4 ms·

The Real 3D Mandelbrot Set

66 points by st3fan 15y ago

3am 15y ago
"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 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 15y ago
  > 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...
tripzilch 15y ago
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 http://www.skytopia.com/project/fractal/mandelbulb.html and in particular http://www.skytopia.com/project/fractal/2mandelbulb.html#epilogue http://www.skytopia.com/project/fractal/2mandelbulb.html#epi...
- colah2 15y ago
  > 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 15y ago
    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.
deleted 15y ago
[deleted]
eps 15y ago
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 http://en.wikipedia.org/wiki/Lyapunov_fractal
Jach 15y ago
I like these better: http://bugman123.com/Hypercomplex/index.html http://bugman123.com/Hypercomplex/index.html (Also includes some 4D ones.)

66 points by st3fan 15y ago

3am 15y ago

"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 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 15y ago
> 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...

tripzilch 15y ago

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

colah2 15y ago
> 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 15y ago
  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.

deleted 15y ago

[deleted]

eps 15y ago

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 http://en.wikipedia.org/wiki/Lyapunov_fractal

Jach 15y ago

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