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Truncated Tetrahedron and Average Vertex Distance

Truncated Tetrahedron and Average Vertex Distance
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2020-01-11 22:45
FunctionSolid GeometryOptimization
This example demonstrates the dependence of the average distance between vertices of a truncated tetrahedron on the truncation parameter, exploring geometric properties and parametric relationships. Core concepts include polyhedra, solid geometry, functions, and optimization. Constructed using GeoGebra commands like Join, Unique, Length, Text, Sum, Sphere, Ray, Zip, TableText, UnicodeToLetter. Suitable for high school mathematics students and researchers interested in exploring 3D shapes 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. Zip(Polygon[a], a, v_poly06) → poly06
  11. Text(text12, C, false, true) → text8
  12. Zip(Segment[lP0(x(a)), lP0(y(a))], a, segN1) → seg1
  13. Zip(Segment[lP0(x(a)), lP0(y(a))], a, segN2) → seg2
  14. Zip(Polygon[a], a, v_poly03) → poly03
  15. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ6) → segΔ6
  16. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ3) → segΔ3
  17. TableText(Append["\;\;\; \fgcolor{#848482}{3-gons}", segΔ3], Append["\;\;\; \fgcolor{#007FFF}{6-gons}", segΔ6], "v") → text3
  18. UnicodeToLetter(34) → u34

AI Drawing Prompt Reference

Draw a truncated tetrahedron with vertices generated by truncation points along each edge of the original tetrahedron, with an adjustable truncation parameter. Show all vertices of the truncated shape and compute the average distance between any two vertices, which varies with the truncation parameter. Use spheres to represent the circumspheres before and after truncation, and a ray to indicate the truncation direction. Color triangular faces and hexagonal faces differently, and label the counts of each face type. Also display comparison segments for edges before and after truncation, along with a table summarizing face counts.
The prompt above was compiled from the original drawing commands and is for reference only.