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Three-Parameter Transformations of the Octahedron and Extreme Distributions

Three-Parameter Transformations of the Octahedron and Extreme Distributions
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2020-03-06 15:56
Solid GeometryTransformationOptimization
This example demonstrates the three-parameter transformations of the octahedron and their extreme distributions. Key mathematical concepts include transformations, extrema, and distributions. Using GeoGebra commands such as Length, Sequence, and Zip, you can dynamically adjust parameters to observe the transformation effects. 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(m8(1, j)), V0c(m8(1, If[j ≟ 10, 1, j + 1]))], j, 1, 10) → seg8_i
  14. Length(V0cc) → n3
  15. Length(V00(1)) → läV00_1
  16. Sequence(Segment[V0c(m6a(1, j)), V0c(m6a(1, If[j ≟ 6, 1, j + 1]))], j, 1, 6) → seg6a_i
  17. KeepIf(x > 0, seg6_i) → seg6_i1
  18. KeepIf(x > 0, seg6a_i) → seg6a_i1
  19. KeepIf(x > 0, seg4_i) → seg4_i1
  20. KeepIf(x > 0, seg8_i) → seg8_i1
  21. Length(V0c(1)) → läV0c_1
  22. Dilate(q0, q, Cmi(1)) → q0'
  23. Sphere(O, läV0c_1) → sphe
  24. Sequence(Polygon[Sequence[V0c(m6(i, k)), k, 1, Length[m6(1)]]], i, 1, Length[m6]) → Poly6
  25. Sequence(Polygon[Sequence[V0c(m8(i, k)), k, 1, Length[m8(1)]]], i, 1, Length[m8]) → Poly8
  26. Sequence(Polygon[Sequence[V0c(m3a(i, k)), k, 1, Length[m3a(1)]]], i, 1, Length[m3a]) → Poly3a
  27. Sequence(Polygon[Sequence[V0c(m3b(i, k)), k, 1, Length[m3b(1)]]], i, 1, Length[m3b]) → Poly3b
  28. Sequence(Polygon[Sequence[V0c(m6a(i, k)), k, 1, Length[m6a(1)]]], i, 1, Length[m6a]) → Poly6a
  29. Sequence(Polygon[Sequence[V0c(m4(i, k)), k, 1, Length[m4(1)]]], i, 1, Length[m4]) → Poly4
  30. Sequence(Polygon[Sequence[V0c(m4a(i, k)), k, 1, Length[m4a(1)]]], i, 1, Length[m4a]) → Poly4a
  31. Sequence(Segment[V0c(m3a(1, j)), V0c(m3a(1, If[j ≟ 3, 1, j + 1]))], j, 1, 3) → seg3a_i
  32. Sequence(Segment[V0c(m3b(1, j)), V0c(m3b(1, If[j ≟ 3, 1, j + 1]))], j, 1, 3) → seg3b_i
  33. Sequence(Segment[V0c(m4a(1, j)), V0c(m4a(1, If[j ≟ 4, 1, j + 1]))], j, 1, 4) → seg4a_i
  34. KeepIf(x > 0, seg4a_i) → seg4a_i1
  35. TableText(Append["segments", 1…8], Append["\fgcolor{red}{ poly3a} ", seg3a_i], Append["\fgcolor{green}{poly3b}", seg3b_i], Append["\fgcolor{#BD33A4}{poly4}", seg4a_i1], Append["\fgcolor{#C19A6B}{poly\; 6\; }", seg6_i1], "v") → text6c
  36. TableText(Append["segments", 1…8], Append["\fgcolor{red}{ poly4} ", seg4_i1], Append["\fgcolor{green}{poly\; 6\; or\; 3}", seg6_i1], Append["\fgcolor{#C19A6B}{poly\; 8\; or\; 4}", seg8_i1], "v") → text6b
  37. Text("" + p) → txt_p
  38. Sum(Flatten[Sequence[Append[Sequence[Element[txt_p, i], i, m1(k), m2(k)], " "], k, 1, 5]]) → txtUm1
  39. Text(text18r, A, false, true) → text18s

AI Drawing Prompt Reference

Draw a three-parameter transformation of an octahedron, showing its extreme distributions. Start by constructing an octahedron, then adjust parameters using length and sequence commands to observe transformation effects. Dilate and rotate the transformed vertices to form new geometric shapes. Finally, label the edges and faces of the new shapes with different colors and add text annotations to compare the before and after transformations.
The prompt above was compiled from the original drawing commands and is for reference only.