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Golden Ratio 120-Polyhedron Animation

Golden Ratio 120-Polyhedron Animation
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2015-07-13 20:23
FunctionSequenceCoordinate GeometrySolid Geometry
In collections Polyeder, whose coordinates are golden numbers
This example demonstrates the dynamic animation of the golden ratio 120-polyhedron, involving core mathematical concepts like the golden ratio p=(sqrt(5)∓1)/2, solid geometry, and coordinate geometry. It utilizes GeoGebra commands and objects such as If, Sequence, Length, Join, Sort, Element, LaTeX, matrix2, a_40, r, a_4, a_4scale, etc. Suitable for high school students and math enthusiasts to explore complex polyhedra and the application of the golden ratio.

GeoGebra Commands & Parameters

  1. If(b, 1, a_4scale) → scalefactor
  2. If(b, q1, 1 + q1) → q
  3. Sequence(((scalefactor * Element[Element[matrix2, i], 1]), (scalefactor * Element[Element[matrix2, i], 2]), (scalefactor * Element[Element[matrix2, i], 3])), i, 1, Length[matrix2]) → A0
  4. If(b, lb, lnb) → l00qqb
  5. Sequence(Segment[((scalefactor * Element[Element[A00, x(Element[l00qqb, i])], 1]), (scalefactor * Element[Element[A00, x(Element[l00qqb, i])], 2]), (scalefactor * Element[Element[A00, x(Element[l00qqb, i])], 3])), ((scalefactor * Element[Element[A00, y(Element[l00qqb, i])], 1]), (scalefactor * Element[Element[A00, y(Element[l00qqb, i])], 2]), (scalefactor * Element[Element[A00, y(Element[l00qqb, i])], 3]))], i, 1, 300) → list1
  6. Length(matrix2) → lämx2
  7. Join(Sequence[Sequence[Segment[Element[A0, i], Element[A0, j]], j, i + 1, lämx2], i, 1, lämx2 - 1]) → l00r
  8. Sort(Union[l01r, l01r]) → l01sortr
  9. Sequence(KeepIf[If[i ≟ 1, x ≤ Element[l01sortr, i], If[i ≟ lämx2, Element[l01sortr, i - 1] < x, Element[l01sortr, i - 1] < x ≤ Element[l01sortr, i]]], l00r], i, 1, lämx2) → la00r
  10. Element(la00r, 1) → la11
  11. Element(la00r, 2) → la12
  12. Element(la00r, 3) → la13
  13. Element(la00r, 4) → la14
  14. Element(la00r, 5) → la15
  15. Element(la00r, 6) → la16
  16. Element(la00r, 7) → la17
  17. Element(la00r, 8) → la18
  18. Element(la00r, a) → la1a
  19. Sequence(((scalefactor * Element[Element[A00, i], 1]), (scalefactor * Element[Element[A00, i], 2]), (scalefactor * Element[Element[A00, i], 3])), i, 1, 62) → A000
  20. Element(A0, Vertex) → AAAr
  21. Length(l01sortr) → läl01sortr
  22. Length(la1a) → läla1a
  23. Element(Element[list2, iq + 1], Vertex) → zzz1
  24. Element(list5, zzz1) → zzz2
  25. If(iq ≟ 0, "120 Polyhedron", If[(iq > 0) ∧ (iq < 11), "Tetrahedra", If[(iq > 10) ∧ (iq < 16), "Cube", If[(iq > 15) ∧ (iq < 21), "Octahedra", If[(iq > 20) ∧ (iq < 26), " rhombic Dodecahedra", If[iq ≟ 26, " Icosahedron", If[iq ≟ 27, " regular Dodecahedron", " rhombic Triacontahedron"]]]]]]) → zzz3
  26. LaTeX(If[b, " \frac{\sqrt{5}-1}{2}", " \frac{\sqrt{5}+1}{2}"], true) → tx3
  27. LaTeX(If[iq ≟ 0, "120\; Polyhedron", If[(iq > 0) ∧ (iq < 11), "Tetrahedron", If[(iq > 10) ∧ (iq < 16), "Cube", If[(iq > 15) ∧ (iq < 21), "Octahedron", If[(iq > 20) ∧ (iq < 26), " rhombic\; Dodecahedron", If[iq ≟ 26, " Icosahedron", If[iq ≟ 27, " regular\; Dodecahedron", " rhombic\; Triacontahedron"]]]]]]], true) → zzz3a

AI Drawing Prompt Reference

Create an animated 3D polyhedron based on the golden ratio, morphing from a 120-cell shape through various regular polyhedra (tetrahedron, cube, octahedron, rhombic dodecahedron, icosahedron, regular dodecahedron, rhombic triacontahedron). The structure consists of many vertices and edges, with vertex coordinates scaled by the golden ratio (√5±1)/2. Edges connect nearby vertices forming a dense network. The object rotates dynamically, with labels showing the current polyhedron name and the golden ratio formula.
The prompt above was compiled from the original drawing commands and is for reference only.