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Dependence of Average Vertex Distance in a Truncated Cube on Truncation Paramete

Dependence of Average Vertex Distance in a Truncated Cube on Truncation Paramete
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2020-01-12 19:56
Solid GeometryOptimizationParametric EquationGeometric Construction
This example demonstrates how the average distance between vertices of a truncated cube varies with the truncation parameter and explores extreme distributions. Key math concepts include solid geometry, optimization, and parametric equations. Construction utilizes GeoGebra commands and objects like Join, Unique, Length, Text, Sum, Sphere, Ray, Zip, UnicodeToLetter, and TableText. Suitable for high school and above to understand polyhedron geometry and optimization problems.

GeoGebra Commands & Parameters

  1. Join(Zip[lP0(x(a)) + ((lP0(y(a)) - lP0(x(a))) * {t, 1 - t}), a, segN]) → lVV
  2. Unique(lVV) → lP'
  3. Length(lP') → n
  4. Length(lVV(1)) → R2
  5. Text("" + p3) → txt_p
  6. Sum(Flatten[Sequence[Append[Sequence[Element[txt_p, i], i, m1(k), m2(k)], " "], k, 1, 5]]) → txtUm1
  7. Sphere(O, R1) → sp1
  8. Sphere(O, R2) → sp2
  9. Ray(B, AA) → h
  10. Text(text12, C, false, true) → text8
  11. Zip(Segment[lP0(x(a)), lP0(y(a))], a, segN1) → seg1
  12. Zip(Segment[lP0(x(a)), lP0(y(a))], a, segN2) → seg2
  13. Zip(Polygon[a], a, v_poly03) → poly03
  14. Zip(Polygon[a], a, v_poly08) → poly08
  15. UnicodeToLetter(34) → u34
  16. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ3) → segΔ3
  17. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ8) → segΔ8
  18. TableText(Append["\;\;\; \fgcolor{#848482}{3-gons}", segΔ3], Append["\;\;\; \fgcolor{#007FFF}{8-gons}", segΔ8], "v") → text3a

AI Drawing Prompt Reference

Draw a truncated cube showing how the average distance between its vertices changes with the truncation parameter. Include two concentric spheres with radii corresponding to the original and truncated shapes, and a ray from a vertex. Use distinct colors for triangular and octagonal faces, and add text labels for the parameter and face counts. Construct the vertex set first, then connect edges and faces, and finally add the spheres, ray, and annotations.
The prompt above was compiled from the original drawing commands and is for reference only.