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毕达哥拉斯树分形

毕达哥拉斯树分形
几何画板及绘图内容加载中，请稍等……
2015-10-05 19:12
多边形角几何作图
所属专题 Pythagoras Fractal Trees constructions
这个范例演示了毕达哥拉斯树分形的构造过程。它基于正方形，通过不断构造直角三角形和正方形来生成分形图形。涉及的核心数学概念包括正方形、直角三角形和分形几何。使用了Polygon、Midpoint、Semicircle、Point、Angle、Rotate、Sequence等GeoGebra命令。适合初中及以上学生学习分形几何和几何作图。

GeoGebra 命令及参数

  1. Polygon(A, B, 4) → square, a, b, c_1, d, C, D
  2. Midpoint(C, D) → E
  3. Semicircle(D, C) → e
  4. Point(e) → F
  5. Angle(C, E, F) → α
  6. Rotate(C, α, E) → C'
  7. Rotate(C, α, E) → C'_1
  8. points(A, B, α) → points1
  9. Polygon(points1) → triangulotest
  10. Join(Sequence[points[Element[points1, k], Element[points1, k + 1], α], k, 1, 4, 2]) → points2
  11. Sequence(Polygon[Element[points1, k], Element[points1, k + 1], 4], k, 1, 4, 2) → squares1
  12. Sequence(Polygon[Element[points2, k], Element[points2, k + 1], Element[points2, k + 3]], k, 1, 8, 4) → triangles2
  13. Join(Sequence[points[Element[points2, k], Element[points2, k + 1], α], k, 1, 8, 2]) → points3
  14. Sequence(Polygon[Element[points2, k], Element[points2, k + 1], 4], k, 1, 8, 2) → squares2
  15. Sequence(Polygon[Element[points3, k], Element[points3, k + 1], Element[points3, k + 3]], k, 1, 16, 4) → triangles3
  16. Join(Sequence[points[Element[points3, k], Element[points3, k + 1], α], k, 1, 16, 2]) → points4
  17. Sequence(Polygon[Element[points3, k], Element[points3, k + 1], 4], k, 1, 16, 2) → squares3
  18. Sequence(Polygon[Element[points4, k], Element[points4, k + 1], Element[points4, k + 3]], k, 1, 32, 4) → triangles4
  19. Sequence(Polygon[Element[points4, k], Element[points4, k + 1], 4], k, 1, 32, 2) → squares4
  20. Join(Sequence[points[Element[points4, k], Element[points4, k + 1], α], k, 1, 32, 2]) → points5
  21. Sequence(Polygon[Element[points5, k], Element[points5, k + 1], Element[points5, k + 3]], k, 1, 64, 4) → triangles5
  22. Sequence(Polygon[Element[points5, k], Element[points5, k + 1], 4], k, 1, 64, 2) → squares5
  23. Join(Sequence[points[Element[points5, k], Element[points5, k + 1], α], k, 1, 64, 2]) → points6
  24. Sequence(Polygon[Element[points6, k], Element[points6, k + 1], Element[points6, k + 3]], k, 1, 128, 4) → triangles6
  25. Sequence(Polygon[Element[points6, k], Element[points6, k + 1], 4], k, 1, 128, 2) → squares6
  26. Join(Sequence[points[Element[points6, k], Element[points6, k + 1], α], k, 1, 128, 2]) → points7
  27. Sequence(Polygon[Element[points7, k], Element[points7, k + 1], Element[points7, k + 3]], k, 1, 256, 4) → triangles7
  28. Sequence(Polygon[Element[points7, k], Element[points7, k + 1], 4], k, 1, 256, 2) → squares7
  29. Join(Sequence[points[Element[points7, k], Element[points7, k + 1], α], k, 1, 256, 2]) → points8
  30. Sequence(Polygon[Element[points8, k], Element[points8, k + 1], Element[points8, k + 3]], k, 1, 512, 4) → triangles8
  31. Sequence(Polygon[Element[points8, k], Element[points8, k + 1], 4], k, 1, 512, 2) → squares8
  32. Join(Sequence[points[Element[points8, k], Element[points8, k + 1], α], k, 1, 512, 2]) → points9
  33. Sequence(Polygon[Element[points8, k], Element[points8, k + 1], 4], k, 1, 512, 2) → squares9
  34. Sequence(Polygon[Element[points9, k], Element[points9, k + 1], 4], k, 1, 1024, 2) → squares10

AI 绘图提示语参考

从一条线段开始，以该线段为边构造一个正方形。取正方形上边中点，以该中点为圆心、上边为直径作半圆，在半圆上任取一点，连接该点与上边两端点形成直角三角形。以该直角三角形的两条直角边为边，分别向外构造两个正方形。重复此过程：在每个新正方形的上边重复上述半圆取点、构造直角三角形和正方形的步骤，逐层迭代，形成一棵由大小递减的正方形和直角三角形组成的分形树，整体呈对称的树状结构。
以上提示语为根据原始绘图命令整理而来，仅供参考。