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Tetrahedron n=4: Polyhedra with Extreme Distribution of Equivalent Vertices

Tetrahedron n=4: Polyhedra with Extreme Distribution of Equivalent Vertices
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2020-10-27 02:42
SequenceSolid GeometryOptimizationGeometric ConstructionSlider Animation
In collections ₈Polyhedra with extreme distribution of equivalent vertices
① This example demonstrates how to construct a family of polyhedra based on a regular tetrahedron (n=4) to explore extreme distributions of equivalent vertices, visualizing how volume, circumsphere radius, and other geometric quantities vary with parameters. ② Core mathematical concepts include centroid (barycenter), volume computation, circumscribed and inscribed spheres, geometric mean, linear programming optimization, and uniform distribution of discrete points. ③ Construction highlights: using GeoGebra commands such as Sequence, Sum, If, Length, Div, Element, TableText, Text, IndexOf, First, and Max to generate vertex lists and compute volumes; sliders control parameters (e.g., n, scaling factors) for dynamic exploration; key objects include r_md, Vold, mm7ad, A00d, r_d, Volsphere_d, Vol, etc., for radii, volumes, and surface areas. ④ Suitable for high school to university-level mathematics teaching, especially in solid geometry, optimization problems, and mathematical modeling, as well as for investigating symmetry and extremal distributions of polyhedra.

GeoGebra Commands & Parameters

  1. If(bu0, A00, A00d) → A00_0
  2. Length(A00_0) → n
  3. Length(l2) → läl2
  4. Sequence(sgn(Determinant[{Element[A00_0, i1] - Element[A00_0, j], Element[A00_0, i2] - Element[A00_0, j], Element[A00_0, i3] - Element[A00_0, j]}]), j, 1, n) → l01
  5. Length(A00d) → zzz
  6. Sum(Zip[(T * F), T, SysTimeAkt1, F, {1, 1000, 60000, 3600000, 86400000}]) → TimeAbs1
  7. Sum(Zip[(T * F), T, SysTimeAkt2, F, {1, 1000, 60000, 3600000, 86400000}]) → TimeAbs2
  8. Div(TimeRelab, 3600) → H
  9. Div(TimeRelab - (H * 3600), 60) → M
  10. Element(ZwZeit, step) → text10
  11. TableText({{"\underline{Polyhedron}"}, Append["Vertices: n=", {Length[A00]}], Append["r=", {r}], Append["\bar{r}=", {r_m}], Append["k-gon : k=", mm5a], Append["their\; numbers: n_k=", mm7a], Append["Volume: V=", {Vol_r}]}, "hr ") → text7
  12. TableText({{"\underline{dual\;Polyhedron}"}, Append["Vertices: n=", {Length[A00d]}], Append["r=", {r_d}], Append[" \bar{r_d}= ", {r_md}], Append["k-gon : k=", mm5ad], Append["their\; numbers: n_k=", mm7ad], Append["Volume: V=", {Vold_r}]}, "hr ") → text20
  13. TableText({Append["a duration", Last[ZwZeit, Min[10, Length[ZwZeit]]]]}, "v") → text2c
  14. Length(A00) → lm1
  15. If(Length[Unique[l01 \ {0}]] ≟ 1, 7, -7) → a7
  16. Sum(Zip[(T * F), T, SysTimeAkt0, F, {1, 1000, 60000, 3600000, 86400000}]) → TimeAbs0
  17. Div(TimeRelab0, 3600) → H0
  18. Div(TimeRelab0 - (H0 * 3600), 60) → M0
  19. Text("" + (TimeRelab0 - (H0 * 3600) - (M0 * 60))) → S0a
  20. Text("" + (TimeRelab - (H * 3600) - (M * 60))) → S
  21. IndexOf(".", S) → index
  22. IndexOf(".", S0a) → indexa
  23. First(S, index + 1) → txtS
  24. First(S0a, indexa + 1) → txtS0a
  25. If(bu0, " ", "d") → txtbuo
  26. Max(Length[A00], Length[A00d]) → maxlä
  27. Element(A00, 24) → m11

AI Drawing Prompt Reference

Draw a family of polyhedra based on a regular tetrahedron to explore the extreme distribution of equivalent vertices. Start by creating a regular tetrahedron, then generate a series of polyhedra by adjusting parameters n and scaling factors. Calculate the length n of each vertex list, generate volumes, circumradius, inradius, and the number of sides and vertices for each face. Finally, display the volumes and geometric properties of these polyhedra, including the centroid, volume, circumscribed sphere, and inscribed sphere.
The prompt above was compiled from the original drawing commands and is for reference only.