Orchid XP v8 wrote:

> So you can't do anything to the generating formula that produces a 
> nontrivially 3D image.

I don't see why not, the generating formula only involves squaring and 
addition (no division) so it don't necessarily matter if you're doing 
the computation over a field or not.  That said, it's possible that some 
of the geometric properties of the set arise for reasons intricately 
linked to the field nature of the complex numbers, but I really don't 
know if this is actually the case or not.

> A more important question: If this mythical set actually exists, would 
> it be interesting to look at?

Well, the author of the webpage had "interesting to look at" as more or 
less the definition of what he was looking for, so I think from his 
perspective the question is more like "is there some number system which 
produces an interesting 3D version of the Mandelbrot set".  There should 
be many ways to define multiplication and addition over 3+D points if 
all you care about is the end result and not what axioms they satisfy, 
so I think it's a pretty open-ended question.

> I rather suspect that any 3D object with an intricate fractal structure 
> to its surface is likely to just look random and chaotic and rather 
> uninteresting. For example, go pick up a sponge and look at it. Does it 
> look interesting? Not really. It just looks like a uniform fuzzy mass. 
> Similarly for a lump of bread.
> 
> For a 3D fractal to *look* good, its surface would have to be 
> sufficiently "simple" that the brain can comprehend it. The brain 
> doesn't seem to respond to surface textures as precisely as it responds 
> to intricate colours.

He gave some interesting 3D pictures of things which had the sort of 
structures he was looking for, so in principle I think it's possible.  I 
do, however, tend to agree with you that it's much harder to get 
something fractally looking good in 3D than in 2D or 2.5D, but in some 
ways that's what makes it an interesting challenge!

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