3d mandelbrot

November 12, 2009 10:06 AM

The Mandelbulb"The original Mandelbrot is an amazing object that has captured the public's imagination for 30 years. It's found by following a relatively simple math formula. But in the end, it's still only 2D and flat - there's no depth, shadows, perspective, or light sourcing. What we have featured in this article is a potential 3D version of the same fractal."
posted by dhruva (117 comments total)

*swoons*
posted by Burhanistan at 10:08 AM on November 12, 2009

Ia! Ia!
posted by Faint of Butt at 10:12 AM on November 12, 2009 [8 favorites]

...

-madness-

...
posted by strixus at 10:17 AM on November 12, 2009

whoa
posted by iamkimiam at 10:17 AM on November 12, 2009

Awesome. I bet that's what was in the trunk of the Malibu.

"Looks like sausage."
posted by bondcliff at 10:17 AM on November 12, 2009 [4 favorites]

Dang. Was just about to post this. Neat stuff.
posted by brundlefly at 10:18 AM on November 12, 2009

Oh.
Hell.
Yes.
posted by Spatch at 10:18 AM on November 12, 2009

Paging Maynard Keenan; Maynard Keenan to the white courtesy phone in kiosk Z_N².
posted by boo_radley at 10:18 AM on November 12, 2009 [6 favorites]

Math is beautiful.
posted by rocket88 at 10:19 AM on November 12, 2009 [2 favorites]

I'm not exactly sure what's going on here - but it sure is purty. Reading that wikipedia entry reminds me of how little math I took.
posted by Slack-a-gogo at 10:25 AM on November 12, 2009 [1 favorite]

"Ice cream from Uranus." (cue hysterical giggles)
posted by chavenet at 10:27 AM on November 12, 2009 [2 favorites]

The buds are growing smaller buds, and at least to the picture on right, there seems to be a great amount of variety too. We're seeing 'branches' with large buds growing around the branch in at least four directions. These in turn contain smaller buds, which themselves contain yet further tiny buds.

Those are buds, alright.
posted by Blazecock Pileon at 10:27 AM on November 12, 2009

The universe is fantastic!
posted by kuatto at 10:27 AM on November 12, 2009

And in case the site is down for others too, the author's posted two short videos to YouTube:

http://www.youtube.com/user/TwinbeeUK
posted by effbot at 10:28 AM on November 12, 2009 [1 favorite]

That's pretty cool. I wish I weren't too stupid to know what I'm looking at.
posted by jefficator at 10:29 AM on November 12, 2009

Oh. My. Fucking. God.

♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥
posted by egypturnash at 10:31 AM on November 12, 2009

These are really quite easy to make yourself, and the real fun isn't just in looking at it, but in exploring the shapes. I pretty much know nothing about the math and am mostly into it for visual jollies, but that said, I didn't find it daunting. Just download POV-ray and spend a few hours figuring out how to move the camera and light around.
posted by Stonestock Relentless at 10:34 AM on November 12, 2009 [1 favorite]

How is it that my heart still beats, and my chest cavity still expands and contracts, given that my mind is completely blown?
posted by CynicalKnight at 10:34 AM on November 12, 2009 [1 favorite]

While this is pretty and all, I think it's worth mentioning that the Mandelbrot Set isn't just an arbitrary simple formula that draws pretty pictures; it is a map of all of the possible Julia sets. Mandelbrot didn't originally try rendering it to get pretty pictures, but to get a sense of how Julia sets are organized in the X-Y plane. That such simple algorithms can draw pretty pictures was a resulting accidental discovery, which led to people looking for more pretty pictures, but unless this set has some important fundamental mathematical function apart from the pretty pictures it draws I think calling it the "Mandel"-anything is a bit wrong.
posted by localroger at 10:35 AM on November 12, 2009 [5 favorites]

This one reminds me a lot of this (wiki).
posted by Rhaomi at 10:35 AM on November 12, 2009

awesome! looks like something out of lovecraft....
posted by condour75 at 10:37 AM on November 12, 2009

Wow, that's cool. Look for it in Guillermo del Toro films and rave visuals…
posted by klangklangston at 10:39 AM on November 12, 2009 [1 favorite]

Interesting - the still images were fantastic, but the videos (at least those on YouTube), as they appear to be two-dimensional creations using shades of the same color instead of a myriad of colors, instead of the dense 3D still visuals.
posted by filthy light thief at 10:40 AM on November 12, 2009

Also from the author's YouTube profile:

Chaotic Bouncefloor of Doom
Super Galactic Mecha Mincer B
posted by kuatto at 10:42 AM on November 12, 2009

Stunning. Gaudi would approve.
posted by weapons-grade pandemonium at 10:43 AM on November 12, 2009 [1 favorite]

Paging Maynard Keenan

I think the name you want paged is actually Alex Grey, who is the artist responsible for much of the imagery associated with TOOL.

