

The Real 3D Mandelbrot Set - st3fan
https://christopherolah.wordpress.com/2011/08/08/the-real-3d-mandelbrot-set/

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tripzilch
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...](http://www.skytopia.com/project/fractal/2mandelbulb.html#epilogue)

~~~
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)

------
3am
"Most likely, this is simply a reflection of me being an unread ignoramus in
the grand schemes of complex dynamics — it is far, far, too obvious to be
novel."

Yes, but great visualization work. The author should contact a decent math
department (<http://www.math.cornell.edu/event/conf/fractals4/index.php> is a
good starting point) and see if he can work with anyone to provide graphics to
something that really is novel. Maybe they can get their name attached to some
published papers.

~~~
colah2
> 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...

------
eps
Some of realistic-looking greyscale renders are remarkably similar to those of
Lyapunov fractal [1], which is 2D (and whose 3D-ish appearance is what makes
it interesting).

[1] <http://en.wikipedia.org/wiki/Lyapunov_fractal>

------
Jach
I like these better: <http://bugman123.com/Hypercomplex/index.html> (Also
includes some 4D ones.)

