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Discs Sphere - 12 Discs: Approximating a Sphere with Rotated Discs

Discs Sphere - 12 Discs: Approximating a Sphere with Rotated Discs
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2020-11-13 23:08
TrigonometrySolid GeometryTransformationParametric EquationGeometric Construction
This GeoGebra example demonstrates how a sphere can be approximated by twelve circular discs arranged in 3D space, providing a visual and interactive representation of a discretized spherical surface. The core mathematical concepts involve the equation of a sphere, parametric surfaces, rotational transformations, and the use of cosine functions to position the discs around the sphere. The construction uses GeoGebra commands such as Sphere to define the base sphere, CurveCartesian to outline each disc's boundary, Surface to generate the disc faces, and Line with Rotate to place and orient the discs in space. Numerous auxiliary objects (e.g., butSetUp, b, d, R, O, e, r) and surface objects (surfAz, surfBz, surf1z, surfRz, etc.) are defined to build the top and bottom faces of each disc. This example is suitable for high school or undergraduate students studying solid geometry, 3D modeling, or mathematical visualization, helping them grasp the relationship between a sphere and its planar approximations, as well as the practical use of parametric surfaces and geometric transformations.

GeoGebra Commands & Parameters

  1. Sphere(O, 1) → e
  2. Sphere(O, R) → r
  3. CurveCartesian((1; (arg(A1) * φ); alt(A1)), φ, -1, 1) → cA1
  4. CurveCartesian((R; (arg(AR) * φ); alt(AR)), φ, -1, 1) → cAR
  5. CurveCartesian((1; (arg(B1) * φ); alt(B1)), φ, -1, 1) → cB1
  6. CurveCartesian((R; (arg(BR) * φ); alt(BR)), φ, -1, 1) → cBR
  7. Surface(((1 - s) * cA1(t)) + (s * cB1(t)), s, 0, 1, t, -1, 1) → s1h
  8. Surface((s * cAR(t)) + ((1 - s) * cBR(t)), s, 0, 1, t, -1, 1) → sRh
  9. Line(O, (1, 1, 1)) → L111
  10. Surface(((1 - s) * cAR(t)) + (s * cA1(t)), s, 0, 1, t, -1, 1) → surfAz
  11. Surface((s * cBR(t)) + ((1 - s) * cB1(t)), s, 0, 1, t, -1, 1) → surfBz
  12. Surface((1; arg(s1h(s, t)); alt(s1h(s, t))), s, 0, 1, t, -1, 1) → surf1z
  13. Surface((R; arg(sRh(s, t)); alt(sRh(s, t))), s, 0, 1, t, -1, 1) → surfRz
  14. Rotate(surfAz, 120°, L111) → surfAx
  15. Rotate(surfBz, 120°, L111) → surfBx
  16. Rotate(surf1z, 120°, L111) → surf1x
  17. Rotate(surfRz, 120°, L111) → surfRx
  18. Rotate(surfAz, (-(120°)), L111) → surfAy
  19. Rotate(surfBz, (-(120°)), L111) → surfBy
  20. Rotate(surf1z, (-(120°)), L111) → surf1y
  21. Rotate(surfRz, (-(120°)), L111) → surfRy
  22. Rotate(surfAz, 180°, xAxis) → surfAz1
  23. Rotate(surfBz, 180°, xAxis) → surfBz1
  24. Rotate(surf1z, 180°, xAxis) → surf1z1
  25. Rotate(surfRz, 180°, xAxis) → surfRz1
  26. Rotate(surfAx, 180°, xAxis) → surfAx1
  27. Rotate(surfBx, 180°, xAxis) → surfBx1
  28. Rotate(surf1x, 180°, xAxis) → surf1x1
  29. Rotate(surfRx, 180°, xAxis) → surfRx1
  30. Rotate(surfAy, 180°, xAxis) → surfAy1
  31. Rotate(surfBy, 180°, xAxis) → surfBy1
  32. Rotate(surf1y, 180°, xAxis) → surf1y1
  33. Rotate(surfRy, 180°, xAxis) → surfRy1
  34. Rotate(surfAz, 180°, yAxis) → surfAz2
  35. Rotate(surfBz, 180°, yAxis) → surfBz2
  36. Rotate(surf1z, 180°, yAxis) → surf1z2
  37. Rotate(surfRz, 180°, yAxis) → surfRz2
  38. Rotate(surfAx, 180°, yAxis) → surfAx2
  39. Rotate(surfBx, 180°, yAxis) → surfBx2
  40. Rotate(surf1x, 180°, yAxis) → surf1x2
  41. Rotate(surfRx, 180°, yAxis) → surfRx2
  42. Rotate(surfAy, 180°, yAxis) → surfAy2
  43. Rotate(surfBy, 180°, yAxis) → surfBy2
  44. Rotate(surf1y, 180°, yAxis) → surf1y2
  45. Rotate(surfRy, 180°, yAxis) → surfRy2
  46. Rotate(surfAz, 180°, zAxis) → surfAz3
  47. Rotate(surfBz, 180°, zAxis) → surfBz3
  48. Rotate(surf1z, 180°, zAxis) → surf1z3
  49. Rotate(surfRz, 180°, zAxis) → surfRz3
  50. Rotate(surfAx, 180°, zAxis) → surfAx3
  51. Rotate(surfBx, 180°, zAxis) → surfBx3
  52. Rotate(surf1x, 180°, zAxis) → surf1x3
  53. Rotate(surfRx, 180°, zAxis) → surfRx3
  54. Rotate(surfAy, 180°, zAxis) → surfAy3
  55. Rotate(surfBy, 180°, zAxis) → surfBy3
  56. Rotate(surf1y, 180°, zAxis) → surf1y3
  57. Rotate(surfRy, 180°, zAxis) → surfRy3

AI Drawing Prompt Reference

Draw a sphere approximated by 12 rotated discs. Start by creating a sphere with a radius of 1, then define the boundary of each disc using parametric curves, and generate the disc surfaces. Rotate these surfaces to place them in 3D space, forming an approximation of the sphere.
The prompt above was compiled from the original drawing commands and is for reference only.