MathPlotAI Logo MathPlotAI

Three-Parameter Model Transformations of the Icosahedron with Extreme Distributi

Three-Parameter Model Transformations of the Icosahedron with Extreme Distributi
Loading the math board and drawing, please wait…
2020-03-06 17:32
Solid GeometryTransformationOptimization
This example demonstrates the extreme distributions of an icosahedron under three-parameter model transformations. Key mathematical concepts include transformations, extreme distributions, and solid geometry. Using GeoGebra commands such as Length, Sequence, and Zip, the transformed icosahedron and its extreme distributions can be constructed. Suitable for high school mathematics education and research.

GeoGebra Commands & Parameters

  1. Length(m00(1)) → n00_1
  2. Length(m00) → n00
  3. Sequence({V00(m00(i, 1) + 1), V00(m00(i, 2) + 1), V00(m00(i, 3) + 1)}, i, 1, n00) → V00'
  4. Sequence(Dilate[Flatten[Sequence[V00'(i, j) + ((If[j < n00_1, V00'(i, j + 1), V00'(i, 1)] - V00'(i, j)) * {t, 1 - t}), j, 1, n00_1]], q, Cmi(i)], i, 1, n00) → V0a
  5. Zip(Rotate[a, α, Line[(0, 0), b]], a, V0a, b, Cmi) → V0b
  6. Join(V0b) → V0c
  7. If(t ≟ 0 ∨ t ≟ 0.5 ∨ q ≟ 1 ∨ q ≟ 0, Unique[V0c], V0c) → V0cc
  8. Length(V0cc) → läV'
  9. Sequence(Polygon[Sequence[V00(m00(i, k) + 1), k, 1, Length[m00(1)]]], i, 1, Length[m00]) → Poly3
  10. Length(V0c) → läV0c
  11. Sequence(Segment[V0c(m4(1, j)), V0c(m4(1, If[j ≟ 4, 1, j + 1]))], j, 1, 4) → seg4_i
  12. Sequence(Segment[V0c(m6(1, j)), V0c(m6(1, If[j ≟ 6, 1, j + 1]))], j, 1, 6) → seg6_i
  13. Sequence(Segment[V0c(m10(1, j)), V0c(m10(1, If[j ≟ 10, 1, j + 1]))], j, 1, 10) → seg10_i
  14. Length(V0cc) → n3
  15. Length(V00(1)) → läV00_1
  16. Sequence(Segment[V0c(m3a(1, j)), V0c(m3a(1, If[j ≟ 3, 1, j + 1]))], j, 1, 3) → seg3a_i
  17. Sequence(Segment[V0c(m3b(1, j)), V0c(m3b(1, If[j ≟ 3, 1, j + 1]))], j, 1, 3) → seg3b_i
  18. Sequence(Segment[V0c(m5a(1, j)), V0c(m5a(1, If[j ≟ 5, 1, j + 1]))], j, 1, 5) → seg5a_i
  19. KeepIf(x > 0, seg6_i) → seg6_i1
  20. KeepIf(x > 0, seg4_i) → seg4_i1
  21. KeepIf(x > 0, seg10_i) → seg10_i1
  22. Length(V0c(1)) → läV0c_1
  23. Dilate(q0, q, Cmi(1)) → q0'
  24. Sphere(O, läV0c_1) → sphe
  25. Sequence(Polygon[Sequence[V0c(m6(i, k)), k, 1, Length[m6(1)]]], i, 1, Length[m6]) → Poly6
  26. Sequence(Polygon[Sequence[V0c(m10(i, k)), k, 1, Length[m10(1)]]], i, 1, Length[m10]) → Poly10
  27. Sequence(Polygon[Sequence[V0c(m3a(i, k)), k, 1, Length[m3a(1)]]], i, 1, Length[m3a]) → Poly3a
  28. Sequence(Polygon[Sequence[V0c(m3b(i, k)), k, 1, Length[m3b(1)]]], i, 1, Length[m3b]) → Poly3b
  29. Sequence(Polygon[Sequence[V0c(m5a(i, k)), k, 1, Length[m5a(1)]]], i, 1, Length[m5a]) → Poly5a
  30. Sequence(Polygon[Sequence[V0c(m4(i, k)), k, 1, Length[m4(1)]]], i, 1, Length[m4]) → Poly4
  31. TableText(Append["segments", 1…6], Append["\fgcolor{#FE6F5E}{ poly3a} ", seg3a_i], Append["\fgcolor{#D19FE8}{ poly3b} ", seg3b_i], Append["\fgcolor{#008000}{poly\; 5\; }", seg5a_i], Append["\fgcolor{#008B8B}{poly\;6\; or\; 3 }", seg6_i1], "v") → text6c
  32. Text("" + p) → txt_p
  33. Sum(Flatten[Sequence[Append[Sequence[Element[txt_p, i], i, m1(k), m2(k)], " "], k, 1, 5]]) → txtUm1
  34. Text(text18r, A, false, true) → text18s
  35. TableText(Append["segments", 1…10], Append["\fgcolor{red}{ poly4} ", seg4_i1], Append["\fgcolor{#66CCFF}{poly\; 6\; or\; 3}", seg6_i1], Append["\fgcolor{#C19A6B}{poly\; 10\; or\; 5}", seg10_i1], "v") → text6b