The first thing I thought when I saw this image from the Mandelbulb site was this image by Alex Grey, which is part of the 10,000 Days artwork also.
posted by hippybear at 10:43 AM on November 12, 2009 [1 favorite]

I'm afaird of looking too closely at any of these, lest an alien burst out a bud and plant eggs in my chest.
posted by The Whelk at 10:43 AM on November 12, 2009

Edible Mandlebulb.
posted by Nelson at 10:43 AM on November 12, 2009 [1 favorite]

Well, that certainly looks complex.
posted by 7segment at 10:44 AM on November 12, 2009 [1 favorite]

localroger: "unless this set has some important fundamental mathematical function apart from the pretty pictures it draws I think calling it the "Mandel"-anything is a bit wrong"

My impression is that this is the same mandelbrot dataset with a 3d rendering rather than the 2d one that we are more familiar with.

Lorenz ran his famous recursive equation from an analog computer connected to a speaker in order to help him conceptualize the properties, mandelbrot preferred a graphic rendering of his equation, they still are worthy of the name even if you are not using them for the math.
posted by idiopath at 10:45 AM on November 12, 2009 [1 favorite]

One day one of me will spend much of his free time in the Mandelbulb.

* Patiently waits for Singularity to get farther along *
posted by everichon at 10:47 AM on November 12, 2009

While this is pretty and all, I think it's worth mentioning that the Mandelbrot Set isn't just an arbitrary simple formula that draws pretty pictures; it is a map of all of the possible Julia sets.

Cool!




....What's a Julia set?
posted by EmpressCallipygos at 10:48 AM on November 12, 2009

previouslyish or, the backstory
posted by carsonb at 10:50 AM on November 12, 2009

....What's a Julia set?

Fractal quiches.
posted by The Whelk at 10:50 AM on November 12, 2009

This is a Julia set.
posted by hippybear at 10:51 AM on November 12, 2009

Sure, it's beautful. That's what they said when the Old Gods cracked open the sky and let loose rains of scorpions and blood. Oooh, they said, beautiful, beautiful, even as they were consumed by fire.
posted by Astro Zombie at 10:51 AM on November 12, 2009 [2 favorites]

This is a Julia set.
posted by The Whelk at 10:53 AM on November 12, 2009 [4 favorites]

Anyone else thinking Gaudi?
posted by jet_silver at 10:58 AM on November 12, 2009

inadequate preview, w-g-p, sorry
posted by jet_silver at 10:59 AM on November 12, 2009

I was told once that if you are studying math and the problems have exhausted the Roman and Greek character sets and have started into Hebrew characters, then you've gone too far. I found this to be true in an advanced prepositional calculus class.

Absolutely beautiful pictures.
posted by double block and bleed at 11:00 AM on November 12, 2009

Must. Obtain. Psychedelics.

I would sacrifice ten men to see a film set in these places.
posted by Drexen at 11:02 AM on November 12, 2009

This is certainly impressive-looking, thanks.
posted by flatluigi at 11:03 AM on November 12, 2009

I was also going to suggest just buying a head of Romanesco. I get it many weeks at the local farmer's market. It sort of tastes halfway between broccoli and cauliflower.
posted by GuyZero at 11:07 AM on November 12, 2009 [1 favorite]

In all seriousness - I'm a poet trying to understand math here -- can I just toss this out and make sure I'm following this correctly? According to what I've just read elsewhere -- A Julia set deals with math formula that determines.....something, and if you take each number and run it through that math formula, you get a Julia set. And then if you make GRAPH showing each result when you run it through that formula, you get THIS.

Do I have that right? Thanks -- I'm just trying to get a handle on what practical application the numbers in this formula would have served, if any, or whether this was just "we're playing with numbers and LOOK WHAT HAPPENED" kind of stuff.
posted by EmpressCallipygos at 11:10 AM on November 12, 2009

This was one of the favorite AskMe posts I've ever asked, and extremely helpful. It might help some of you who are confused by the math behind the Mandlebrot set.

(Does it count as a self-link if it's going to AskMe?)
posted by bondcliff at 11:11 AM on November 12, 2009 [3 favorites]

EmpressCallipygos: it is a recursive formula, that means you put a number into it, and then that gives you a new number, and you plug that one back in etc. etc. The visualizations are ways to look at properties of that equation over time with certain starting numbers, mainly mapping out which numbers end up being part of that string of values that goes through the equation, and which do not.
posted by idiopath at 11:19 AM on November 12, 2009

self correction: ignore what I said in my comment above and just read the comment by edd in the thread bondcliff links to, that is a much better explanation.
posted by idiopath at 11:23 AM on November 12, 2009

This was one of the favorite AskMe posts I've ever asked, and extremely helpful. It might help some of you who are confused by the math behind the Mandlebrot set.

that helps a bit, but I think my question is more like "what does one use a Mandelbrot set FOR"? As in, I understand that percentages can help me, say, calculate tips or plan a budget, and addition and subtraction can help me balance checkbook, and division can help me either double or halve the quanitities of ingredients in a recipe or re-size a knitting pattern. I'm just falling down on "to what PURPOSE do we use a Mandelbrot set for?" Does it relate to different chaos-theory physics in some way ("the random-movements of a flock of birds ain't that random"), is it a predictor of some behavior of economics ("we didn't look at how the Mandelbrot set would have affected the stock market over time and so that's part of why it crashed"), or...is it just a pure-math thing ("we were just playing with numbers and came up with something that just looked really freakin' cool")?

