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120-Polyhedron with Golden Number Coordinates Animation

120-Polyhedron with Golden Number Coordinates Animation
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2015-07-14 19:58
TrigonometrySequenceCoordinate GeometrySolid GeometryMatrix
This example demonstrates an animation of a 120-polyhedron whose vertices are defined by golden numbers. The polyhedron features 120 triangular faces, 180 edges, and 62 vertices. Key mathematical concepts include golden numbers, coordinate geometry, solid geometry, and trigonometric functions. The construction utilizes GeoGebra commands and objects such as If, Sequence, Length, Join, Sort, Element, and LaTeX, with elements like matrix2, a_40, r, a_4, etc. It is suitable for high school and advanced learners exploring complex geometric constructions and the application of golden numbers.

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