Tracing Hyperbolic Components of the Mandelbrot

Tracing Hyperbolic Components of the Mandelbrot
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2019-02-13 09:02
This example demonstrates how to trace the hyperbolic components of the Mandelbrot set, specifically those with period 3 or less. It involves core mathematical concepts like complex numbers, fractal geometry, function iteration, and periodicity. The construction uses GeoGebra commands such as Cartesian curves, segments, and circles. Suitable for high school students and above, or as a teaching demonstration in complex analysis and fractal geometry.

GeoGebra Commands & Parameters

  1. CurveCartesian(cos(t) / 2 - cos((2 * t)) / 4, sin(t) / 2 - sin((2 * t)) / 4, t, 0, (2 * π)) → b_{1}
  2. CurveCartesian(cos(t) / 4 - 1, sin(t) / 4, t, 0, (2 * π)) → b_{2}
  3. Segment(z_1, z_2) → s_{1}
  4. Segment(z_1, z_3) → s_{2}
  5. Circle((-1 / 8, (3 * sqrt(3) / 8) + 256 / 2709), 256 / 2709) → c