Why did we take the time to derive the Julia set and the Mandelbrot, in other words? Or was it just for the hell of it?
posted by EmpressCallipygos at 11:23 AM on November 12, 2009

I'm glad I'm not alone in finding these beautiful but somehow terrifying. Truly, if the Old Gods lay sleeping below the earth, their palaces must look as this.
posted by luftmensch at 11:25 AM on November 12, 2009 [1 favorite]

And I hope that's not taken in a sneering "why am I ever going to have to learn this stuff, I'm never going to use it" sense. I just know I'm somehow missing out on some additional layer that would let me appreciate it, and that's the layer I'm missing.
posted by EmpressCallipygos at 11:25 AM on November 12, 2009

There are things that for one reason or another stir within me a kind of mindless, irrational revulsion. Just terror, at a very small scale, in some little corner of my brain. It turns out that this visualization is one of those things.

Gorgeous, though, for the part of my brain that's not currently trying to gnaw its own leg off to get away. Old gods indeed. This is the eye of Cthulhu right here.
posted by cortex at 11:27 AM on November 12, 2009 [4 favorites]

EmpressCallipygos: many a mathematician would glibly say that if it has a point it is not real math anymore.

One reason many people find it interesting because it is a simple procedure for deriving seemingly complex results, with implies for many that other seemingly complex things in the real world may have a very simple mechanism behind them, which would mean that our intellectual accounting for the behavior of the world around us may be much closer to complete than we suspected.

Mr. Mandelbrot tried to apply his ideas to the shapes of mountains and coastlines and the behavior of the stock market, and there are many people who still think that using fractal theory they can find the magic simple equation that will let them beat the complexity of the stock market.
posted by idiopath at 11:28 AM on November 12, 2009

So it sounds like it's some combination of "we were just playing around and THIS happened and it was freakin' cool" combined with "if something so complicated came from running just one single formula, maybe there really IS a Grand Unified Field Theory -- keen!"

Something like that?
posted by EmpressCallipygos at 11:31 AM on November 12, 2009

cortex: I have to agree. I was actually getting vaguely nauseated looking at the renderings, which didn't make any sense to me other than there seems to be something about those kind of things which just set my brain on the wrong edge. I'm glad I'm not the only one who found that deep within the Mandelbulb lurks the screaming heebie jeebies.
posted by hippybear at 11:33 AM on November 12, 2009 [1 favorite]

EmpressCallipygos: one interesting thing about all these sets, Mandelbrot and Julia etc, is that we didn't have a clue about their real existence or what they contained until the advent of modern computing. They are too complex, with too many datapoints to do without advanced calculatory aid. Fractal science is one of the few truly new fields which has emerged in recent generations, and it's really very cool.

I don't think they really know what to do with a lot of it yet, overall.
posted by hippybear at 11:37 AM on November 12, 2009

there seems to be something about those kind of things which just set my brain on the wrong edge

I think there's an aspect of organic wrongness about it—it looks like it could be something that exists in nature, but it also looks like something that didn't come out right.

To put it another way: imagine if that thing took a breath.
posted by cortex at 11:38 AM on November 12, 2009 [4 favorites]

It's mandelbulbs all the way down.
posted by contessa at 11:39 AM on November 12, 2009 [5 favorites]

EmpressCallipygos: "So it sounds like it's some combination of "we were just playing around and THIS happened and it was freakin' cool" combined with "if something so complicated came from running just one single formula, maybe there really IS a Grand Unified Field Theory -- keen!""

Yeah, I think so, pretty much. Before fractals, there seemed to be for the most part* this common sense feeling that a complex shape or behavior must have some pretty complex math behind it. Then fractals kind of blew everyone's mind with how organic, complex, and natural looking they could be, with a very simple mathematical operation (simple, but repeated many millions of times to see the kind of images we are seeing here).

* I was mentored by some cyberneticists who had a pretty big chip on their shoulder about chaos theory, cybernetics had been addressing complexity and recursion since the 1940's or so, and then this Fractal shit comes around in the '70s or something and everyone loves it so much and it is so sexy and it gets all this popular press attention that cybernetics never really got (and don't even get a cyberneticist started on the way we use the prefix "cyber" these days).
posted by idiopath at 11:46 AM on November 12, 2009

I was told once that if you are studying math and the problems have exhausted the Roman and Greek character sets and have started into Hebrew characters, then you've gone too far. I found this to be true in an advanced prepositional calculus class.

If you use subscripts, you can just use a single letter and have at least Aleph null variables!

Oh crap.
posted by kmz at 11:46 AM on November 12, 2009 [3 favorites]

Nova's program "Hunting The Hidden Dimension" is a good overview of fractals and their discovery and some of why they are important.

