MathPlotAI Logo MathPlotAI

Extreme Distribution of 72 Points on a Sphere: Polyhedron Coloring and Dual Comp

Extreme Distribution of 72 Points on a Sphere: Polyhedron Coloring and Dual Comp
Loading the math board and drawing, please wait…
2020-10-22 19:32
PolygonSolid GeometryTransformationOptimizationSlider Animation
This GeoGebra example demonstrates the extreme distribution of 72 points on the surface of a sphere, constructing the convex hull polyhedron and comparing it with two known but non-extreme configurations having the same number of vertices. The core mathematical concepts include spherical point distribution optimization (such as Thomson or Tammes problems), convex hull construction, polyhedral duality, and visualization of vertices, edges, and faces through coloring. The construction uses GeoGebra commands including Sphere for the base sphere, Text, LaTeX, and TableText for dynamic labels, Zip, If, and Sum for list operations on vertex coordinates and distance calculations, and Append with UnicodeToLetter for data sequencing. Interactive sliders (n, r, i0, j0) and input boxes (InputBox1, InputBox2) allow users to adjust parameters in real time. This resource is suitable for high school to university-level mathematics, particularly for exploring solid geometry, optimization problems, and computational geometry. It serves as an excellent tool for mathematical modeling, competition training, or classroom demonstrations to visualize the relationship between extreme distributions and polyhedral structures.

GeoGebra Commands & Parameters

  1. Text("" + Um1) → txt_p
  2. LaTeX(Um1) → lll1_1
  3. LaTeX(Um0) → lll0
  4. Sphere(O, R) → p1
  5. Textfield(n) → InputBox1
  6. Textfield(i0) → InputBox2
  7. Text(r) → Txtr
  8. Textfield(k_1) → InputBox4
  9. UnicodeToLetter(34) → u34
  10. Textfield(k_2) → InputBox5
  11. TableText(ltxt0, ltxt1, "v") → textltxt01
  12. Append(A, Sequence[((random(), random(), random()) * R), i, 2, n]) → l2s
  13. Text(("(up\; to\; 10^{-15})\;\\ accuracy= " * ScientificText[cc, 3]), C, true, true) → text5r
  14. Zip(Vector[a, a + UnitVector[a]], a, lP') → lVector1
  15. Textfield(scale) → InputBox6
  16. TableText(Append["i", 1…n], Append[" \fgcolor{orange}{GM :Δφ_i}, rad", l1], Append["\fgcolor{magenta}{GC :Δφ_i} , rad", l1c], "v") → text23
  17. Text("\;\;\;\;\;\;\;\;\;\;\;\;\;\fgcolor{" + (If[c', "orange", "brown"]) + "} \boxed{ " + c' + "}\;\;\;\; \;\;\;\;\; \\\;\; ( accuracy- up\; to\;\; 2.*10^{" + k_2 + ("\;})\;\\\;\; ➤➤➤\;\;\;\scalebox{1.3}{ " + (ScientificText[cc, 3])) + "}\;█ ", C, true, true) → text9
  18. If(bu, "Animation", "Stop") → bue0
  19. Zip(Vector[a, a + (1.5 * UnitVector[b])], a, lP', b, Wi_P) → lVector3
  20. Zip(Vector[a, a + (1.5 * UnitVector[b])], a, lP', b, Wi_Pc) → lVector2
  21. TableText({Join[{{"n"}, Sequence[i, i, 4, 69 + 3]}], Join[{{" p_n=P_n/[R*n*(n-1)/2]"}, lp}]}, "vl") → txt_lp
  22. Sum(Flatten[Sequence[Append[Sequence[Element[txt_p, i], i, m1(k), m2(k)], " "], k, 1, 5]]) → txtUm1
  23. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(2)) → ll2
  24. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(3)) → ll3
  25. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(4)) → ll4
  26. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(5)) → ll5
  27. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(6)) → ll6
  28. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(7)) → ll7
  29. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(8)) → ll8
  30. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(9)) → ll9
  31. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(10)) → ll10
  32. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(1)) → ll1

AI Drawing Prompt Reference

Draw a sphere with 72 optimally distributed points, forming a convex polyhedron. Color the vertices, edges, and faces of the polyhedron differently. Adjust parameters n, i0, j0, k1, and k2 to change the number and distribution of points, and observe the changes in the polyhedron.
The prompt above was compiled from the original drawing commands and is for reference only.