This is ancient history - 3D/4D Julibrots have been known of and rendered since the early days of Fractint. Here's an animation of many ways of viewing the 4D Julibrot: http://www.youtube.com/watch?v=gr-ul7sZDwc Also when Daniel White and others at http://www.fractalforums.com/ say "the true 3D Mandelbrot" (or 4D+) they actually mean a (Mandelbrot-style) fractal using the standard iteration of z^2+c that is fractal in all directions on the surface i.e. it has no "whipped cream" - clearly so far there is no 3D+ number form for which z^2+c fulfils this - in fact such a thing may not be possible in 3D+ - at least it seems unlikely with respect to "sensible" number forms ;) Daniel White and Paul Nylander's Mandelbulb (z^p+c) does however come close at higher powers - say p>=6 or so: http://makinmagic.deviantart.com/gallery/?offset=72#/d2f2kkz On 9 Aug 2011, at 00:13, Tom Karzes wrote:
Me neither. There is, however, a very natural 4-dimensional set that combines both the standard Mandelbrot set and all of its corresponding Julia sets into a single entity.
Let one plane determine c0, and an orthogonal plane determine c1. The product is 4-dimensional, and the iteration is:
z(0) = c0 z(i) = z(i-1)^2 + c1
The planar slice obtained by setting c0 to 0 and varying c1 is the Mandelbrot set. The planar slice obtained by setting c1 to some constant and varying c0 is the Julia set corresponding to c1.
Unfortunately, having a 2-dimensional retina makes it difficult for me to visualize such a set. Being three dimensional can be very frustrating.
Tom
I cannot understand the basis for the assumption that there is a "real" 3D Mandelbrot set.
--Dan
Sometimes the brain has a mind of its own.
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