There is also Arthur C. Clarke's "The Colors Of Infinity" Part 1 Part 2 Part 3 Part 4 Part 5 Part 6.
posted by hippybear at 11:49 AM on November 12, 2009 [1 favorite]

To put it another way: imagine if that thing took a breath.

Fine. Thanks. I'm sending you my therapy bills.
posted by hippybear at 11:52 AM on November 12, 2009 [1 favorite]

"I think there's an aspect of organic wrongness about it—it looks like it could be something that exists in nature, but it also looks like something that didn't come out right.

To put it another way: imagine if that thing took a breath."

I think it's kind of amusing that your brain went to, I assume, huge horror; my brain went to electron microscopes. If this took a breath, it wouldn't freak me out, because I'd assume that I shed billions of them from my skin every day.
posted by klangklangston at 11:59 AM on November 12, 2009

I think it's kind of amusing that your brain went to, I assume, huge horror; my brain went to electron microscopes.

I think the horror just comes from not being able to neatly package something to resolve into some kind of discrete phenomenon.
posted by Burhanistan at 12:01 PM on November 12, 2009

Triiiiiipppiiiiiiiiiiin' baaaaalllllssss...
posted by infinitywaltz at 12:13 PM on November 12, 2009 [1 favorite]

I'm with cortex and hippybear, something about these is really unsettling. If I wandered into the desert and came across the 'Cave of Lost Secrets,' I'd probably do no better than a Lovecraft character in describing the undescribableness of it.
posted by usonian at 12:20 PM on November 12, 2009

And now I've had the Mandelbrot Set song stuck in my head for HOURS now.
posted by strixus at 12:23 PM on November 12, 2009 [1 favorite]

The many-angled ones, as they say, live at the bottom of the Mandelbrot set,
posted by fearfulsymmetry at 12:48 PM on November 12, 2009 [3 favorites]

it's worth mentioning that the Mandelbrot Set isn't just an arbitrary simple formula that draws pretty pictures; it is a map of all of the possible Julia sets

Part of the point there is whose name deserves to mentioned and for what. People can be inventors, and they can be popularizers -- both are important. As mentioned above, the interest in these things is mainly in the way they stimulate human visual perception. The Julia set happened in a forest where nobody saw it, and in that sense it didn't really happen, not in the same way that the Mandelbrot set did.
posted by StickyCarpet at 12:49 PM on November 12, 2009

So that's what the crocheted acrylic monstrosity draped over grandma's couch wants to be when it grows up.
posted by casarkos at 12:51 PM on November 12, 2009 [1 favorite]

EmpressCallipygos: I don't think the Mandelbrot Set itself has a particular physical significance; it was more of a "we were just messing around with this equation and WTF?!?" event. But the kind of recursive function that gives rise to the Mandelbrot (and Julia) sets is actually pretty common in practical applied math. In particular, the question that actually leads to the set is "if I apply this formula over and over, what parameters lead to a stable situation and what parameters lead to the numbers zooming off to infinity?". Answering that question for other formulas can help you figure out, e.g., under what circumstances a machine or bridge might shake itself apart. (It's also the kind of question that has historically lead to some broadly useful mathematical insights and discoveries.)

The unexpected thing is the complex shape of the set, though. Usually, the answer to the stability/convergence question is simple, like "if X is less than 3, you're fine, otherwise kapow". The "mandelbrot set" in that case would be a boring circle or half-plane. But instead it's a shape that's not just complex, it's in a sense infinitely complex, and it's complex in weird and interesting ways. This is like crack to mathematicians.
posted by hattifattener at 12:53 PM on November 12, 2009

EmpressCallipygos: Well, the Mandebrot set on its own isn't very useful. It's just a mathematical function that produces beautiful graphs when transformed in a specific way. The concept behind it is useful because a wide variety of phenomena in the world appears to have a fractal nature. There was a nice bit of work about 10 years ago exploring the ways in which Jackson Pollock drip paintings imitated the same fractal aesthetics of coastlines.
posted by KirkJobSluder at 12:57 PM on November 12, 2009

Re: Practical applications...
To paraphrase Mr. Feynman: This type of mathematics is like sex...it may give some practical results, but that's not why we do it.
posted by rocket88 at 1:09 PM on November 12, 2009 [2 favorites]

And on second thought, there were some ideas floating around out there once upon a time that because natural scenes and textures are fractal in nature, that this could be a shortcut for more realistic electronic texture and sound. Of course what happened is that memory and storage capacity exploded along with computing power so it's usually easier to just sample the natural phenomena you want the computer to use. But it's still an interesting idea.
posted by KirkJobSluder at 1:10 PM on November 12, 2009

I would sacrifice ten men to see a film set in these places.

Ten not required... just Decker and Ilia
posted by CynicalKnight at 1:18 PM on November 12, 2009

They do look a bit like - growths. I suppose the natural forms we are most used to are constrained by function, and we appreciate, as it were, the design qualities. Something alive but growing unconstrainedly according to its own internal maths is probably a cancer or a nasty fungus, so our old reflexes assign it repugnance. Well, it's a theory.

