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Fractal Curve: Iterated Semicircles with Scaling and Rotation

Fractal Curve: Iterated Semicircles with Scaling and Rotation
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2019-10-27 20:31
FunctionSequenceTransformationGeometric ConstructionLocus
This GeoGebra example demonstrates the construction of a fractal curve through iterative application of semicircles, points, dilation, and rotation. It visually explores self-similarity and infinite complexity typical of fractal geometry. Core mathematical concepts include fractal geometry, iterative transformations, composite scaling and rotation, and the relationship between perimeter growth and fractal dimension. The construction uses commands such as Semicircle, Point, Dilate, and Rotate, with objects including A, B, c, C, d, D, e, A1 through C7, forming nested semicircular arcs. Suitable for high school and university students studying fractal geometry, geometric transformations, and iterative processes, as well as for exploring statistical properties of perimeter in fractal curves.

GeoGebra Commands & Parameters

  1. Semicircle(A, B) → c
  2. Semicircle(A, C) → d
  3. Point(xAxis) → D
  4. Semicircle(C, D) → e
  5. Semicircle(E, F) → f
  6. Dilate(Mirror[{A1, B1, C1}, xAxis], 0.5, B) → A2
  7. Dilate(Rotate[A2, (5 * π / 4)], sqrt(2)) → B2
  8. Rotate(Translate[A2, (1, 1)], π / 2, C) → C2
  9. Dilate(Mirror[{A2, B2, C2}, xAxis], 0.500000000000002, B) → A3
  10. Dilate(Rotate[A3, (5 * π / 4)], sqrt(2)) → B3
  11. Rotate(Translate[A3, (1, 1)], π / 2, C) → C3
  12. Dilate(Mirror[{A3, B3, C3}, xAxis], 0.500000000000004, B) → A4
  13. Dilate(Rotate[A4, (5 * π / 4)], sqrt(2)) → B4
  14. Rotate(Translate[A4, (1, 1)], π / 2, C) → C4
  15. Dilate(Mirror[{A4, B4, C4}, xAxis], 0.500000000000006, B) → A5
  16. Dilate(Rotate[A5, (5 * π / 4)], sqrt(2)) → B5
  17. Rotate(Translate[A5, (1, 1)], π / 2, C) → C5
  18. Dilate(Mirror[{A5, B5, C5}, xAxis], 0.500000000000008, B) → A6
  19. Dilate(Rotate[A6, (5 * π / 4)], sqrt(2)) → B6
  20. Rotate(Translate[A6, (1, 1)], π / 2, C) → C6
  21. Dilate(Mirror[{A6, B6, C6}, xAxis], 0.50000000000001, B) → A7
  22. Dilate(Rotate[A7, (5 * π / 4)], sqrt(2)) → B7
  23. Rotate(Translate[A7, (1, 1)], π / 2, C) → C7

AI Drawing Prompt Reference

Draw a fractal curve by iteratively applying semicircles, points, scaling, and rotation. Start by creating a point D on the x-axis, then draw a semicircle with D as the starting point. For each subsequent semicircle, perform scaling, rotation, and mirroring, using the center of the previous semicircle as the new starting point, with a scaling factor that gradually decreases and a fixed rotation angle of 5π/4. After each rotation, translate by one unit. Repeat this process six times to form a complex fractal pattern.
The prompt above was compiled from the original drawing commands and is for reference only.