Meta-Mandelbrots

Ian McDonald came up with a new novel way to render Mandelbrot (actually Julia) Set fractals.

The usual Mandelbrot fomula is
z=z*z+c

Taking the z*z+c part, replace the z’s with (z*z+c) and replace the c’s with (c*c+z)

After one level of replacement you get
((z*z+c)*(z*z+c)+(c*c+z))

Level 2 is
(((z*z+c)*(z*z+c)+(c*c+z)) * ((z*z+c)*(z*z+c)+(c*c+z)) + ((c*c+z)*(c*c+z)+(z*z+c)))

and Level 3 is
((((z*z+c)*(z*z+c)+(c*c+z))*((z*z+c)*(z*z+c)+(c*c+z))+((c*c+z)*(c*c+z)+(z*z+c)))*(((z*z+c)*(z*z+c)+(c*c+z))*((z*z+c)*(z*z+c)+(c*c+z))+((c*c+z)*(c*c+z)+(z*z+c)))+(((c*c+z)*(c*c+z)+(z*z+c))*((c*c+z)*(c*c+z)+(z*z+c))+((z*z+c)*(z*z+c)+(c*c+z))))

Then you use the level 3 formula and render it as a Julia Set.

Complex C (-0.2,0.0)

Complex C (-0.14 0.0)

Complex C (-0.141 0.0)

The following movie shows the complex C changing slowly from 0 to -0.2 and three zooms into Meta-Mandelbrots. Unfortunately because these are Julia sets the shapes deeper in are virtually identical to the original fractal. You don’t get those totally different looking areas as you do with Mandelbrot fractals.

For more information see the original Fractal Forums post here.

The GLSL shader to generate these fractals is now included with Visions of Chaos.