[colah2]

> 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 :)

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

>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.

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.

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.

>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.

>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).

As I explained in the essay (in particular, the several pages of explanation of what the Mandelbrot set is mathematically and why we care about it), we're interested in z₀=0 because it differentiates Julia sets into two classes with very different properties (in particular, topologically). If our goal is to understand Julia sets better however, including their whole real axis does give us a lot more information.

[colah2]

I've added the following to the post:

Update

It’s been suggested that ZRXC is not the “Real 3D Mandelbrot Set” because it fails to achieve certain visual standards.

This is a legitimate concern. I understand that to most people, the visual aesthetics of a fractal are the most important part. One of the goals of this essay was to show the reader something else, more important and beautiful beneath that.

Different people will have different standards for judging whether something is the “Real 3D Mandelbrot Set” — to me the Mandelbrot set is a step on the way to understanding a mystery, to solving a puzzle. So my generalization tried to fulfill that role better.

You are welcome to disagree, and I think a very strong argument can be made that the Mandelbulb set is more visually appealing than ZRXC… But I’d ask you make sure you understand the math outlined in this essay — if you skipped the sections on what the Julia Set and Mandelbrot set are, you missed the point of this essay (if you had trouble following, that’s to be expected since I’m not always the best at explaining things, please feel free to ask below).

Perhaps I should have titled this “the Natural 3D Mandelbrot Set.” Hindsight is always 20/20.

[tripzilch]

I think that is a good addition. If I may make one more suggestion, is that you state at the start of your article, how your goal differs from that of the Mandelbulb project: that you seek for mathematical elegance, while the Mandelbulb project first and foremost went for finding a 3D visual equivalent of the classic Mandelbrot shapes.

About skipping sections: the only bits that I skipped were the bits that I already knew :) I did love the visualisations of transformations in the complex plane, however. Very nicely done.

I did not understand what you were trying to say with the animated GIF about branches being arbitrary. Obviously something about the complex square root having two solutions, but how that relates to the image with rotating colours isn't quite obvious to me.

Also the part about the operations on sets isn't as clear as it could be. It's very intriguing though, yet another way of thinking of complex numbers that I didn't know yet :) I think it would become a bit clearer if you'd add axes to the image. Even better if you can also put angles (0, pi/2, pi, ..) in it.

BTW one very cool way in which complex numbers were explained to me when I was 16, does explain something you gloss over a bit in the "complex analysis" section. Why is the imaginary axis perpendicular to the real axis?

The way it was explained to me, is to think of the (real) number line, and how a multiplication by -1 is visually the same as a rotation by 180 degrees around the zero.

Now what if you'd decide to rotate 90 degrees instead? Such a crazy idea! You'd get a second number line. Let's call a rotation of 90 degrees to multiply by i.

So if we multiply 3 by i we get 3i, and if we multiply it again, it rotates 90 degrees further and we get -3. So that means 3 * i * i = -3 and i * i = -1, so i is the square root of minus one! Insanity! (the 16 year old me was giggling like a madman at this point)

(so yeah the explanation I got kind of started the other way around, with the 90 degrees angle first, and only "discovering" that this implied the square root of minus one after that)

[tripzilch]

> 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.

[eru]

You may be delighted to hear that there's a Linux version of fractint called xfractint.