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Julia Set Boundary Visualization

Julia Set Boundary Visualization
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2016-04-25 22:07
FunctionComplex NumberGeometric ConstructionGraphing
This example demonstrates the boundary of stability in the complex plane for the iteration z -> z^2 + C, allowing users to explore how moving the parameter C affects the fractal's structure. Key math concepts include complex number operations, iterative functions, and fractal geometry. The construction leverages GeoGebra objects A, A1, C, A2, ..., A38 to visualize and animate the Julia set's behavior. Ideal for high school students, complex analysis beginners, or those interested in fractal art and mathematics.

GeoGebra Commands & Parameters

  1. Point A(0.6020044513897775, 1.9253924034948553)
  2. A
  3. Point C(0.0, 0.0)
  4. (((Random[0, 1] * 2) - 1) * sqrt(A1 - C))
  5. (((Random[0, 1] * 2) - 1) * sqrt(A2 - C))
  6. (((Random[0, 1] * 2) - 1) * sqrt(A3 - C))
  7. (((Random[0, 1] * 2) - 1) * sqrt(A4 - C))
  8. (((Random[0, 1] * 2) - 1) * sqrt(A5 - C))
  9. (((Random[0, 1] * 2) - 1) * sqrt(A6 - C))
  10. (((Random[0, 1] * 2) - 1) * sqrt(A7 - C))
  11. (((Random[0, 1] * 2) - 1) * sqrt(A8 - C))
  12. (((Random[0, 1] * 2) - 1) * sqrt(A9 - C))
  13. (((Random[0, 1] * 2) - 1) * sqrt(A10 - C))
  14. (((Random[0, 1] * 2) - 1) * sqrt(A11 - C))
  15. (((Random[0, 1] * 2) - 1) * sqrt(A12 - C))
  16. (((Random[0, 1] * 2) - 1) * sqrt(A13 - C))
  17. (((Random[0, 1] * 2) - 1) * sqrt(A14 - C))
  18. (((Random[0, 1] * 2) - 1) * sqrt(A15 - C))
  19. (((Random[0, 1] * 2) - 1) * sqrt(A16 - C))
  20. (((Random[0, 1] * 2) - 1) * sqrt(A17 - C))
  21. (((Random[0, 1] * 2) - 1) * sqrt(A18 - C))
  22. (((Random[0, 1] * 2) - 1) * sqrt(A19 - C))
  23. (((Random[0, 1] * 2) - 1) * sqrt(A20 - C))
  24. (((Random[0, 1] * 2) - 1) * sqrt(A21 - C))
  25. (((Random[0, 1] * 2) - 1) * sqrt(A22 - C))
  26. (((Random[0, 1] * 2) - 1) * sqrt(A23 - C))
  27. (((Random[0, 1] * 2) - 1) * sqrt(A24 - C))
  28. (((Random[0, 1] * 2) - 1) * sqrt(A25 - C))
  29. (((Random[0, 1] * 2) - 1) * sqrt(A26 - C))
  30. (((Random[0, 1] * 2) - 1) * sqrt(A27 - C))
  31. (((Random[0, 1] * 2) - 1) * sqrt(A28 - C))
  32. (((Random[0, 1] * 2) - 1) * sqrt(A29 - C))
  33. (((Random[0, 1] * 2) - 1) * sqrt(A30 - C))
  34. (((Random[0, 1] * 2) - 1) * sqrt(A31 - C))
  35. (((Random[0, 1] * 2) - 1) * sqrt(A32 - C))
  36. (((Random[0, 1] * 2) - 1) * sqrt(A33 - C))
  37. (((Random[0, 1] * 2) - 1) * sqrt(A34 - C))
  38. (((Random[0, 1] * 2) - 1) * sqrt(A35 - C))
  39. (((Random[0, 1] * 2) - 1) * sqrt(A36 - C))
  40. (((Random[0, 1] * 2) - 1) * sqrt(A37 - C))
  41. (((Random[0, 1] * 2) - 1) * sqrt(A38 - C))
  42. (((Random[0, 1] * 2) - 1) * sqrt(A39 - C))
  43. (((Random[0, 1] * 2) - 1) * sqrt(A40 - C))
  44. (((Random[0, 1] * 2) - 1) * sqrt(A41 - C))
  45. (((Random[0, 1] * 2) - 1) * sqrt(A42 - C))
  46. (((Random[0, 1] * 2) - 1) * sqrt(A43 - C))
  47. (((Random[0, 1] * 2) - 1) * sqrt(A44 - C))
  48. (((Random[0, 1] * 2) - 1) * sqrt(A45 - C))
  49. (((Random[0, 1] * 2) - 1) * sqrt(A46 - C))
  50. (((Random[0, 1] * 2) - 1) * sqrt(A47 - C))
  51. (((Random[0, 1] * 2) - 1) * sqrt(A48 - C))
  52. (((Random[0, 1] * 2) - 1) * sqrt(A49 - C))
  53. (((Random[0, 1] * 2) - 1) * sqrt(A50 - C))
  54. (((Random[0, 1] * 2) - 1) * sqrt(A51 - C))
  55. (((Random[0, 1] * 2) - 1) * sqrt(A52 - C))
  56. (((Random[0, 1] * 2) - 1) * sqrt(A53 - C))
  57. (((Random[0, 1] * 2) - 1) * sqrt(A54 - C))
  58. (((Random[0, 1] * 2) - 1) * sqrt(A55 - C))
  59. (((Random[0, 1] * 2) - 1) * sqrt(A56 - C))
  60. (((Random[0, 1] * 2) - 1) * sqrt(A57 - C))

AI Drawing Prompt Reference

Draw the boundary of a Julia set in the complex plane centered at point A(0.602, 1.925) with point C at the origin. Use the iteration formula z -> z^2 + C to observe the dynamic changes of point A and its surrounding points, showcasing the structure of the Julia set's boundary. Randomly generate multiple points A1 to A38, each connected to point C through the formula (((Random[0, 1] * 2) - 1) * sqrt(Ai - C)), forming a dynamic visualization of the Julia set.
The prompt above was compiled from the original drawing commands and is for reference only.