Multiple Critical Points Part 2   Leave a comment


In the previous part the example fractals with multiple critical points contained Mandelbrot islands, there were no cunbic Mandelbrot or “multibrot” islands. In this part there is a fractal which has both Mandelbrot and Cubic Mandelbrot islands.

The formula:

zn+1 = zn4 – zn3 + c

f(z) = z4 – z3 + c
f'(z) = 4z3 – 3z2

the critical points are the roots of

4z3 – 3z2 = 0

or

z2(4z – 3) = 0

so the roots are 0 and 0.75.

Critical Point 0.75 showing Mandelbrots

Critical Point 0.75 showing Mandelbrots

Critical Point 0 showing Cubic Mandelbrots

Critical Point 0 showing Cubic Mandelbrots

Combination of both critical Points

Combination of both critical Points

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