>> 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?
> 
> Are you talking about extruding a 2D fractal along a third axis and 
> varying the values? Is this not what ends up producing those bubble gum 
> shapes?

No. Those are produced by iterating Z = Z^2 + C, but with Z and C as 
hypercomplex or quaternion numbers instead of the usual complex numbers.

What *I* am talking about is iterating Z = Z^3 - 3 A^2 Z + B, where Z, A 
and B are all normal complex numbers. A Julia set is rendered by varying 
the start value for Z - which still has 2 components (Re(Z) and Im(Z)). 
However, the Mandelbrot set is drawn by varying the parameters, which 
gives us 4 axies: Re(A), Im(A), Re(B) and Im(B).

This is not new, just not very widely known. A few people have drawn it 
before. If you search *waaay* back through the POV-Ray images newsgroup 
you'll find some renderings I did.

As I say, it turns out that the higher the iteration count (and hence 
the more complex the surface), the less "interesting" the image actually 
becomes.

> Perhaps shading is the key. The 2D Mandelbrot makes sense to the eye 
> primarily because of the black basin, and then the colors depicting 
> iterations after that. So how would this work in three dimensions? The 
> basin might extend from itself with branching structures, with certain 
> areas of prominence. The whole thing would look confusing unless you 
> applied a light shining down upon it. Or maybe each iteration could be 
> made translucent, which might work fairly well, though you would have to 
> increase the transparency if you wanted to zoom in further.

All of that seems at least plausible.

> Of course 
> all this is moot unless you figure out out how to apply true 
> transcendental complexity to the third dimension. Those taffy-like 
> quaternions don't seem like the ultimate destination to me.

Agreed.

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