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Truncated Biscribed Pentakis: Dependence of Vertex Average Distance on Truncatio

Truncated Biscribed Pentakis: Dependence of Vertex Average Distance on Truncatio
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2020-01-09 17:55
Solid GeometryOptimizationParametric EquationGeometric Construction
This example demonstrates the dependence of the average distance between vertices of a truncated biscribed pentakis polyhedron on the truncation parameter, along with extreme distributions. Key mathematical concepts include solid geometry, parametric equations, and optimization. The construction utilizes GeoGebra commands and objects such as Join, Unique, Length, Sphere, Ray, Zip, UnicodeToLetter, TableText, KeepIf, Text, and Sum to visualize how parameter changes affect the geometric shape. Suitable for high school students and math enthusiasts exploring complex polyhedra 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. Sphere(O, R1) → sp1
  6. Sphere(O, R2) → sp2
  7. Ray(B, AA) → h
  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_poly03) → poly03
  11. Zip(Polygon[a], a, v_poly06) → poly06
  12. Zip(Polygon[a], a, v_poly05) → poly05
  13. UnicodeToLetter(34) → u34
  14. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ5) → segΔ5
  15. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ6) → segΔ6
  16. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ31) → segΔ31
  17. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ32) → segΔ32
  18. TableText(Append["\;\;\; \fgcolor{#CD7F32}{3_1-gons}", segΔ31], Append["\;\;\; \fgcolor{#CD7F32}{3_2-gons}", segΔ32], Append["\;\;\; \fgcolor{red}{5-gons}", segΔ5], Append["\;\;\; \fgcolor{#6E7F80}{6-gons}", segΔ6], "v") → text3a
  19. Unique(round(poly03, 14)) → poly03round
  20. KeepIf(a ≟ poly03round(1), a, poly03) → poly03_2
  21. KeepIf(a ≟ poly03round(2), a, poly03) → poly03_1
  22. Text("" + p3) → txt_p
  23. Sum(Flatten[Sequence[Append[Sequence[Element[txt_p, i], i, m1(k), m2(k)], " "], k, 1, 5]]) → txtUm1
  24. Text(text12, C, false, true) → text8