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Biscribed Truncated Cuboctahedron, n=48: Polyhedra with Extreme Distribution of

Biscribed Truncated Cuboctahedron, n=48: Polyhedra with Extreme Distribution of
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2020-10-28 01:24
PolygonSequenceSolid GeometryOptimizationGeometric Construction
In collections ₈Polyhedra with extreme distribution of equivalent vertices
This GeoGebra example demonstrates a biscribed truncated cuboctahedron with 48 equivalent vertices, illustrating a polyhedron with extreme vertex distribution on a sphere. The core mathematical concepts include polyhedral symmetry, vertex equivalence, circumscribed and inscribed spheres (biscribed property), and optimization of vertex placement. The construction uses GeoGebra commands such as Sequence, Sum, Div, Element, If, Length, and TableText to generate and organize vertex coordinates, along with sliders and buttons for interactive exploration. Suitable for high school and university students studying solid geometry, polyhedral constructions, and optimization problems, as well as for mathematical modeling and computational geometry.

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({Append["a duration", Last[ZwZeit, Min[10, Length[ZwZeit]]]]}, "v") → text2c
  12. Length(A00) → lm1
  13. If(Length[Unique[l01 \ {0}]] ≟ 1, 7, -7) → a7
  14. Sum(Zip[(T * F), T, SysTimeAkt0, F, {1, 1000, 60000, 3600000, 86400000}]) → TimeAbs0
  15. Div(TimeRelab0, 3600) → H0
  16. Div(TimeRelab0 - (H0 * 3600), 60) → M0
  17. Text("" + (TimeRelab0 - (H0 * 3600) - (M0 * 60))) → S0a
  18. Text("" + (TimeRelab - (H * 3600) - (M * 60))) → S
  19. IndexOf(".", S) → index
  20. IndexOf(".", S0a) → indexa
  21. First(S, index + 1) → txtS
  22. First(S0a, indexa + 1) → txtS0a
  23. If(bu0, " ", "d") → txtbuo
  24. Max(Length[A00], Length[A00d]) → maxlä
  25. Element(A00, 24) → m11
  26. 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
  27. 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

AI Drawing Prompt Reference

Draw a biscribed truncated cuboctahedron with 48 equivalent vertices, illustrating extreme vertex distribution on a sphere. First, create 48 vertices, then truncate and bisect the cube, ensuring each vertex lies on the sphere, and finally adjust the vertex positions to optimize the distribution.
The prompt above was compiled from the original drawing commands and is for reference only.