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Finding and coloring edges and faces of a polyhedron and its dual image

Finding and coloring edges and faces of a polyhedron and its dual image
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2020-09-22 20:11
Solid GeometryOptimizationGeometric Construction
This example demonstrates how to find and color the edges and faces of a polyhedron and its dual image. Key mathematical concepts include geometric transformations, volume calculations, and optimization problems. Construction involves using GeoGebra commands such as Sequence, Sphere, Circle, and others to build objects, with parameter adjustments for dynamic visualization. Suitable for high school students or college undergraduates learning solid geometry and optimization techniques.

GeoGebra Commands & Parameters

  1. Sequence(((zoom * Element[Element[A00, i], 1]), (zoom * Element[Element[A00, i], 2]), (zoom * Element[Element[A00, i], 3])), i, 1, Length[A00]) → A0
  2. Sequence(((zoom * Element[Element[A00d, i], 1]), (zoom * Element[Element[A00d, i], 2]), (zoom * Element[Element[A00d, i], 3])), i, 1, Length[A00d]) → A0d
  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. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segn, 5], Segments) → la5
  35. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segn, 6], Segments) → la6
  36. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segn, 7], Segments) → la7
  37. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segn, 8], Segments) → la8
  38. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segn, 9], Segments) → la9
  39. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segn, 10], Segments) → la10
  40. 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}
  41. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[poly_{3}S, 1], poly_{3}) → poly_{3}1
  42. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[poly_{3}S, 2], poly_{3}) → poly_{3}2
  43. Sequence(Polygon[Sequence[A0(mm6(1, f, k)), k, 1, Length[mm6(1, 1)]]], f, 1, Length[mm6(1)]) → poly_{5}
  44. Sequence(Polygon[Sequence[A0(mm6(2, f, k)), k, 1, Length[mm6(2, 1)]]], f, 1, Length[mm6(2)]) → poly_{6}
  45. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segnd, 1], Segmentsd) → lad1
  46. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segnd, 2], Segmentsd) → lad2
  47. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segnd, 3], Segmentsd) → lad3
  48. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segnd, 4], Segmentsd) → lad4
  49. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[Segnd, 5], Segmentsd) → lad5
  50. 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}
  51. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[poly_{3}Sd, 1], polyd_{3}) → polyd_{3}1
  52. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[poly_{3}Sd, 2], polyd_{3}) → polyd_{3}2
  53. KeepIf(round((x * 10^(k))) / 10^(k) ≟ Element[poly_{3}Sd, 3], polyd_{3}) → polyd_{3}3

AI Drawing Prompt Reference

Draw a polyhedron and its dual image, and mark their edges and faces. Start by drawing the polyhedron, then draw its dual image. Next, identify and mark the edges and faces of the polyhedron and its dual image. Finally, adjust parameters to achieve dynamic visualization.
The prompt above was compiled from the original drawing commands and is for reference only.