> I figured they weren't new, but couldn't find preexisting discussion of them. Is there a name for them?

Not that I know of. Most of the 3D slices objects are Quaternion julia sets, which I suppose you already are familiar with.

I used to explore fractals using the oldskool DOS program called FRACTINT, which had an extensive hypertext-like documentation system, maybe they have something:

http://www.nahee.com/spanky/www/fractint/findex.html#search_...

... I think what comes quite close is what they call a "julibrot" type fractal, which consists of layered julia fractals:
http://www.nahee.com/spanky/www/fractint/juliabrot_type.html

It's not as pretty as your renderings, though. But remember the Fractint software is decades old :)

Also see the other skytopia link below, where he explains his attempts at making a 3D Mandelbrot before the Mandelbulb was discovered, and why he considered those other 3D Mandelbrots not "the real thing".

> I suppose the aesthetic shortcomings are what most people are concerned with, though some people, such as myself, were disappointed by how mathematically arbitrary it is.

Yes, I can completely understand that. Personally, I care about both. I truly think that the shapes you get when making 3D slices of mandelbrot/julia parameter space are too warped, stretched and unrecognizable.

I suppose the biggest problem I have with them is that the stacked taffy-like chewing gum structures actually obscure the familiar beauty of the spirals/coral/tree structures. They are still in there, but only if you know how to slice the object, from the outside it looks like a (very complex and beautiful in its own way) tangled mess.

And that is the problem that I think the Mandelbulb solves beautifully. Even though the maths are sadly arbitrary. The sheer visual overkill of eyecandy does tip the scale for me, in the short term, but it doesn't mean we shouldn't keep looking for something that is mathematically more "right", and possibly even prettier.

> The sad thing is that most people don't even understand the math behind the Mandel* sets enough to be able to care about such things.

Well, you wouldn't be able to implement a Mandelbulb, or even come up with the formulas for it without a good solid understanding of complex math. In fact, the Mandelbulb formulae went slightly over my head [I could probably understand them if I took a bit more time, though].

> >So, the blog-article's author's implication that his is the "real" 3D Mandelbrot set, specifically referencing the Mandelbulb fractal, is just plain inaccurate.

> Again, it would seem that this would depend on the metric you apply.

Yeah sorry I think I wasn't quite clear. I got the idea that you were trying to improve on the Mandelbulb fractal. And the Mandelbulb fractal's formula was designed with the purpose of producing the same visual complexity in 3D as the classic Mandelbrot does in 2D. They already realized that just using Quaternions or stacking the parameters the way you have done, results in stretched-taffy objects that lack any of the visual beauty of the classic Mandelbrot. They felt that this couldn't be the "right" 3D representation of the Mandelbrot.

As you can see, they already went there, and tried loads of variations:
http://www.skytopia.com/project/fractal/mandelbrot.html

So where I got confused is that you started out referencing the Mandelbulb project, and I assumed you had the same goals as them but did a better job at it.

And as you can see from the above article, it's almost as if you retraced their steps backwards, than if you actually improved the Mandelbulb.