I know nothing, but aren't fractals a bit last-century now? Wasn't there was time when they were supposed to be basis of new file formats and stock market algorithms, but in the end it turned out they were only good for making pretty patterns, and not even in a way you could use in a game unless it was especially trippy?
posted by Phanx at 1:25 PM on November 12, 2009

Isn't the Julia set a 3d fractal already?
posted by delmoi at 1:30 PM on November 12, 2009

Here's something regarding the Julia set and disputed attribution. He was interviewed on WFMU in 1998, but I don't see that in their recording archive. He was effusively generous in praising his collaborators, and all the CG pioneers that made his work possible. Seems to be a very nice man.
posted by StickyCarpet at 1:34 PM on November 12, 2009

Julia sets are really about complex numbers, which are inherently useful in a lot of ways that are not intuitive at all. A Julia set is a set of points which is created by starting somwhere on the x,y plane (actually the real,imaginary plane, but you can use x,y and almost everybody does). You apply a simple operation to that x,y pair to get another x,y pair. Then you apply the same operation to that pair. As you keep rolling, you either converge on a set of points that goes around in a circle, or zoom out to infinity.

The Mandelbrot set is a map of all the points that start Julia sets that converge. Some fractal drawing programs have a mode where you can cursor around the Mandelbrot set, seeing the Julia sets that result in each point in a smaller window.

What bugs me about calling this thing a MandelSomething is that Mandelbrot didn't have to tweak or pull on anything to get it to come out -- it comes out naturally from assumptions that have nothing to do with the pretty pictures. Once people realized how much complexity was there they started looking at other operations similar to the quadratic polynomial that draws the Mandelbrot set, and they found other examples; but while there are many fractals, only one fractal is the Mandelbrot Set. Mandelbrot didn't go looking for ways to represent the set looking for pretty ones; there is only one way to represent the set that makes sense mathematically, and that's an x-y plane. If you do that, it's math. If you're taking that as an input and messing around with it to get some particular type of display, that's art.

And for me, the most important thing about fractals is that they prove mathematically that complex things like weathered terrain and life do not need to be created by similarly complex sources. So fractals completely undermine the most common argument for the existence of God.
posted by localroger at 1:37 PM on November 12, 2009 [1 favorite]

there was time when they were supposed to be basis of new file formats

Fractal resolution zooming programs are available as plugins for Photoshop, and they do work better than zooming without them, but they can be slow for video or animation. I guess the catch is that digital SLR images are now too big to begin with.
posted by StickyCarpet at 1:38 PM on November 12, 2009

One thing people have used fractals for is Iimage Compression. However, because the person who came up with the idea patented it, almost no one actually uses fractal image compression in practice.
posted by delmoi at 1:38 PM on November 12, 2009

I think it's kind of amusing that your brain went to, I assume, huge horror; my brain went to electron microscopes.

I get the same heebie jeebies from a lot of electron microscopy images, actually. The stuff is fundamentally out-of-scale to the part of my brain that reacts badly—a rational assessment of just how small the thing I'm looking at doesn't doesn't come into it for a disturbing peek at something 1,000,000x magnified, so it definitely doesn't come into it for something entirely abstract like the mandelbulb renderings.

Something alive but growing unconstrainedly according to its own internal maths

KAAAANEEEEEEDDAAAAAA, etc
posted by cortex at 1:43 PM on November 12, 2009

The enduring importance of fractal geometry is that it provides a way of imagining organic structures and processes as mathematical functions. It is also important to remember that Mandelbrot single-handedly restored geometry and mathematical visualization to a place of honour.
posted by No Robots at 1:52 PM on November 12, 2009

Oh man, I looooove the Mandelbrot set and fractal geometry from just a visual point of view and I fully admit I have no mental capacity to understand any of the math or mechanics of it. But I used to just watch those animated zooms of a Mandelbrot set simply because they were so damn creepy. It just seemed so infinite when you look at all the nooks and crannies and pick out the shapes. I was probably imagining someone having to traverse one and how long would it take and such. But the images from the Mandelbulb seriously started giving me that itchy feeling all across my scalp that you get when seeing an ant colony with holes in the ground and ants crawling in and out or maggots squirming about on a dead bird.
posted by kkokkodalk at 2:10 PM on November 12, 2009

Ye gods, this is amazing. It looks just like my college dorm room looked about every other Saturday.
posted by newmoistness at 2:15 PM on November 12, 2009 [1 favorite]

Just download POV-ray and spend a few hours figuring out how to move the camera and light around.

Err, did you read the text?

Eager to get a better look at this thing, I set about trying to find software to render it, preferably with full shadowing and even global illumination, and at least something that was fairly nippy. But it turns out that there are probably no 3D programs out there on the market that can render arbitrary functions, at least not with while loops and local variables (a prerequisite for anything Mandelbrot-esque!). So I set out to create my own specialized voxel-ish raytracer. Results could be slow (perhaps a week for 4000x4000 pixel renders!), but it'll be worth it right?

