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雪花分形

雪花分形
几何画板及绘图内容加载中，请稍等……
2019-11-02 20:43
函数三角函数几何作图
①本范例演示了如何通过迭代构造雪花分形图案；②涉及的核心数学概念包括分形几何、切线函数、对数函数、多边形构造与旋转；③作图要点——从下方『构造摘要』提炼用到了Segment（线段）、PolyLine（多边形线）、Rotate（旋转）等GeoGebra命令与对象；④适合高中及以上数学学习者，用于理解分形图案的生成过程与数学原理。

GeoGebra 命令及参数

  1. Segment(A, B) → f
  2. Segment(B, C) → g
  3. Segment(C, D) → h
  4. Segment(D, E) → i
  5. Segment(E, F) → j
  6. Segment(F, G) → k
  7. Segment(A, G) → l
  8. Segment(G, H) → m
  9. Segment(H, A) → n
  10. PolyLine(A, B, C, D, E, F, G) → p
  11. Rotate(PolyLine[Dilate[A, f / l, A], Dilate[B, f / l, A], Dilate[C, f / l, A], Dilate[D, f / l, A], Dilate[E, f / l, A], Dilate[F, f / l, A], Dilate[G, f / l, A]], Angle[l, f], A) → A1
  12. Rotate(Translate[PolyLine[Dilate[A, g / l, A], Dilate[B, g / l, A], Dilate[C, g / l, A], Dilate[D, g / l, A], Dilate[E, g / l, A], Dilate[F, g / l, A], Dilate[G, g / l, A]], B - A], Angle[l, g], B) → B1
  13. Rotate(Translate[PolyLine[Dilate[A, h / l, A], Dilate[B, h / l, A], Dilate[C, h / l, A], Dilate[D, h / l, A], Dilate[E, h / l, A], Dilate[F, h / l, A], Dilate[G, h / l, A]], C - A], Angle[l, h], C) → C1
  14. Rotate(Translate[PolyLine[Dilate[A, i / l, A], Dilate[B, i / l, A], Dilate[C, i / l, A], Dilate[D, i / l, A], Dilate[E, i / l, A], Dilate[F, i / l, A], Dilate[G, i / l, A]], D - A], Angle[l, i], D) → D1
  15. Rotate(Translate[PolyLine[Dilate[A, j / l, A], Dilate[B, j / l, A], Dilate[C, j / l, A], Dilate[D, j / l, A], Dilate[E, j / l, A], Dilate[F, j / l, A], Dilate[G, j / l, A]], E - A], Angle[l, j], E) → E1
  16. Rotate(Translate[PolyLine[Dilate[A, k / l, A], Dilate[B, k / l, A], Dilate[C, k / l, A], Dilate[D, k / l, A], Dilate[E, k / l, A], Dilate[F, k / l, A], Dilate[G, k / l, A]], F - A], Angle[l, k], F) → F1
  17. Rotate(Dilate[{A1, B1, C1, D1, E1, F1}, f / l, A], Angle[l, f], A) → A2
  18. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, g / l, A], B - A], Angle[l, g], B) → B2
  19. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, h / l, A], C - A], Angle[l, h], C) → C2
  20. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, i / l, A], D - A], Angle[l, i], D) → D2
  21. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, j / l, A], E - A], Angle[l, j], E) → E2
  22. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, k / l, A], F - A], Angle[l, k], F) → F2
  23. Rotate(Dilate[{A2, B2, C2, D2, E2, F2}, f / l, A], Angle[l, f], A) → A3
  24. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, g / l, A], B - A], Angle[l, g], B) → B3
  25. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, h / l, A], C - A], Angle[l, h], C) → C3
  26. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, i / l, A], D - A], Angle[l, i], D) → D3
  27. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, j / l, A], E - A], Angle[l, j], E) → E3
  28. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, k / l, A], F - A], Angle[l, k], F) → F3
  29. Rotate(Dilate[{A3, B3, C3, D3, E3, F3}, f / l, A], Angle[l, f], A) → A4
  30. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, g / l, A], B - A], Angle[l, g], B) → B4
  31. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, h / l, A], C - A], Angle[l, h], C) → C4
  32. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, i / l, A], D - A], Angle[l, i], D) → D4
  33. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, j / l, A], E - A], Angle[l, j], E) → E4
  34. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, k / l, A], F - A], Angle[l, k], F) → F4
  35. Rotate(Translate[PolyLine[Dilate[A, m / l], Dilate[B, m / l], Dilate[C, m / l], Dilate[D, m / l], Dilate[E, m / l], Dilate[F, m / l], Dilate[G, m / l]], G - A], Angle[l, m], G) → q
  36. Rotate(Translate[PolyLine[Dilate[A, n / l, G], Dilate[B, n / l, G], Dilate[C, n / l, G], Dilate[D, n / l, G], Dilate[E, n / l, G], Dilate[F, n / l, G], Dilate[G, n / l, G]], A - G], Angle[l, n], A) → r
  37. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, m / l], G - A], Angle[l, m], G) → A8
  38. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, n / l, G], A - G], Angle[l, n], A) → B8
  39. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, m / l], G - A], Angle[l, m], G) → A9
  40. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, n / l, G], A - G], Angle[l, n], A) → B9
  41. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, m / l], G - A], Angle[l, m], G) → A10
  42. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, n / l, G], A - G], Angle[l, n], A) → B10
  43. Rotate(Translate[Dilate[{A4, B4, C4, D4, E4, F4}, m / l], G - A], Angle[l, m], G) → A11
  44. Rotate(Translate[Dilate[{A4, B4, C4, D4, E4, F4}, n / l, G], A - G], Angle[l, n], A) → B11