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Dependence of Vertex Average Distance on Truncation Parameter in a Truncated Oct

Dependence of Vertex Average Distance on Truncation Parameter in a Truncated Oct
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2020-01-10 19:12
Solid GeometryOptimization
This example demonstrates the relationship between the average distance of vertices and the truncation parameter in a truncated octahedron. Key mathematical concepts include solid geometry, optimization, and parametric equations. The construction utilizes GeoGebra commands/objects such as Join, Unique, Length, Text, Sum, Sphere, Ray, Zip, UnicodeToLetter, and TableText. Suitable for high school and above, it serves as a teaching tool for solid 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. Zip(Polygon[a], a, v_poly06) → poly06
  11. Zip(Polygon[a], a, v_poly05) → poly05
  12. Text(text12, C, false, true) → text8
  13. Zip(Segment[lP0(x(a)), lP0(y(a))], a, segN1) → seg1
  14. Zip(Segment[lP0(x(a)), lP0(y(a))], a, segN2) → seg2
  15. UnicodeToLetter(34) → u34
  16. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ6) → segΔ6
  17. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ4) → segΔ4
  18. TableText(Append["\;\;\; \fgcolor{#848482}{4-gons}", segΔ4], Append["\;\;\; \fgcolor{#007FFF}{6-gons}", segΔ6], "v") → text3a

AI Drawing Prompt Reference

Create a visual representation of a truncated octahedron, illustrating the relationship between the average vertex distance and the truncation parameter. Construct the vertices and edges of the truncated octahedron, calculate the average vertex distance, and highlight the faces as quadrilaterals and hexagons in different colors. Draw concentric spheres to represent the inner and outer radii, and add text annotations for parameter values. Finally, use rays and line segments to connect key points, completing the overall structure.
The prompt above was compiled from the original drawing commands and is for reference only.