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Constructing and Visualizing Polyhedra with Surface Triangulation

Constructing and Visualizing Polyhedra with Surface Triangulation
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2015-11-14 20:31
TrigonometrySolid GeometryGeometric Construction
This example demonstrates the construction of polyhedra, surface triangulation, and 3D visualization. Key math concepts include solid geometry, triangulation, and geometric transformations. GeoGebra commands and objects used in the construction include Sequence, Sort, Element, Length, and TableText. Suitable for high school students and math enthusiasts learning 3D graphics construction and visualization.

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. Sort(Sequence[Segment[Element[A0, Element[Element[nom, i], 1]], Element[A0, Element[Element[nom, i], 2]]], i, 1, Length[nom]]) → Segments
  3. Sequence(Element[Segments, i], i, 1, i_s) → Segments_o
  4. Sort(Union[{}, ceil((Segments * 10^(k))) / 10^(k)]) → Segn
  5. Sequence(KeepIf[If[i ≟ 1, x ≤ Element[Segn, i], If[i ≟ Length[Segn], Element[Segn, i - 1] < x, Element[Segn, i - 1] < x ≤ Element[Segn, i]]], Segments], i, 1, Length[Segn]) → la0
  6. Sequence(Length[Element[la0, i]], i, 1, Length[Segn]) → lLength
  7. Element(la0, 1) → la1
  8. Element(la0, 2) → la2
  9. Element(la0, 3) → la3
  10. Element(la0, 4) → la4
  11. Sequence(((zoom * Element[Element[A00d, i], 1]), (zoom * Element[Element[A00d, i], 2]), (zoom * Element[Element[A00d, i], 3])), i, 1, Length[A00d]) → A0d
  12. Sort(Sequence[Segment[Element[A0d, Element[Element[nomd, i], 1]], Element[A0d, Element[Element[nomd, i], 2]]], i, 1, Length[nomd]]) → Segmentsd
  13. Sequence(Element[Segmentsd, i], i, 1, id_s) → Segmentsd_o
  14. Sort(Union[{}, ceil((Segmentsd * 10^(k_d))) / 10^(k_d)]) → Segnd
  15. Sequence(KeepIf[If[i ≟ 1, x ≤ Element[Segnd, i], If[i ≟ Length[Segnd], Element[Segnd, i - 1] < x, Element[Segnd, i - 1] < x ≤ Element[Segnd, i]]], Segmentsd], i, 1, Length[Segnd]) → la0d
  16. Sequence(Length[Element[la0d, i]], i, 1, Length[Segnd]) → lLengthd
  17. Length(Segments) → läSegments
  18. Length(Segmentsd) → läSegmentsd
  19. TableText({mm5ad, mm7ad}, "hr") → text2
  20. Element(la0d, 1) → la1d
  21. Element(la0d, 2) → la2d
  22. Element(la0d, 3) → la3d
  23. Element(la0d, 4) → la4d
  24. TableText({Segn, lLength}, "hr") → text8
  25. TableText({Segnd, lLengthd}, "hr") → text9
  26. TableText({Append["n=", mm5a], Append["N=", mm7a]}, "hr") → text1
  27. 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
  28. Sequence(Polygon[Element[A0d, Element[Element[Element[mm6d, 1], i], 1]], Element[A0d, Element[Element[Element[mm6d, 1], i], 2]], Element[A0d, Element[Element[Element[mm6d, 1], i], 3]], Element[A0d, Element[Element[Element[mm6d, 1], i], 4]], Element[A0d, Element[Element[Element[mm6d, 1], i], 5]]], i, 1, 12) → polyd_{5}
  29. Sequence(Polygon[Element[A0d, Element[Element[Element[mm6d, 2], i], 1]], Element[A0d, Element[Element[Element[mm6d, 2], i], 2]], Element[A0d, Element[Element[Element[mm6d, 2], i], 3]], Element[A0d, Element[Element[Element[mm6d, 2], i], 4]], Element[A0d, Element[Element[Element[mm6d, 2], i], 5]], Element[A0d, Element[Element[Element[mm6d, 2], i], 6]]], i, 1, 2) → polyd_{6}
  30. 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

AI Drawing Prompt Reference

Visualize a polyhedron with surface triangulation and 3D rendering. Construct the vertices and edges of the polyhedron, then perform surface triangulation, and finally combine the triangulated triangles with the original polyhedron to form a complete 3D model, labeling key data.
The prompt above was compiled from the original drawing commands and is for reference only.