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Rhombicosidodecahedron and Its Dual Deltoidal Hexecontahedron: Edge/Face Colorin

Rhombicosidodecahedron and Its Dual Deltoidal Hexecontahedron: Edge/Face Colorin
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2021-01-02 01:10
PolygonVectorSolid GeometryTransformationGeometric Construction
This GeoGebra example visualizes the geometry of a Rhombicosidodecahedron-c (V=60) together with its dual, the Deltoidal Hexecontahedron (V=62), and highlights their structural symmetry by systematically coloring edges and faces in 3D. Key mathematical ideas include dual polyhedra, solid geometry, and quantitative measures referenced to a sphere (e.g., volume computations using Sphere as a scale). Construction-wise, it uses Sequence to generate collections of geometric objects, Sphere and volume-related objects (Vol, Vold, Vol_{sphere}, Vol_r, Vold_r) to set the metric context, PointIn and Angle to support spatial selection and angular relationships, and Segment for edge creation. Objects are filtered and organized with If, Sort, First, and KeepIf so that only selected edges/faces are included in the coloring scheme. Finally, Textfield, TableText, UnicodeToLetter, and multiple text objects label and display parameters/results. Suitable for advanced high school through university-level geometry visualization, symmetry/adversarial duality learning, and math-software demonstrations.

GeoGebra Commands & Parameters

  1. Sequence(((zoom * Element[Element[A00d, i], 1]), (zoom * Element[Element[A00d, i], 2]), (zoom * Element[Element[A00d, i], 3])), i, 1, Length[A00d]) → A0d
  2. Sequence(((zoom * Element[Element[A00, i], 1]), (zoom * Element[Element[A00, i], 2]), (zoom * Element[Element[A00, i], 3])), i, 1, Length[A00]) → A0
  3. Sphere(O, r_o) → sp02d
  4. Textfield(R1sphere) → InputBox5
  5. Sphere(O, R1sphere) → sp01
  6. Textfield(r_o) → InputBox6
  7. Textfield(jarc) → InputBox8
  8. Textfield(jarc_d) → InputBox9
  9. Textfield(f_{Poly}) → InputBox11
  10. Textfield(k) → InputBox15
  11. Textfield(k_d) → InputBox14
  12. TableText({Append["i", 1…Length[mm5a]], Append["\fgcolor{orange}{n}", mm5a], Append["\fgcolor{orange}{N}", mm7a], Append["\fgcolor{#007FFF}{n/dual}", mm5ad], Append["\fgcolor{#007FFF}{N}", mm7ad]}, "hr") → text24
  13. Textfield(iseg) → InputBox17
  14. Textfield(f_{Poly}d) → InputBox18
  15. Textfield(isegd) → InputBox19
  16. UnicodeToLetter(34) → u34
  17. TableText({Append["i", 1…Max[Length[Segn], Length[Segnd]]], Append["\fgcolor{orange}{s_i}", Segn / Segments_o1], Append["\fgcolor{orange}{N}", lLength], Append["\fgcolor{#007FFF}{s_id}", Segnd / Segments_o1d], Append["\fgcolor{#007FFF}{N}", lLengthd]}, "hr") → text23
  18. Textfield(n_3) → InputBox20
  19. If(n_3 ≟ 0, "", Text["" + n_3]) → txt_3
  20. Textfield(nd_3) → InputBox21
  21. If(nd_3 ≟ 0, "", Text["" + nd_3]) → txtd_3
  22. Circle(A_{farbe}, rfabe) → farbekreis
  23. PointIn(farbekreis) → A
  24. Angle(A) → α
  25. Segment(C, D) → h
  26. Sort(Sequence[Segment[Element[A0, Element[Element[nom, i], 1]], Element[A0, Element[Element[nom, i], 2]]], i, 1, Length[nom]]) → Segments
  27. First(Segments, i_s) → Segments_o
  28. Sort(Sequence[Segment[Element[A0d, Element[Element[nomd, i], 1]], Element[A0d, Element[Element[nomd, i], 2]]], i, 1, Length[nomd]]) → Segmentsd
  29. First(Segmentsd, id_s) → Segmentsd_o
  30. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segn, 1], Segments) → la1
  31. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segn, 2], Segments) → la2
  32. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segn, 3], Segments) → la3
  33. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segn, 4], Segments) → la4
  34. Sequence(Polygon[Element[A0, Element[Element[mm1, f], 1]], Element[A0, Element[Element[mm1, f], 2]], Element[A0, Element[Element[mm1, f], 3]]], f, 1, Length[mm1]) → poly_{3}
  35. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[poly_{3}S, 1], poly_{3}) → poly_{3}1
  36. Sequence(Polygon[Sequence[A0(mm6(1, f, k)), k, 1, Length[mm6(1, 1)]]], f, 1, Length[mm6(1)]) → poly_{4}
  37. Sequence(Polygon[Sequence[A0(mm6(2, f, k)), k, 1, Length[mm6(2, 1)]]], f, 1, Length[mm6(2)]) → poly_{5}
  38. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segnd, 1], Segmentsd) → lad1
  39. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segnd, 2], Segmentsd) → lad2
  40. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segnd, 3], Segmentsd) → lad3
  41. Sequence(Polygon[Element[A0d, Element[Element[mm1d, f], 1]], Element[A0d, Element[Element[mm1d, f], 2]], Element[A0d, Element[Element[mm1d, f], 3]]], f, 1, Length[mm1d]) → polyd_{3}
  42. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[poly_{3}Sd, 1], polyd_{3}) → polyd_{3}1

AI Drawing Prompt Reference

Draw a rhombicosidodecahedron and its dual deltoidal hexecontahedron, both sharing a common circumsphere. Color the original solid's edges and faces in orange, and the dual's in blue. Use input boxes to adjust scale, radius, and color parameters, and display tables of volumes and edge lengths. Construct edges as segments between vertices, fill faces as polygons, and filter specific edges/faces by length for coloring. Include a circular color wheel with an angle indicator for interactive color selection. Show the symmetry and quantitative measures of the two dual polyhedra.
The prompt above was compiled from the original drawing commands and is for reference only.