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.

> 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.

> Yes, but great visualization work.

Thank you, but that's actually fairly easy with modern software, and the most difficult part (choosing angles and colors) was done by a friend.

I'm not terribly interested in Complex Dynamics any more and mostly wrote this essay because it had been on my todo list for almost a year... Most of my interest is directed towards 3D printers and design of CAD software now.

In any case, the part of this that I was happiest with was the explanation of what Julia and Mandelbrot sets are. Most people, even those who are very interested in fractals have no idea what they are, just the software (or maybe equations that make them) and what they look like. Which I find frustrating. So I used the fact that people don't seem to be aware of more natural generalizations mandelbrot sets to 3 dimensions to teach it. :) I was fairly sure they were already known of. Though the proper name of such fractals would be appreciated...

And I was! Can't believe I made HN!

You're right, and, in addition to being the creator of this vacuum (which I designed in a few hours), I'm a core developer of the hacklab-reprap (https://github.com/hacklabto/Hacklab-RepRap; you can see my last update on it at http://christopherolah.wordpress.com/2011/01/24/hacklab-repr... ). Unfortunately, I can't focus on that 24/7 and occasionally work on other projects...

You can also get 3D printers, eg https://github.com/hacklabto/Hacklab-RepRap

I bought the DC motor for the second one I'm making at active surplus for $1.95. The cloth was under hacklabs sink in a package of 10 that cost ~ $5, and I didn't use a full cloth. The tape was sitting around, but probably cost about $2 (the one I used this time was taken out of an old vacuum cleaner, ironically enough). The power supply was a fancy one and would cost >$100 but I could have bought a cheap converter for $5. And I used a dollar or two of plastic.

So, even if I'd had to buy everything from scratch, it only would have cost me ~ $15. Since that wasn't the case, it only cost me my time and some negligible expenses. Meanwhile, the cheapest portable vacuum cleaner I could find at the stores I went to was $39.99 + tax...

Also, we're not that far from printing motors. Fab@home has printed metal by dissolving it into a thick paste and others have printed grooves and run solder through them. Once I can print metal coils, I can print a motor... (Some motors don't need an internal magnet -- see http://en.wikipedia.org/wiki/Induction_motor)