The author had to write his own software to get decent renderings. This is not the same as the old 2D mandlebrot zooms that everyone's been playing with for years.
posted by Rhomboid at 2:30 PM on November 12, 2009

The Mandelbrot Set is a fun way of visualizing what's between 0 and 1 (using 0's and 1's).
posted by vectr at 2:48 PM on November 12, 2009

Rhomboid: "The author had to write his own software to get decent renderings."

blender is 100% python scriptable, but any software that tries to interact with 3d fractally generated forms in real time has its work cut out for it, and python is not exactly a speed-demon of a language.
posted by idiopath at 2:49 PM on November 12, 2009

Do you guys also feel like you've seen it before?

Somewhere.
posted by krilli at 3:10 PM on November 12, 2009

Beautiful, thanks dhruva.
posted by tellurian at 3:17 PM on November 12, 2009

cortex: "There are things that for one reason or another stir within me a kind of mindless, irrational revulsion. Just terror, at a very small scale, in some little corner of my brain. It turns out that this visualization is one of those things.

Gorgeous, though, for the part of my brain that's not currently trying to gnaw its own leg off to get away. Old gods indeed. This is the eye of Cthulhu right here."

I felt the same way, actually. It was sort of the mathematical visualization equivalent of looking at an operating nuclear reactor from inside it. It's fascinating to look at, it's something no one has seen before, and you just know you're not going to live long enough to tell anyone what it looked like.
posted by FishBike at 3:35 PM on November 12, 2009 [1 favorite]

The Mandelbrot Man has shambled out of the pages of Last Call to shoot the face of Moon.
posted by robocop is bleeding at 3:51 PM on November 12, 2009

Complex numbers are freakishly useful, and, as localroger mentioned above, this is for somewhat obscure reasons. One can think of it this way, though: If you think of the integers, dividing one number by another often doesn't have an answer. (1/2 isn't an integer, after all.) But by moving to the rational numbers, this is taken care of. Now we can divide anything by anything. But clever people then notice that there's space between the rational numbers. For example, the square root of two is certainly a number, but it's not tough to show that it can't possibly be a rational number. So to fill in that space, we invent the real numbers.

But then you notice that no negative number has a (real) square root, since the square of any real number is positive. And then the complex numbers are introduced to solve this problem. This ends up solving a much bigger problem: The associated theorem (the Fundamental Theorem of Algebra, it's sometimes called) says that every polynomial can be completely factored over the complex numbers. (In fact, finding a root of the polynomial x^2+1=0 is equivalent to finding the sqrt of negative one. Over the complex numbers, this polynomial factors into (x+i)(x-i).) To introduce some jargon, this theorem is saying that the complex numbers are 'algebraically closed.' This property ends up making all kinds of math work out incredibly nicely over the complex numbers, because we can often make interesting problems into statements about roots of polynomials. In spite of the name, complex numbers pretty much always make math more simple!

(In my field, we study (tangentially) objects called Lie Groups (pronounced LEE), which are big collections of matrices. If we let these matrices have complex numbers in them, we get an extremely nice classification of the (simple) Lie groups: four infinite families and five exceptional types. Over the reals, it's a big old mess.)

When we look at the Mandelbrot set, we're really looking at colored points in the complex plane. Let's think a bit about what that means: A complex number is something like a+bi where a and b are real numbers and i is sqrt(-1). So we can think of a+bi as a point in the plane with coordinates (a,b).

There are two nice things about the complex numbers: One is that they're algebraically closed (as I mentioned above), and the second is that they have a natural addition and multiplication. The second doesn't seem terribly profound, since they're just numbers. But if you think of points in the plane as pairs of real numbers, there's a natural way to add them (just add the coordinates), but no really obvious way to multiply them.

The Mandelbrot set is a statement about the multiplication and addition of the complex numbers. Given a number x, if I keep replacing x by x^2+1, over and over and over again, will I shoot off to infinity, or stick around near the origin forever? The Mandelbrot set is the set of numbers that stay near the origin. All the fancy colors are generally used to tell how fast the other numbers OUTSIDE of the Mandelbrot set shoot off to infinity.

So what's the problem with 3D Mandelbrot? Well, just as there isn't a natural multiplication in the plane with real coordinates, there also isn't a natural multiplication in three dimensional space. Before, we came up with complex numbers, which we can draw in the plane and multiply naturally, but NO SUCH THING exists for three dimensions; as such, any mutliplication is going to be a bit arbitrary. This is what the 'degree eight' stuff in the webpage is about: there isn't a natural port of the problem to three dimensions, so they came up with something arbitrary that yields pretty pictures.

There IS a natural multiplication in four dimensions, given by the Quaternions. The quaternions extend the idea of the complex numbers a bit more, but at a heavy price: their multiplication doesn't commute. (xy!=yx) Also, they don't simplify anything, so next to no one uses them. (That said, I do have a little paper cube that models the multiplication of the quaternions on my desk.) As such, Quaternion Mandelbrot sets do make sense mathematically, but (according to the link author at least) don't look pretty enough to be considered on their own; they give rotationally-symmetric sets when projected into three dimensions.

