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Koch雪花的3D立体几何构造

Koch雪花的3D立体几何构造
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2020-10-17 05:47
立体几何变换几何作图
Koch雪花的3D立体几何构造展示了分形几何在三维空间中的应用。该范例涉及的核心数学概念包括分形几何、几何变换和立体图形构造。作图要点包括使用GeoGebra命令如Polygon、Cube、Tetrahedron、Sequence、Mirror和Translate来构建四面体立方体结构。适合高中生或大学低年级学生用于学习几何变换和分形概念。

GeoGebra 命令及参数

  1. Polygon(Take[T0, 1, 3]) → G00
  2. Cube(T0(4), (T0(1) + T0(2) - T0(3) + T0(4)) / 2, O) → Cubus, O2, faceAEDO, faceBCDE, edge3, edge2, edge1, edgeAO, edge4, edgeAE, edgeBE, edgeBC, O7, edge5, edgeCD_{1}, edgeDO, edgeDE, O6, O4, O3, face1, face4, face3, face2
  3. Tetrahedron(G00, true) → P0, A, face6, face5, face7, edge6, edge8, edge7
  4. Polygon(Take[T1, 1, 3]) → G10
  5. Tetrahedron(G10, true) → P1, B, face9, face8, face10, edge9, edge11, edge10
  6. Sequence(Polygon[Element[T2, j]], j, 1, 8) → G20
  7. Sequence(Tetrahedron[Element[G20, j], ¬(j ≟ 7 ∨ j ≟ 2)], j, 1, 8) → P2
  8. Sequence(Polygon[Element[T31, j]], j, 1, Length[T31]) → G31
  9. Sequence(Tetrahedron[Element[G31, j], j ≟ 4 ∨ j ≟ 5], j, 1, 8) → P31
  10. Mirror(P31, Plane[O6, O, O2]) → P32
  11. Mirror(P31, Plane[O6, O3, O5]) → P33
  12. Mirror(P31, Plane[O, O3, O5]) → P34
  13. Mirror(P31, Plane[(O5 + O6) / 2, (O8 + O7) / 2, (O2 + O3) / 2]) → P35
  14. Mirror(P31, Plane[(O3 + O7) / 2, (O8 + O2) / 2, (O5 + O) / 2]) → P36
  15. Mirror(P31, Plane[(O + O2) / 2, (O8 + O5) / 2, (O7 + O6) / 2]) → P37
  16. Sequence(Polygon[Element[T41, j]], j, 1, Length[T41]) → G51
  17. Sequence(Tetrahedron[Element[G51, j], j ≟ 4 ∨ j ≟ 5], j, 1, 8) → P41
  18. Translate(P41, Vector[(O2 - O8) / 4]) → P42
  19. Translate(P41, Vector[(O2 - O8) / 2]) → P43
  20. Translate(P41, Vector[((O2 - O8) * 3 / 4)]) → P44
  21. Translate(P41, Vector[(O7 - O8) / 4]) → P45
  22. Translate(P45, Vector[(O2 - O8) / 4]) → P46
  23. Translate(P45, Vector[(O2 - O8) / 2]) → P47
  24. Translate(P45, Vector[((O2 - O8) * 3 / 4)]) → P48
  25. Translate(P41, Vector[(O7 - O8) / 2]) → P49
  26. Translate(P49, Vector[(O2 - O8) / 4]) → P4A
  27. Translate(P49, Vector[(O2 - O8) / 2]) → P4B
  28. Translate(P49, Vector[((O2 - O8) * 3 / 4)]) → P4C
  29. Translate(P41, Vector[((O7 - O8) * 3 / 4)]) → P4D
  30. Translate(P4D, Vector[(O2 - O8) / 4]) → P4E
  31. Translate(P4D, Vector[(O2 - O8) / 2]) → P4F
  32. Translate(P4D, Vector[((O2 - O8) * 3 / 4)]) → P4G
  33. Cube(O8, (O8 + O2) / 2, Vector[O - O2]) → Cubus_3, C, face13, faceCDHG, faceEFGH, edgeDO8, edge12, edge13, edgeCD, edgeEO8, edgeDH, edgeEH, D, edgeEF, edge14, edgeFG, edgeCG, edgeGH, E, F, G, H, face11, faceDHEO8, face12
  34. Cube(O8, O8 + (O2 - O8) / 4, Vector[O - O2]) → Cubus_4, I, face16, faceIJNM, faceKLMN, edgeJO8, edge15, edge16, edgeIJ, edgeKO8, edgeJN, edgeKN, J, edgeKL, edge17, edgeLM, edgeIM, edgeMN, K, L, M, N, face14, faceJNKO8, face15
  35. Cube(O8, O8 + (O2 - O8) / 8, Vector[O - O2]) → Cubus_5, P, face19, facePQUT, faceRSTU, edgeO8Q, edge18, edge19, edgePQ, edgeO8R, edgeQU, edgeRU, Q, edgeRS, edge20, edgeST, edgePT, edgeTU, R, S, T, U, face17, faceO8QUR, face18
  36. Sequence(Polygon[Element[T51, j]], j, 1, Length[T51]) → G61
  37. Sequence(Tetrahedron[Element[G61, j], j ≟ 4 ∨ j ≟ 5], j, 1, 8) → P50
  38. Sequence(Translate[Translate[P50, Vector[(O7 - O8) / 8]], Vector[((O2 - O8) * j / 8)]], j, 0, 7) → P52
  39. Sequence(Translate[Translate[P50, Vector[((O7 - O8) * 2 / 8)]], Vector[((O2 - O8) * j / 8)]], j, 0, 7) → P53
  40. Sequence(Translate[Translate[P50, Vector[((O7 - O8) * 3 / 8)]], Vector[((O2 - O8) * j / 8)]], j, 0, 7) → P54
  41. Sequence(Translate[Translate[P50, Vector[((O7 - O8) * 4 / 8)]], Vector[((O2 - O8) * j / 8)]], j, 0, 7) → P55
  42. Sequence(Translate[Translate[P50, Vector[((O7 - O8) * 5 / 8)]], Vector[((O2 - O8) * j / 8)]], j, 0, 7) → P56
  43. Sequence(Translate[Translate[P50, Vector[((O7 - O8) * 6 / 8)]], Vector[((O2 - O8) * j / 8)]], j, 0, 7) → P57
  44. Sequence(Translate[Translate[P50, Vector[((O7 - O8) * 7 / 8)]], Vector[((O2 - O8) * j / 8)]], j, 0, 7) → P58
  45. Sequence(Translate[P50, Vector[((O2 - O8) * j / 8)]], j, 0, 7) → P51
  46. Translate(P31, Vector[(O4 - O8) / 2]) → P38

AI 绘图提示语参考

创建一个三维科赫雪花分形的立体构造：先给定一个多边形作基底并生成一个立方体结构，再在基底上形成一个四面体，并对其对应的顶点面进行复制生成多层分段体。接着把同一组四面体/多边形通过关于若干平面的镜像得到多份对称体，使整体呈现沿空间方向展开的分形雏形。随后使用沿特定方向的平移，把镜像后的分段体按等比例步长重复堆叠到更深的层级，形成由多个尺度单元组成的三维分形雪花结构。
以上提示语为根据原始绘图命令整理而来，仅供参考。