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Soddy Spheres in a Spiral

Soddy Spheres in a Spiral
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2022-02-12 12:40
TrigonometrySolid GeometryGeometric ConstructionGraphing
In collections Curves and Spirals on Spheres
This example demonstrates Soddy spheres arranged in a spiral structure, illustrating the packing of spheres in three-dimensional space. Key mathematical concepts include sphere, geometric mean, cone, function graph, and trigonometry. The construction involves commands and objects such as Segment, OrthogonalLine, Circle, Intersect, Midpoint, Line, Center, Sphere, Distance, Surface, and CurveCartesian from GeoGebra. It is suitable for high school and above students to explore spatial geometry and properties of spheres along with related mathematical applications.

GeoGebra Commands & Parameters

  1. Segment(B, C) → g
  2. OrthogonalLine(A, g) → i
  3. Circle(A, B) → c
  4. Segment(C, A) → h
  5. Intersect(c, h, 1) → X
  6. Circle(C, X) → d
  7. Intersect(d, g, 1) → Y
  8. Midpoint(B, Y) → E
  9. Circle(B, E) → e
  10. Segment(A, B) → f
  11. Intersect(e, f, 1) → D
  12. Circle(A, D) → a
  13. Intersect(a, i, 1) → U'
  14. Circle(U', A) → k
  15. Intersect(k, i, 2) → P'
  16. Line(P', E) → l
  17. Line(U', E) → j
  18. Intersect(a, j, 2) → U
  19. Line(U, A) → m
  20. Intersect(l, m) → P
  21. Circle(P, U) → q
  22. Center(q) → H
  23. Intersect(a, h, 1) → F
  24. Circle(C, F) → p
  25. Intersect(a, i, 2) → U''
  26. OrthogonalLine(B, h) → n
  27. Intersect(e, n, 2) → V''
  28. Circle(V'', B) → r
  29. Intersect(r, n, 2) → Q''
  30. Line(F, Q'') → s
  31. Circle(U'', A) → t
  32. Intersect(t, i, 2) → P''
  33. Line(P'', E) → b
  34. Intersect(s, b) → S
  35. Intersect(e, s, 1) → R
  36. Circle(S, R) → c_1
  37. Sphere(B, D) → o
  38. Sphere(A, D) → u
  39. Sphere(C, F) → v
  40. Intersect(p, c_1, 1) → G
  41. Sphere(S, G) → w
  42. Distance(H, U) → RL
  43. Surface(x(H) + ((RL * cos(u)) * sin(v)), y(H) + ((RL * sin(u)) * sin(v)), z(H) + (RL * cos(v)), u, 0, (2 * π), v, 0, π) → a_1
  44. CurveCartesian(x(H) + (RL * cos(t) / sqrt(1 + (0.1^(2) * t^(2)))), y(H) + (RL * sin(t) / sqrt(1 + (0.1^(2) * t^(2)))), z(H) + (RL * (-(((0.1 * t)) / sqrt(1 + (0.1^(2) * t^(2)))))), t, (-30 * π), (30 * π)) → d_1

AI Drawing Prompt Reference

Create a 3D visualization of Soddy spheres arranged along a spiral. First set up two perpendicular reference lines in space and determine a central reference point. Construct several key control points by taking intersections of circles in those perpendicular cross-sections, then refine additional points using midpoints and lines connecting them to the reference positions. Use the resulting control points as centers to place multiple spheres (so you can see a sequence of spheres progressing along the central direction). Build a spiral guideline by drawing lines that link key sphere centers, then add a parametric spiral curve and a ruled/parametric surface around the spiral so the spheres appear packed along the spiral structure in three dimensions.
The prompt above was compiled from the original drawing commands and is for reference only.