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Fractal Spirograph

Fractal Spirograph
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2020-04-10 21:11
TransformationParametric EquationGeometric Construction
This example demonstrates the generation of fractal spirographs. It involves the concepts of parametric curves and similarity transformations. Using GeoGebra commands such as Line, Circle, Intersect, Point, Rotate, Ray, Segment, and CurveCartesian, complex fractal patterns can be constructed. Suitable for high school mathematics education and learning about parametric equations and geometric transformations.

GeoGebra Commands & Parameters

  1. Line(A, xAxis) → n
  2. Circle(A, R) → c
  3. Intersect(c, n) → E, F
  4. Point(c) → B
  5. Circle(A, R - r1) → d
  6. Rotate(F, (t * w), A) → B'
  7. Ray(A, B') → a
  8. Intersect(d, a) → C,
  9. Circle(C, B') → e
  10. Rotate(B', (t * w1) + 3.141592653589793, C) → B''
  11. Circle(C, r1 - r2) → f
  12. Ray(C, B'') → b
  13. Intersect(f, b) → D, K
  14. Circle(D, B'') → g
  15. Rotate(B'', (t * w2) + 3.141592653589793, D) → B'''
  16. Segment(D, B''') → h
  17. Segment(C, B'') → i
  18. Segment(A, B') → j
  19. Line(G, H) → p
  20. Point(p) → I
  21. Segment(I, J) → r
  22. Segment(A, C) → f_1
  23. Segment(C, D) → g_1
  24. CurveCartesian(x(A) + ((R - r1) * cos((w * i))) + ((r1 - r2) * cos(((w1 + w) * i) + π)) - (r2 * cos(((w + w1 + w2) * i) + π)), y(A) + ((R - r1) * sin((w * i))) + ((r1 - r2) * sin(((w1 + w) * i) + π)) - (r2 * sin(((w + w1 + w2) * i) + π)), i, 0, t) → u
  25. Intersect(g, g_1) → L,
  26. Segment(C, L) → k

AI Drawing Prompt Reference

Draw a fractal spirograph demonstrating parametric curves and similarity transformations. Start with a line and a circle, find their intersections and a point on the circle. Create a smaller circle with the intersection point as the center, and connect one intersection to the rotated point. Repeat by drawing circles and rotating points, connecting them with segments to form intricate geometric patterns. Finally, use line segments to connect key points and plot a parametric curve to highlight the fractal structure.
The prompt above was compiled from the original drawing commands and is for reference only.