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Mandelbrot Iteration Orbits Visualization

Mandelbrot Iteration Orbits Visualization
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2019-10-17 09:11
Complex NumberGeometric ConstructionLocusGraphing
In collections Complex Analysis: A visual and interactive introduction
This GeoGebra example dynamically visualizes iteration orbits of points in the Mandelbrot set, illustrating the fractal boundary behavior under repeated complex squaring. It demonstrates how points may converge, diverge, or exhibit periodic orbits depending on their initial complex value. Core mathematical concepts include complex number iteration, the Mandelbrot set, fractal geometry, escape radius, and complex dynamics. The construction employs GeoGebra commands such as Polygon, If, TableText, and PolyLine, utilizing the max object and a sequence of points A1 through A38 to trace and label orbital paths. It is suitable for high school enrichment, university complex analysis, or fractal geometry courses, and for learners interested in dynamic systems and complex plane visualization.

GeoGebra Commands & Parameters

  1. Polygon((1.5, -1.5), (-2.5, -1.5), (-2.5, 1.5), (1.5, 1.5)) → q1, g, h, i, j
  2. If(condition > 100000 ∨ A100 ≠ A100, 1, 0) → coloring
  3. If(condition > 10000 ∨ condition ≠ condition ∧ c, 1, 0) → d
  4. If(abs(A2) > max ∨ A2 ≠ A2, "\infty", "" + A2 + "") → text1
  5. If(abs(A3) > max ∨ A3 ≠ A3, "\infty", "" + A3 + "") → text2
  6. If(abs(A4) > max ∨ A4 ≠ A4, "\infty", "" + A4 + "") → text3
  7. If(abs(A5) > max ∨ A5 ≠ A5, "\infty", "" + A5 + "") → text4
  8. If(abs(A6) > max ∨ A6 ≠ A6, "\infty", "" + A6 + "") → text5
  9. If(abs(A7) > max ∨ A7 ≠ A7, "\infty", "" + A7 + "") → text6
  10. If(abs(A8) > max ∨ A8 ≠ A8, "\infty", "" + A8 + "") → text7
  11. If(abs(A9) > max ∨ A9 ≠ A9, "\infty", "" + A9 + "") → text8
  12. If(abs(A10) > max ∨ A10 ≠ A10, "\infty", "" + A10 + "") → text9
  13. If(abs(A11) > max ∨ A11 ≠ A11, "\infty", "" + A11 + "") → text10
  14. If(abs(A21) > max ∨ A21 ≠ A21, "\infty", "" + A21 + "") → text20
  15. If(abs(A51) > max ∨ A51 ≠ A51, "\infty", "" + A51 + "") → text50
  16. If(abs(A101) > max ∨ A101 ≠ A101, "\infty", "" + A101 + "") → text100
  17. If(abs(A1) > max ∨ A1 ≠ A1, "\infty", "" + A1 + "") → text0
  18. If(abs(A401) > max ∨ A401 ≠ A401, "\infty", "" + A401 + "") → text400
  19. If(abs(A201) > max ∨ A201 ≠ A201, "\infty", "" + A201 + "") → text200
  20. TableText({C1:C17, D1:D17}, "|_|vc") → text
  21. If(abs(A501) > max ∨ A501 ≠ A501, "\infty", "" + A501 + "") → text500
  22. PolyLine(A1:A500) → f

AI Drawing Prompt Reference

Visualize the Mandelbrot set's iteration orbits by first drawing a quadrilateral boundary, then marking the convergence, divergence, and periodicity of orbits with different colors, and finally connecting a series of points with a polyline to show the orbit paths.
The prompt above was compiled from the original drawing commands and is for reference only.