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Steve's Fives Fractal

Steve's Fives Fractal
几何画板及绘图内容加载中，请稍等……
2019-11-03 22:53
微积分几何作图
①本范例演示了一个由Maestro Steve Phelps创建的分形图形，通过迭代构造出复杂的几何图案；②涉及的核心数学概念包括分形几何、百分比计算、上下和/黎曼和、项、圆锥截面、组合数学；③作图要点——从下方『构造摘要』提炼用到了PolyLine（多段线）、Segment（线段）、Rotate（旋转）、Polygon（多边形）等GeoGebra命令/对象；④适合高中及以上数学学习者，用于理解分形构造和几何变换的原理。

GeoGebra 命令及参数

  1. PolyLine(A, B, C, D, E) → f
  2. Segment(A, B) → g
  3. Segment(B, C) → h
  4. Segment(C, D) → i
  5. Segment(D, E) → j
  6. Segment(A, E) → k
  7. Rotate(PolyLine[Dilate[A, g / k, A], Dilate[B, g / k, A], Dilate[C, g / k, A], Dilate[D, g / k, A], Dilate[E, g / k, A]], Angle[k, g], A) → A1
  8. Rotate(Translate[PolyLine[Dilate[A, h / k, A], Dilate[B, h / k, A], Dilate[C, h / k, A], Dilate[D, h / k, A], Dilate[E, h / k, A]], B - A], Angle[k, h], B) → B1
  9. Rotate(Translate[PolyLine[Dilate[A, i / k, A], Dilate[B, i / k, A], Dilate[C, i / k, A], Dilate[D, i / k, A], Dilate[E, i / k, A]], C - A], Angle[k, i], C) → C1
  10. Rotate(Translate[PolyLine[Dilate[A, j / k, A], Dilate[B, j / k, A], Dilate[C, j / k, A], Dilate[D, j / k, A], Dilate[E, j / k, A]], D - A], Angle[k, j], D) → D1
  11. Rotate(Dilate[{A1, B1, C1, D1}, g / k, A], Angle[k, g], A) → A2
  12. Rotate(Translate[Dilate[{A1, B1, C1, D1}, h / k, A], B - A], Angle[k, h], B) → B2
  13. Rotate(Translate[Dilate[{A1, B1, C1, D1}, i / k, A], C - A], Angle[k, i], C) → C2
  14. Rotate(Translate[Dilate[{A1, B1, C1, D1}, j / k, A], D - A], Angle[k, j], D) → D2
  15. Rotate(Dilate[{A2, B2, C2, D2}, g / k, A], Angle[k, g], A) → A3
  16. Rotate(Translate[Dilate[{A2, B2, C2, D2}, h / k, A], B - A], Angle[k, h], B) → B3
  17. Rotate(Translate[Dilate[{A2, B2, C2, D2}, i / k, A], C - A], Angle[k, i], C) → C3
  18. Rotate(Translate[Dilate[{A2, B2, C2, D2}, j / k, A], D - A], Angle[k, j], D) → D3
  19. Rotate(Dilate[{A3, B3, C3, D3}, g / k, A], Angle[k, g], A) → A4
  20. Rotate(Translate[Dilate[{A3, B3, C3, D3}, h / k, A], B - A], Angle[k, h], B) → B4
  21. Rotate(Translate[Dilate[{A3, B3, C3, D3}, i / k, A], C - A], Angle[k, i], C) → C4
  22. Rotate(Translate[Dilate[{A3, B3, C3, D3}, j / k, A], D - A], Angle[k, j], D) → D4
  23. Rotate(Dilate[{A4, B4, C4, D4}, g / k, A], Angle[k, g], A) → A5
  24. Rotate(Translate[Dilate[{A4, B4, C4, D4}, h / k, A], B - A], Angle[k, h], B) → B5
  25. Rotate(Translate[Dilate[{A4, B4, C4, D4}, i / k, A], C - A], Angle[k, i], C) → C5
  26. Rotate(Translate[Dilate[{A4, B4, C4, D4}, j / k, A], D - A], Angle[k, j], D) → D5
  27. Rotate(Dilate[{A5, B5, C5, D5}, g / k, A], Angle[k, g], A) → A6
  28. Rotate(Translate[Dilate[{A5, B5, C5, D5}, h / k, A], B - A], Angle[k, h], B) → B6
  29. Rotate(Translate[Dilate[{A5, B5, C5, D5}, i / k, A], C - A], Angle[k, i], C) → C6
  30. Rotate(Translate[Dilate[{A5, B5, C5, D5}, j / k, A], D - A], Angle[k, j], D) → D6
  31. Polygon(F, G, 5) → poly1, l, m, p, q, r, H, I, J