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Pythagoras Tree Fractal

Pythagoras Tree Fractal
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2015-10-05 19:12
PolygonAngleGeometric Construction
In collections Pythagoras Fractal Trees constructions
This example demonstrates the construction of the Pythagoras tree fractal, which is based on squares and involves right triangles. Key mathematical concepts include squares, right triangles, and fractal geometry. It uses commands like Polygon, Midpoint, Semicircle, Point, Angle, Rotate, and Sequence in GeoGebra. Suitable for middle school and above students to learn fractal geometry and geometric construction.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Start with a line segment and build a square on it. Take the midpoint of the square's top side, draw a semicircle with that side as diameter, and pick a point on the semicircle. Connect this point to the two endpoints of the top side to form a right triangle. Then construct squares outward on the two legs of this triangle. Repeat the process: on each new square's top side, again draw a semicircle, pick a point, form a right triangle, and build squares on its legs. Continue iterating this procedure layer by layer, producing a symmetric fractal tree composed of progressively smaller squares and right triangles.
The prompt above was compiled from the original drawing commands and is for reference only.