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String Sphere: Icosahedron and Spherical Chord Network

String Sphere: Icosahedron and Spherical Chord Network
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2020-06-27 14:30
PolygonAngleSolid GeometryTransformationGeometric Construction
① This example demonstrates how to construct a network of chords on a sphere by projecting the edges of a regular icosahedron onto the sphere's surface, creating a visually striking 'string sphere'. ② Core mathematical concepts include regular icosahedron geometry, spherical geometry, midpoints, plane intersections, rotations and reflections, along with related topics like exponential functions and linear programming as tagged. ③ Construction highlights: using GeoGebra commands such as Point, Icosahedron, Midpoint, Sphere, PointIn, OrthogonalLine, Plane, Intersect, Angle, Sequence, Rotate, and Mirror to define faces (e.g., faceABC, faceACF) and edges (e.g., areteAC, areteAB), then generating chords via midpoints, orthogonal lines, and plane intersections. ④ Suitable for high school or college students studying solid geometry, spatial reasoning, and advanced GeoGebra usage, as well as for mathematical art projects.

GeoGebra Commands & Parameters

  1. Point(Circle[Midpoint[A, B], (Distance[A, B] * sqrt(3) / 2), Segment[A, B]]) → C
  2. Icosahedron(A, B, C) → a, D, faceABC, faceACF, faceAFG, faceADG, faceABD, faceBDH, faceBEH, faceBCE, faceCEI, faceCFI, E, faceEHK, faceEIK, faceIKL, faceFIL, faceHJK, faceJKL, faceGJL, faceFGL, faceDHJ, faceDGJ, F, areteAC, areteAB, areteBC, areteAF, areteCF, areteAG, areteFG, areteAD, areteDG, areteBD, G, areteBH, areteDH, areteBE, areteEH, areteCE, areteCI, areteEI, areteFI, areteEK, areteHK, H, areteIK, areteIL, areteKL, areteFL, areteJK, areteHJ, areteJL, areteGL, areteGJ, areteDJ, I, J, K, L
  3. Midpoint(D, I) → M
  4. Sphere(M, A) → b
  5. PointIn(b) → P
  6. OrthogonalLine(M, xOyPlane) → f
  7. Plane(J, f) → c
  8. OrthogonalLine(M, c) → g
  9. Intersect(f, b) → N, O
  10. Angle(P, M, O) → α
  11. Sequence(Rotate[Rotate[J, (-a), g], (3 * a), f], a, 0, α, α / 9) → L_1
  12. Rotate(L_1, (2 * π / 3), f) → L_2
  13. Sequence(Polygon[M, Element[L_1, i], Element[L_2, i]], i, 1, Length[L_1], n) → L_3
  14. Rotate(L_3, (2 * π / 3), f) → L_4
  15. Rotate(L_3, (-2 * π / 3), f) → L_5
  16. Sequence(Rotate[{L_3, L_4, L_5}, ((a * 2) * π / 5), Line[M, J]], a, 1, 4) → L_6
  17. Sequence(Rotate[{L_3, L_4, L_5}, ((a * 2) * π / 5), Line[M, K]], a, 2, 4) → L_7
  18. Mirror(L_6, M) → L_8
  19. Sequence(Rotate[{L_3, L_4, L_5}, ((a * 2) * π / 5), Line[M, L]], a, 2, 3) → L_9
  20. Mirror(L_9, M) → L_{10}
  21. Rotate({L_3, L_4, L_5}, (4 * π / 5), Line[M, J]) → L_{11}
  22. Rotate(L_{11}, (4 * π / 5), Line[M, D]) → L_{12}
  23. Rotate(L_{11}, (2 * π / 5), Line[M, D]) → L_{13}
  24. Rotate(L_{11}, (6 * π / 5), Line[M, G]) → L_{14}
  25. Rotate(L_{12}, (-2 * π / 5), Line[M, A]) → L_{15}

AI Drawing Prompt Reference

Draw a 'string sphere' by projecting the edges of a regular icosahedron onto a sphere's surface to form a network of chords. Start by creating the icosahedron, then identify its vertices and faces. Use midpoints, orthogonal lines, and plane intersections to generate the chords. Finally, apply rotation and reflection operations to create the entire chord network, forming a visually striking spherical structure.
The prompt above was compiled from the original drawing commands and is for reference only.