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Truncated Cuboctahedron: Vertex Average Distance vs. Truncation Parameter

Truncated Cuboctahedron: Vertex Average Distance vs. Truncation Parameter
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2020-01-08 20:58
Solid GeometryOptimizationParametric EquationGeometric Construction
This example demonstrates the dependence of the average distance between vertices of a truncated cuboctahedron on its truncation parameter, highlighting extreme distribution patterns. Key mathematical concepts include polyhedron geometry, parameterization, average distance calculation, and optimization problems. GeoGebra commands like Join, Unique, Length, Sphere, Zip, Ray, and others are used to construct the model dynamically, showing how parameter changes affect vertex distances. Suitable for high school students and math enthusiasts exploring polyhedral properties and parametric geometry.

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. Sphere(O, R1) → sp1
  6. Sphere(O, R2) → sp2
  7. Zip(Polygon[a], a, v_poly06) → poly06
  8. Zip(Segment[lP0(x(a)), lP0(y(a))], a, segN1) → seg1
  9. Zip(Segment[lP0(x(a)), lP0(y(a))], a, segN2) → seg2
  10. Zip(Polygon[a], a, v_poly04) → poly04
  11. Zip(Polygon[a], a, v_poly08) → poly08
  12. Ray(B, AA) → h
  13. UnicodeToLetter(34) → u34
  14. Text("" + p3) → txt_p
  15. Sum(Flatten[Sequence[Append[Sequence[Element[txt_p, i], i, m1(k), m2(k)], " "], k, 1, 5]]) → txtUm1
  16. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ4) → segΔ4
  17. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ6) → segΔ6
  18. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ8) → segΔ8
  19. TableText(Append["\;\;\; \fgcolor{red}{4-gons}", segΔ4], Append["\;\;\; \fgcolor{#FFBF00}{6-gons}", segΔ6], Append["\;\;\; \fgcolor{#5D8AA8}{8-gons}", segΔ8], "v") → text3a