These are three-dimensional slices made by varying certain parameters pertaining to the classic Z^2+C mandelbrot set. This is nothing new, except for varying the exponent, which indeed I had not seen before. Pretty pictures, though :)

In the search for the "real" 3D mandelbrot, which recently lead to the discovery of the "Mandelbulb" fractal, the goal was to find a three-dimensional object with as much visual complexity as the classic Mandelbrot fractal, and which does not have mostly regions all stretched out like taffy.

Especially the latter requirement is a problem with most "naive" representations of 3D Mandelbrot or Julia fractals. They're all stretched and skewed, and in some sense that's pretty, but it's a completely different type of "pretty" than the amazing spirals, flowerheads and branching lightning forks that are found in the classic 2D Mandelbrot and Julia sets.

This is why people were so enthusiastic about the Mandelbulb fractal: Even though some regions of it were stretched and skewed like the previous attemps at 3D Mandelbrot fractals, there were also many, many regions that, for the first time actually did show three-dimensional romanesco broccoli flowers, organic spiral staircases and all sorts of amazing 3D visual complexity.

And as described near the end of the (wonderful) essay on the Mandelbulb* it is not considered "the Real McCoy" 3D Mandelbrot set, not just because, as the author of this article implies, it is not mathematically elegant enough, but rather because the Mandelbulb still contains "smeared 'whipped cream' sections", and that the power-two version isn't as interesting (they mostly investigated the power-eight Mandelbulb). And also that (as far as anyone's found) it doesn't contain copies of itself, like the classic 2D Mandelbrot.

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.

The sorts of structures he created, while again, pretty, are exactly the kinds of structures that the people that discovered the Mandelbulb fractal (and a whole bunch of other 3D fractals along the way) were trying to avoid.

So it's definitely a step back, and frankly I was a littlebit disappointed when I got to the end of the article, finding he had really done nothing more than what I had been doing 14 years ago with FRACTINT and POV-ray. It took many days to render and the smeared chewing gum shapes were sorely disappointing :)

What's wrong with the smeared whipped cream taffy structures? Well, for starters, the classic Mandelbrot and Julia sets don't have them at all.
Second, the Mandelbrot set's iteration starts at Z=0+0i. This value is not arbitrarily just the origin, it's the point where the derivative of the formula is zero (or something like that, correct me if I'm wrong). The result of this is, for different kinds of polynomials or formula, you may need a different starting point than 0+0i in order to get the "correct" Mandelbrot set corresponding with the Julia sets of the same formula. Guess what structures such a Mandelbrot set often looks like if you picked the wrong starting point? It contains a lot of smeared stretched taffy-like regions.

* http://www.skytopia.com/project/fractal/mandelbulb.html and in particular http://www.skytopia.com/project/fractal/2mandelbulb.html#epi...