Print Page - Generalizing the Mandelbulb

The Mandelbulb set is a "generalization" of the Mandelbrot set, but it is not general enough. The idea or rotating and dilating vectors written in spherical coordinates is a good one, and is a little restricted by having $(r\, \cos\, \theta\, \sin\, \phi\,, r\, \sin\, \theta\, \sin\, \phi\,, r\, \cos\, \phi\,) \rightarrow (r^n\, \cos\, n\, \theta\, \cos\, n\, \phi\,, r^n\, \sin\, n\, \theta\, \cos\, n\, \phi\,, r^n\, \cos n\, \phi)$ .

I have found that there is a lot of interesting territory that can be discovered by adding two parameters p and q that multiply theta and phi, so that the transformation is
$(r\, \cos\, \theta\, \sin\, \phi\,, r\, \sin\, \theta\, \sin\, \phi\,, r\, \cos\, \phi\,) \rightarrow (r^n\, \cos\, p\, n\, \theta\, \cos\, q\, n\, \phi\,, r^n\, \sin\, p\, n\, \theta\, \cos\, q\, n\, \phi\,, r^n\, \cos\, q\, n\, \phi)$ . When p = q =1, we have the usual Mandelbulb, and when either parameter is greater than one, it crinkles the surface, while values between 0 and 1 smooth the surface. I have had good success with these, and some of the results can be seen in this gallery

http://www.aurapiercing.com/gallery/main.php

as well as some videos on my YouTube channel

http://www.youtube.com/user/jstarret#p/u

The surface is very sensitive to small changes in s, t, u, but varies quite smoothly with changes in p and q. The other videos of mandelbulbs use changes in the p and q parameters.

I have been experimenting with these in Chaos Pro 4, and will be glad to post the function files if anyone is interested.

Tater

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Fractal Math, Chaos Theory & Research => Theory => Topic started by: Tater on December 13, 2010, 10:36:42 PM