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Julia Sets: Visualizing Divergence of Complex Iteration x² + C

Julia Sets: Visualizing Divergence of Complex Iteration x² + C
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2020-09-06 05:29
FunctionSequenceComplex NumberGraphing
This GeoGebra example visualizes Julia sets by determining which complex numbers cause the iterative series z_{n+1} = z_n² + C to diverge for a fixed complex parameter C. Core mathematical concepts include complex numbers, power function iteration, convergence and divergence criteria, fractal geometry, and complex dynamical systems. The construction relies on GeoGebra commands such as IterationList to generate the iteration sequence, Sequence to build list terms, and Circle together with Intersect for geometric intersection testing, involving objects A, B, C, a, c, l1, and l2. It is suitable for advanced high school, undergraduate complex analysis, and fractal geometry courses, as well as classroom demonstrations and mathematical inquiry activities that help learners intuitively grasp the dynamic behavior of complex iteration.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → B
  2. Circle(B, 1) → c
  3. IterationList(z^(2) + C, z, {A}, a) → l1
  4. Sequence(Segment[l1(i), l1(i + 1)], i, 0, a) → l2

AI Drawing Prompt Reference

Create a visualization of a Julia set by drawing a complex coordinate system with the real and imaginary axes crossing at the origin, a unit circle centered at the origin, a chosen complex parameter C, and the trajectory of a starting point under the quadratic map z → z² + C. Show the first several iterates as points and connect each consecutive pair with straight line segments to form a piecewise linear path that reveals whether the orbit stays bounded or escapes to infinity.
The prompt above was compiled from the original drawing commands and is for reference only.