Finally, there's another even stranger algebra out there. The Octonions give a natural multiplication on 8-dimensional space. John Baez said this of the Octonions:

The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.
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There seems to be very, very little on the web relating to Octonion realizations of the Mandelbrot set. Perhaps this is as it should be: This is the number set that was itself driven mad by the secret knowledge of the Old Ones.
posted by kaibutsu at 6:20 PM on November 12, 2009 [11 favorites]

When I first discovered Mandlebrot fractals about 6 or 7 years ago, I was talking with a friend about how this is a good benchmark of computing power. Specifically, about video processing and real time rendering capacity. I will be impressed when I can fly around in one of these 3D Mandelbrot things, complete with high resolution, textures, etc.

Think of a Mandelbulb mod to Half Life II. Now that would be some seriously impressive computing power. Wonder how long it'll take to happen.
posted by zardoz at 6:46 PM on November 12, 2009

When I think of Quaternions I think of Gestalt.
posted by namagomi at 6:52 PM on November 12, 2009 [1 favorite]

There needs to be some sort of Godwin's Law for quaternions and math stuff.
posted by GuyZero at 7:47 PM on November 12, 2009

This is in reference to something pretty far back upthread, but I sort of figure that if you get bored waiting around for a Singularity to happen, it's not a Singularity at all...
posted by You Can't Tip a Buick at 7:56 PM on November 12, 2009

How many watts is a Mandelbulb, and where can I buy one?
posted by Eideteker at 8:32 PM on November 12, 2009

"How many watts is a Mandelbulb"

All of them!
posted by Eideteker at 8:33 PM on November 12, 2009 [1 favorite]

I will be impressed when I can fly around in one of these 3D Mandelbrot things, complete with high resolution, textures, etc.

Oh absolutely. I would love a 3D version of Xaos (even moreso if there were some naturally 3D fractal like the Mandelbrot Set that didn't resort to a hack like this one). But I seriously wonder what such a program would do to my sanity.
posted by straight at 8:51 PM on November 12, 2009

As exquisite as the detail is in our discovery, there's good reason to believe that it isn't the real McCoy. Sure, there are incredible patterns, and I for one could be fooled at first glance. However, it would seem that the real thing will have even more exquisite detail, surpassing even the pictures we've seen! (That's if it exists, but hey, there seems less doubt about that now!)

Evidence it's not the holy grail? Well, the most obvious is that the standard quadratic version isn't anything special. Only higher powers (around after 3-5) seem to capture the detail that one might expect. The original 2D Mandelbrot has organic detail even in the standard power/order 2 version.

Maybe you just need to have a power equal to or greater than the number of dimensions you are rendering in? A power of 1 Mandelbrot looks pretty lame in 2D (it's just a single point at the origin, not even a fractal).
posted by L.P. Hatecraft at 9:08 PM on November 12, 2009

There are plenty of 3d fractals out there, lots of recursive functions should be able to produce them. This kind of gets into more computer science (in the Turing sense) then algebra: predicting the output of a recursive function. As in, can you mathematically define a "pretty" n-dimensional fractal pattern, and then determine whether or not a given recursive function will produce one?
posted by delmoi at 9:54 PM on November 12, 2009

There seems to be very, very little on the web relating to Octonion realizations of the Mandelbrot set. Perhaps this is as it should be: This is the number set that was itself driven mad by the secret knowledge of the Old Ones.

According to wikipedia there is also a Sedenions which retain power associativity, but are not even an integral domain. According to Wikipedia, you can keep extending these infinitely using Cayley Dickson Construction
posted by delmoi at 10:08 PM on November 12, 2009 [1 favorite]

Over/under on how long it takes to find a Langford Basilisk in there?
posted by hades at 12:22 AM on November 13, 2009 [2 favorites]

Lovely. And great comment, kaibutsu -- very informative.
posted by painquale at 2:45 AM on November 13, 2009

How many watts is a Mandelbulb, and where can I buy one?

The more closely you examine it, the more watts it consumes. Fractal power!

And you can buy one here.

(Well, OK, it's just a print that he's selling to help cover bandwidth costs, apparently. Personally I am hoping to see some of the prettier/creepier close-up images available there... I'd really like a giant high-resolution print of some of these.)
posted by FishBike at 5:41 AM on November 13, 2009

delmoi, As in, can you mathematically define a "pretty" n-dimensional fractal pattern, and then determine whether or not a given recursive function will produce one?

Perhaps you can (this is what Fractal compression aims for), but there are a couple of limits you run into (bear with me here as my understanding is enthusiastic-amateur-level):

The first is that the julia and mandelbrot sets are like infinite coastlines. They have a bounded area (assuming a 2D set) - you can just from looking at them that a lower and upper bound on their area can be worked out relatively simply. However, their perimeter is unbounded, ie there is, in theory, infinite detail between any two points. So specifying your pretty fractal pattern can tricky in the first place. In some simple iterative cases like the Koch curve it isn't so hard though.

