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Internal Coloring of the Mandelbrot Set

Internal Coloring of the Mandelbrot Set
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2019-04-26 10:05
FunctionComplex NumberGeometric Construction
This example demonstrates the internal coloring of the Mandelbrot set using iteration value coloring, revealing its intricate fractal structure. Key mathematical concepts include complex numbers, iteration, fractal geometry, and numerical computation. The construction leverages GeoGebra commands and objects such as b, C, r, A, B, B₁, A₁, z₁, A1, a, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, A15, A16, A17, A18, A19, A20, B20, C20, and slider animations. It is suitable for high school students and beyond, particularly those interested in complex numbers and fractal geometry.

GeoGebra Commands & Parameters

  1. Numeric b(0.0001)
  2. Numeric C(70)
  3. (b * C)
  4. (1 / 2 * sqrt(r))
  5. A
  6. (B * cos(C)) + (A * sin(C))
  7. (A * cos(C)) - (B * sin(C))
  8. A_{1} + (B_{1} * ί)
  9. z_1
  10. Numeric a(2)
  11. A1^(a) + A$1
  12. A2^(a) + A$1
  13. A3^(a) + A$1
  14. A4^(a) + A$1
  15. A5^(a) + A$1
  16. A6^(a) + A$1
  17. A7^(a) + A$1
  18. A8^(a) + A$1
  19. A9^(a) + A$1
  20. A10^(a) + A$1
  21. A11^(a) + A$1
  22. A12^(a) + A$1
  23. A13^(a) + A$1
  24. A14^(a) + A$1
  25. A15^(a) + A$1
  26. A16^(a) + A$1
  27. A17^(a) + A$1
  28. A18^(a) + A$1
  29. A19^(a) + A$1
  30. x(A20)
  31. y(A20)