cube. Key mathematical concepts include solid geometry and volume calculation. Using GeoGebra commands such as Surface, Translate, Point, Cube, Sphere, and Segment, a model of sphere packing within a cube is constructed. Suitable for high school or university-level solid geometry studies.

GeoGebra Commands & Parameters

  1. Surface((sin(u) * cos(v)), (sin(u) * sin(v)), cos(u), u, 0, π / 2, v, 0, (2 * π)) → HK
  2. Translate(Rotate[HK, 90°, xAxis], Vector[(0, sqrt(2), 0)]) → HK_1
  3. Translate(Rotate[HK, -90°, xAxis], Vector[(0, (-sqrt(2)), 0)]) → HK_2
  4. Point(Circle[B, Distance[A, B], Segment[A, B]]) → E
  5. Cube(A, B, E) → a, F, FlächeEFJI, FlächeGHIJ, KanteAF, KanteAB, KanteBE, KanteEF, KanteAG, KanteFJ, KanteGJ, KanteGH, G, KanteBH, KanteHI, KanteEI, KanteIJ, H, I, J, FlächeABEF, FlächeAFJG, FlächeABHG, FlächeBEIH
  6. Translate(Rotate[HK, 90°, yAxis], Vector[((-sqrt(2)), 0, 0)]) → HK_3
  7. Translate(Rotate[HK, -90°, yAxis], Vector[(sqrt(2), 0, 0)]) → HK_4
  8. Sphere((0, sqrt(2), 0), 1) → K_1
  9. Sphere((0, (-sqrt(2)), 0), 1) → K_2
  10. Sphere(((-sqrt(2)), 0, 0), 1) → K_3
  11. Sphere((sqrt(2), 0, 0), 1) → K_4
  12. Translate(HK, Vector[(0, 0, (-sqrt(2)))]) → HK_5
  13. Translate(Rotate[HK, 180°, yAxis], Vector[(0, 0, sqrt(2))]) → HK_6
  14. Sphere((0, 0, (-sqrt(2))), 1) → K_5
  15. Sphere((0, 0, sqrt(2)), 1) → K_6
  16. Surface((-sqrt(2)) + (sin(u) * sin(v)), (-sqrt(2)) + (sin(u) * cos(v)), (-sqrt(2)) + cos(u), u, 0, π / 2, v, 0, π / 2) → VK_1
  17. Surface((-sqrt(2)) + (sin(u) * sin(v)), (-sqrt(2)) + (sin(u) * cos(v)), sqrt(2) - cos(u), u, ((-π)) / 2, 0, v, π, (3 * π / 2)) → VK_2
  18. Surface(sqrt(2) + (sin(u) * sin(v)), (-sqrt(2)) + (sin(u) * cos(v)), sqrt(2) - cos(u), u, 0, π / 2, v, ((-π)) / 2, 0) → VK_3
  19. Surface(sqrt(2) + (sin(u) * sin(v)), (-sqrt(2)) + (sin(u) * cos(v)), (-sqrt(2)) + cos(u), u, 0, π / 2, v, ((-π)) / 2, 0) → VK_4
  20. Surface(sqrt(2) + (sin(u) * sin(v)), sqrt(2) + (sin(u) * cos(v)), (-sqrt(2)) + cos(u), u, ((-π)) / 2, 0, v, 0, π / 2) → VK_5
  21. Surface((-sqrt(2)) + (sin(u) * sin(v)), sqrt(2) + (sin(u) * cos(v)), (-sqrt(2)) + cos(u), u, ((-π)) / 2, 0, v, ((-π)) / 2, 0) → VK_6
  22. Surface(sqrt(2) + (sin(u) * sin(v)), sqrt(2) + (sin(u) * cos(v)), sqrt(2) - cos(u), u, ((-π)) / 2, 0, v, 0, π / 2) → VK_7
  23. Surface((-sqrt(2)) + (sin(u) * sin(v)), sqrt(2) + (sin(u) * cos(v)), sqrt(2) - cos(u), u, ((-π)) / 2, 0, v, ((-π)) / 2, 0) → VK_8
  24. Sphere((sqrt(2), sqrt(2), sqrt(2)), 1) → K_7
  25. Sphere(((-sqrt(2)), sqrt(2), sqrt(2)), 1) → K_8
  26. Sphere(((-sqrt(2)), (-sqrt(2)), sqrt(2)), 1) → K_9
  27. Sphere((sqrt(2), (-sqrt(2)), sqrt(2)), 1) → K_a
  28. Sphere((sqrt(2), (-sqrt(2)), (-sqrt(2))), 1) → K_b
  29. Sphere((sqrt(2), sqrt(2), (-sqrt(2))), 1) → K_c
  30. Sphere(((-sqrt(2)), sqrt(2), (-sqrt(2))), 1) → K_d
  31. Sphere(((-sqrt(2)), (-sqrt(2)), (-sqrt(2))), 1) → K_e
  32. Segment(A, A + (-1, 0, 0)) → f
  33. Segment(A, A + (0, -1, 0)) → g
  34. Segment(A, A + (0, 0, 1)) → h
  35. Segment(B, B + (0, 0, 1)) → i
  36. Segment(B, B + (0, -1, 0)) → j
  37. Segment(B, B + (1, 0, 0)) → k
  38. Segment(C, C + (-1, 0, 0)) → l
  39. Segment(C, C + (0, 1, 0)) → m
  40. Segment(C, C + (0, 0, 1)) → n
  41. Segment(E, E + (0, 0, 1)) → p
  42. Segment(E, E + (1, 0, 0)) → q
  43. Segment(E, E + (0, 1, 0)) → r
  44. Segment(G, G + (0, -1, 0)) → s
  45. Segment(H, H + (0, -1, 0)) → t
  46. Segment(G, G + (0, 0, -1)) → e
  47. Segment(G, G + (-1, 0, 0)) → f_1
  48. Segment(H, H + (1, 0, 0)) → g_1
  49. Segment(H, H + (0, 0, -1)) → h_1
  50. Segment(I, I + (0, 0, -1)) → i_1
  51. Segment(I, I + (1, 0, 0)) → j_1
  52. Segment(I, I + (0, 1, 0)) → k_1
  53. Segment(J, J + (0, 1, 0)) → l_1
  54. Segment(J, J + (0, 0, -1)) → m_1
  55. Segment(J, J + (-1, 0, 0)) → n_1