stbenge wrote:

> I wonder if anyone here thinks 
> there is really a 3D Mandelbrot as the author describes?

It's a basic fact that you can't generalise complex algebra to more 
dimensions and still have a "field"; one or other of the axioms must be 
broken. The result is the hypercomplex and quaternion algebras, which 
just look like (optionally twisted) surfaces of revolution.

So you can't do anything to the generating formula that produces a 
nontrivially 3D image. But the 2D set has many obvious geometric 
properties (particularly the prominent appearence of circles). Could you 
not manually reproduce those same relationships with spheres instead? I 
think perhaps you could. Working out what to do with all the "filaments" 
would be harder, but not in principle impossible.

There is another possibility to consider as well: the general 3rd-order 
complex equation has *two* unknowns instead of one, resulting in 2D 
Julia sets but a 4D Mandelbrot set. So here we have a 4D set based on a 
true field algebra, which structure in all directions. And it follows 
the same kind of patterns as the 2nd order set. Maybe this could be 
interesting to explore?



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

Draw a tangled mess of lines on a sheet of paper and the human brain is 
very good at untangling it. But draw a tangle of lines in 3D and 
suddenly it just looks like a mess.

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.



Just my thoughts on the matter...

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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