Orchid XP v8 wrote:
> 
> 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? It sounds like it, but then again, my math skills do not allow 
me to visualize what you are saying. Is this a new concept? If so, you 
might find your name in a fractal news journal somewhere if you apply 
the concept :)

> 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.

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. 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.

Sam

Post a reply to this message