The second is information-theoretic as related to all compression. Depending on the pattern, at some point the fractal function you create might be larger than the raw "fractal pattern" you are trying to get it to generate.
posted by vanar sena at 6:30 AM on November 13, 2009

Over/under on how long it takes to find a Langford Basilisk in there?

There are an infinite number of them, but fortunately each is only harmful when viewed from one particular angle.
posted by straight at 8:00 AM on November 13, 2009

There are an infinite number of them [...]

I know that's probably a joke, but it raises an interesting point. Has there been any mathematical analysis one way or another to say if every possible shape must be present, somewhere, within a particular fractal?

I'm thinking of a comparison with random and irrational numbers. An infinite series of truly random digits must contain every possible finite series of digits somewhere within it. But a decimal representation of an irrational number is infinitely long, non-repeating, and yet there are trivial examples that show it does not have to contain every possible sequence of digits somewhere within it.

Fractals seem more like the latter in that there's no true randomness to them, so there's no reason why every possible shape would have to appear in there eventually. But I'm also just an "enthusiastic amateur", as vanar sena put it, so it would be cool if an expert had actually studied this.

Obviously there are fractal shapes that are easy to analyze and show that they can't possibly contain every other shape, I'm wondering specifically about more complex ones like those shown here, that aren't so easy to analyze visually.

I am also having one of those "imagine trying to have this discussion in Youtube comments" moments.
posted by FishBike at 8:39 AM on November 13, 2009

(And now I realize my question above is silly, because the neat feature about these fractal curves and shapes is no matter how much you magnify them, there's always more detail there to be seen, which cannot be said of most shapes. So almost by definition it would seem that most shapes cannot be found within such a fractal, but only other fractal shapes are there. And maybe in many cases only the *same* fractal shape can be there, with them being self-similar at all scales.)
posted by FishBike at 9:25 AM on November 13, 2009

So almost by definition it would seem that most shapes cannot be found within such a fractal, but only other fractal shapes are there.

Yeah, but $100 says that if we ever did find a Langford Basilisk, it would be a fractal.

XaoS ought to have a disclaimer popup: WARNING extended of use of this software virtually guarantees that you will see shapes never before seen by a human being. One or more of these might induce nausea, headaches, insanity, and/or death. But probably not.
posted by straight at 9:59 AM on November 13, 2009 [1 favorite]

I am amazed at the proportion of links in this thread that my browser shows as visited.
posted by jimfl at 7:16 PM on November 13, 2009

It's possible to print things in 3D, right?
even things like this?

Why can't I buy this as a hat?
posted by Acari at 7:25 PM on November 13, 2009

I know that's probably a joke, but it raises an interesting point. Has there been any mathematical analysis one way or another to say if every possible shape must be present, somewhere, within a particular fractal?

You could ask as to whether a close approximation to a particular shape is present. Also, a fractal as a recursive function could theoretically have non-fractal sub-units. For example a Sierpinski Triangle has plenty of straight lines and large, empty triangles. A Menger Sponge has empty cubes and straight lines. It seems obvious that a Sierpinski triangle and Menger cube haven't got any shapes besides cubes and triangles.
posted by delmoi at 1:19 AM on November 14, 2009

straight: Yeah, but $100 says that if we ever did find a Langford Basilisk, it would be a fractal.

It's a great short thriller story that I'll have to add to my collection of Pseudosicence Fiction. (It joins the illustrious company of Asimov and Clarke there.)
posted by KirkJobSluder at 10:05 AM on November 14, 2009

delmoi: "You could ask as to whether a close approximation to a particular shape is present."

For the purposes of finding neat stuff in fractal imagery, I agree that close enough is good enough. I was thinking, though, that from the point of view of some sort of mathematical proof, allowing for "close enough" shapes pushes it from the perhaps just barely possible well into "you gotta be kidding me"¹ territory.

"It seems obvious that a Sierpinski triangle and Menger cube haven't got any shapes besides cubes and triangles."

Yeah, these were the kinds of simple fractal shapes I had in mind. Obviously if I'm looking for anything with round edges, I'm not going to find it in one of those. But I think it goes farther, because no matter how much you zoom in on one of those shapes, all you get are smaller and smaller versions of it.

I think the same may be true of more complex shapes like the Mandelbrot fractals -- those probably do not contain anything but smaller and slightly transformed versions of themselves². I think I remember reading about that somewhere, actually, that somebody had found a complete but tiny copy of the whole thing somewhere in it.

1 - not an actual mathematical term as far as I know
2 - I'm sure to actual mathematicians, this sounds like a serious discussion of what type of cheese the moon is made of
posted by FishBike at 6:07 PM on November 14, 2009

I'm glad I wasn't the only one who thought of Cthulhu when I saw these.
posted by malaprohibita at 7:49 PM on November 15, 2009

This post prompted me to plot the Mandelbrot set on the complex plane. It was, as people usually say, surprisingly easy. So, thanks!

Now I have to resist the temptation to ask probably-already-answered questions ...
posted by fantabulous timewaster at 1:33 PM on November 16, 2009