Rudy Sets and Mandelbrots of Higher Degree

[Rudy]

I've been playing with UltraFractal, and I implemented some lesser known higher-order Mandelbrot style fractals in here.  These fractals were including in the Autodesk CHAOS packaget that I co-authered in the 1990s, but at that time (due to the slowness of computers) it wasn't so practical to explore these forms.

Recently I've been look at degree four and degree five (quartic and quintic) versions of the generalized Mandelbrot set.  The image below is from a generalized quintic Mandelbrot set...it's reassuring to find a copy of the quadratic Mandelbrot set as a detail, albeit within gnarly quintic leopard skin patterns here.



Even the generalized cubic Julia set Jkc should be better known...this is the set of all z such that z doesn't go to infinity under the iterated map z^3 + k*z + c.  (By adjusting your coordinate system, you can always avoid having a degree n-1 term in a degree n polynomial, so the Jkc show the full range of cubic possibilities.)  And the generalized cubic Mandelbrot set Mk is the set of all c such that Jkc is connected.

What interests me more is what I've been calling the (cubic) Rudy set ever since CHAOS, this is the set of all c such that the cubic Julia set Jcc is connected.  That is, we look for those points such that if you use that point value for both the k and the c of a cubic Julia set, that set doesn't break into pieces.

As I say, recently, using Ultra Fractal's power, I've bumped this up to look at quartic and quintic Julias, Mandelbrots and Rudys.  

I have a long post about this with many images and links at "Cubic Mandelbrots and the Rudy Sets" http://www.rudyrucker.com/blog/2010/04/02/the-rudy-set-fractal/.  The post includes a detailed explanation and links to my Ultra Fractal formula file rvr.ufm and parameter file rvr.upr, including the params for the MandQuinticLeopard shown above.

For introductory purposes, I'll include two YouTube videos here.

The first is a zoom into the (cubic) Rudy Set.  I used Ultra Fractal to compute the images, Windows Live Movie Maker for the titles and YouTube upload, VirtualDub to adjust the frame rate, and AVISynth to "pingpong" the video with the second half being the time-reversed version of the first half.

<a href="http://www.youtube.com/v/eGmhiu074Mo&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/eGmhiu074Mo&rel=1&fs=1&hd=1</a>

The second is an exploration of the range of cubic Mandelbrot sets Mk.  In this animation, I vary the values of k and show the changing shapes of the Mk.

<a href="http://www.youtube.com/v/ytCIGUS4DKI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/ytCIGUS4DKI&rel=1&fs=1&hd=1</a>

[stigomaster]

I have done similar stuff before, here is a minibrot with other interior shapes:


http://www.fractalforums.com/index.php?action=gallery;sa=view;id=1072

I think the formula was

i have also rendered a video varying values for different powers of z:
<a href="http://www.youtube.com/v/9mTSdAtlhRY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/9mTSdAtlhRY&rel=1&fs=1&hd=1</a>