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Example of Non-uniqueness of Extreme Distribution for n=16 Particles on a Sphere

Example of Non-uniqueness of Extreme Distribution for n=16 Particles on a Sphere
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2020-10-21 23:07
SequenceCoordinate GeometrySolid GeometryOptimizationGeometric Construction
① This example demonstrates that for 16 particles placed on the surface of a sphere, the extreme distribution (e.g., minimizing potential energy or maximizing symmetry) is not unique, with multiple equivalent configurations existing. ② Core mathematical concepts include spherical geometry, combinatorial optimization, extreme distributions, symmetry, non-uniqueness, and balanced arrangement of points on a curved surface. ③ Construction highlights: The GeoGebra commands used include Sphere to create the sphere, PointIn to generate particles on the surface, Sequence and Zip for batch processing coordinates, Textfield and InputBox for interactive parameter input, TableText for data display, and buttons to control calculations. Key objects include spheres, points, texts, sliders, and input boxes. ④ This is suitable for high school to university-level teaching in mathematics or physics, useful for exploring spherical point distribution problems, optimization algorithms, and geometric symmetry, and can serve as a demonstration for mathematical modeling or computational geometry.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, 0), R) → d_1
  2. Textfield(n) → InputBox2
  3. Textfield(h) → InputBox3
  4. Textfield(R) → InputBox1
  5. PointIn(d_1) → A
  6. Append(A, Sequence[((random(), random(), random()) * R), i, 2, n]) → l2s
  7. TableText({Join[{{"n"}, Sequence[i, i, 4, 57 + 3]}], Join[{{" p_n=P_n/[R*n*(n-1)/2]"}, l1}]}, "vl") → text8
  8. Textfield(i) → InputBox5
  9. Textfield(Um002) → InputBox4
  10. Sequence(Sequence[Segment[Element[R01'_1, k], Element[R01'_1, t]], t, k + 1, n], k, 1, n - 1) → segments1
  11. UnicodeToLetter(34) → u34
  12. Text("\;\;\;\;\;\;\;\;\;\;\;\;\;\bgcolor{#F2F3F4}{\fgcolor{" + (If[cc1', "orange", "#536872"]) + "} \boxed{ " + cc1' + ("}}\;\;\;\; \;\;\;\;\; \\ \;\; ( accuracy- up\; to\;\; 2*10^{" - k_2) + ("\;})\;\\\;\; ➤➤➤\;\;\;\scalebox{1.3}{ " + (ScientificText[cc1, 3])) + "}\;█ ", B01, true, true) → text01
  13. Text("\;\;\;\;\;\;\;\;\;\;\;\;\;\bgcolor{#F2F3F4}{\fgcolor{" + (If[cc2', "orange", "#536872"]) + "} \boxed{ " + cc2' + ("}}\;\;\;\; \;\;\;\;\; \\ \;\; ( accuracy- up\; to\;\; 2*10^{" - k_2) + ("\;})\;\\\;\; ➤➤➤\;\;\;\scalebox{1.3}{ " + (ScientificText[cc2, 3])) + "}\;█ ", B02, true, true) → text02
  14. Sequence(Sequence[Segment[Element[R01'_2, k], Element[R01'_2, t]], t, k + 1, n], k, 1, n - 1) → segments2
  15. Sum(Flatten[Sequence[Append[Sequence[Element[Um1, i], i, m101(k), m201(k)], " "], k, 1, 5]]) → txtUm11
  16. ptxt(Text[Sum11]) → txtsum11
  17. ptxt(Text[Sum12]) → txtsum12
  18. ptxt(Text[Sum21]) → txtsum21
  19. ptxt(Text[Sum22]) → txtsum22
  20. ptxt(Text[Element[l1, n - 3]]) → txttable
  21. TableText(lp00, lp11, lp22, "v") → text6d
  22. Zip(Segment[Element[R01'_1, y(a)], Element[R01'_1, z(a)]], a, lr_1b(1)) → ll_11
  23. Zip(Segment[Element[R01'_1, y(a)], Element[R01'_1, z(a)]], a, lr_1b(2)) → ll_12
  24. Zip(Segment[Element[R01'_1, y(a)], Element[R01'_1, z(a)]], a, lr_1b(3)) → ll_13
  25. Zip(Segment[Element[R01'_1, y(a)], Element[R01'_1, z(a)]], a, lr_1b(4)) → ll_14
  26. Zip(Length[a], a, {ll_11, ll_12, ll_13, ll_14}) → lä_1
  27. Zip(Segment[Element[R01'_2, y(a)], Element[R01'_2, z(a)]], a, lr_2b(1)) → ll_21
  28. Zip(Segment[Element[R01'_2, y(a)], Element[R01'_2, z(a)]], a, lr_2b(2)) → ll_22
  29. Zip(Segment[Element[R01'_2, y(a)], Element[R01'_2, z(a)]], a, lr_2b(3)) → ll_23
  30. Zip(Segment[Element[R01'_2, y(a)], Element[R01'_2, z(a)]], a, lr_2b(4)) → ll_24
  31. Zip(Length[a], a, {ll_21, ll_22, ll_23, ll_24}) → lä_2

AI Drawing Prompt Reference

In a 3D scene, place a sphere as the main geometric object, with user inputs for the particle count n, a height parameter h, and the sphere radius R. Generate n points randomly on the sphere’s surface to represent particles. Display a related normalized quantity depending on n in a table-like text area. Then construct and draw multiple groups of short line segments that illustrate the structured connections between corresponding points, using an indexed sequence so the segment sets appear in parallel. Add on-screen text elements to present the extreme-distribution discussion and accuracy-related numeric items, and include controls that let the viewer trigger the displayed computations.
The prompt above was compiled from the original drawing commands and is for reference only.