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I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties).

Regardless, I'm wondering if there might be a 'trick' to create a 3D Mandelbrot which has detail running through each dimension without any discontinuities or 'smeared' sections. In 2008 I wrote a short article discussing the possibility, and went on to help discover the Mandelbulb in 2009.

As mentioned in those articles, variations on the quaternion Julia 4D fractal unfortunately resemble 'whipped cream' and have detail running through only 1 or 2 dimensions. Other attempts at a 3D analogue are mere extrusions or lathes of a 2D Mandelbrot. The real thing (if it exists) would look MUCH more interesting and beautiful.

Even the new 'Mandelbulb' isn't perfect as it too contains 'whipped cream' and has less variety than even the 2D Mandelbrot.

Is a Mandelbrot-looking equivalent in 3D space even remotely possible? Here's an artist's impression (created by Marco Vernaglione):

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Relevant: this and this, I guess. I'd recommend first cataloging what you think are the 'essential' features of the Mandelbrot fractal, that sets it apart from others, in mathematical language. (Also, privileging three dimensions over say four or eight may prove fatally anthropocentric. Math doesn't always turn out user-friendly, but good luck anyway!) –  anon May 26 '12 at 18:12
    
It's hard to pin down the essential aesthetic features mathematically. But lack of 'smeared' sections, lack of discontinuities, and (nearly) spheres on the surface seems like a good place to start from an informal perspective. –  Dan W May 26 '12 at 18:23
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It's probably going to be tough to give a satisfactory answer to your question until you can describe what you want without using quotation marks. Does the shape you get by rotating the Mandelbrot set in $\mathbb{R}^3$ count? Why or why not? You seem to be aware that the Mandelbrot set is closely related to a certain dynamical system. Maybe you can express what you want in terms of dynamics rather than geometry? –  Paul Siegel May 26 '12 at 21:00
    
See the formula for 3d Mandelbrot fractals: fractal.org/Formula-Mandelbulb.pdf –  julesruis Dec 30 '13 at 21:08

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I was wondering something similar, I was trying to find a 3D mandelbrot. Not a mandelbulb, but a 3D equivilant of the mandelbrot set. When searching for it, I came across some 3D 'representations,' but they didn't really appear to be a true 3D mandelbrot set.

Then I accidentally made a 3D mandelbrot, in the program Mandelbulber. I made a hybrid fractal of the mandelbulb, and the 'general mandelbox fold' and came up with this: http://img.photobucket.com/albums/v252/TMTIGD/3Dmandelbrot.jpg

The fun thing about this model is I can enter the X and Y coordinates from the traditional mandelbrot set, and it will zoom in on the same portion of the 3D model. You can even see the dendrites, and self-similar mandelbrots on the dendrites, exactly were they would be in the 2D set.

I'm not completely sure this is what you were looking for, I have been having trouble finding julias sets in the 3D model were I find them in the 2D model, but that could simply be because of the quality settings I'm using. Either way, this is the closest match to a true 3D mandelbrot I've come across. Any others have strange artifacts or artistic fluff that makes me wonder if it's just the 2D set with a lot of pretty effects.

edit:: After playing around more, this 3D mandelbrot is puzzling. With certain settings, I can see the outline of the 2D mandelbrot from a top-down view. If I use too many iterations, the spiraling branches and arms visible in the 2D set disappear and I only see smaller and smaller self-similar mandelbrots. Then if I set certain settings (mainly "DE detection") too too high of a quality on a point were you expect spirals and intricate branches, I am only shown an "oil spill". Then on other combinations of quality settings, I see wispy clouds of semi-transparent smoke over the mandelbrot set itself. (http://img.photobucket.com/albums/v252/TMTIGD/image00384.jpg)

It seems most obvious that with the best of quality, the features from the 2D mandelbrot set are clouded and usually completely hidden when viewing on a microscopic scale.

I would really love to hear more input from someone (especially the question asker). I have a feeling this probably isn't what he was looking for since he appears to be somewhat of an expert in this area, and he hasn't found a 3D mandelbrot-esque figure. I only found this by chance, but it doesn't look like anything he has dismissed yet, so I'd love it my dumb luck happened to find a way to view a 'true' 3D mandelbrot.

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