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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](http://math.stackexchange.com/questions/2710/why-does-the-mandelbrot-set-contain-slightly-deformed-copies-of-itself) and [this](http://math.stackexchange.com/questions/46573/what-is-known-about-nice-automorphisms-of-the-mandelbrot-set/46582#46582), 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!)2012-05-26
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    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.2012-05-26
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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?2012-05-26
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    See the formula for 3d Mandelbrot fractals: http://www.fractal.org/Formula-Mandelbulb.pdf2013-12-30

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