4-D fractal film

June 16, 2005 1:21 PM

Video: 4-dimensional quaternions (group of fractals) are visualized by projecting them into three-dimensional space. (x[n+1]=x[n]^p, baby.
posted by signal (21 comments total)

That should, of course, have read: "x[n+1]=x[n]^p, baby."
posted by signal at 1:24 PM on June 16, 2005

This is pretty cool, too, if you into this sort of thing. Look for the links in the bottom right div.
posted by signal at 1:26 PM on June 16, 2005

I'd like to state for the record that quaternions are cool. They are like complex numbers, but rather than being of the form a+bi, they are of the form a+bi+cj+dk. Thus there are three 'imaginary' numbers rather than one in a quaternion.

Develepod by Hamilton. The MathWorld article is cool, if you are so inclined.
posted by teece at 1:26 PM on June 16, 2005

i want a develepod! i am not sure what that exactly is, but i could sure use a pod to assist in my development needs.
posted by gnutron at 1:49 PM on June 16, 2005

)

God, without that bracket you nearly killed us all!! Fractal edges are infinitely sharp!
posted by Kickstart70 at 1:52 PM on June 16, 2005

The quaternions are indeed really interesting -- they represent a bold leap forward into mathematical systems where multiplication is non-commutative (that is, a multiplied by b is not the same thing as b multiplied by a.)

That said, the quaternions are definitely not the "group of fractals," and I'm not at all sure what this movie actually depicts. Maybe he's computed some fractal subset of the quaternions, which he's then projecting onto 3-space in various ways?
posted by escabeche at 2:05 PM on June 16, 2005

How can you have a 4-dimensional anything with only one variable?
posted by delmoi at 2:12 PM on June 16, 2005

delmoi: I'm assuming that n is a step in an iteration, p and c are parameters, and x is a quaternion. Which would make the problem 4-dimensional, because of the 4-D nature of quaternions.

He doesn't really spell it out, though, and that's just a guess.

And I agree with escabeche. The language on the site is confused. I can't decide if he doesn't know what he is talking about, he is a non-native speaker, or if he's just a bad writer.
posted by teece at 2:18 PM on June 16, 2005

delmoi: I'm assuming that n is a step in an iteration, p and c are parameters, and x is a quaternion. Which would make the problem 4-dimensional, because of the 4-D nature of quaternions.

Wll, except c dosn't show up the equation, giving us just n and p. But I guess n, p, and x give us 3 dimensions.
posted by delmoi at 2:23 PM on June 16, 2005

These are four-dimensional fractals, in the same sense as an image of the Mandelbrot set is two-dimensional on the complex plane (not talking about fractal dimensions here). You draw the 4D image by starting from all initial values (combinations of a, b, c and d) and iterate to find out if the series remains bounded. Those 4D initial values which do, define the fractal set.
posted by ikalliom at 2:30 PM on June 16, 2005

By the looks of this, this is analagous to the Mandelbrot set - the set of complex numbers c such that the following iteration remains unbounded: Z_n+1 = Z_n² + c.

He seems to use on the webpage a very analagous formula, x_n+1 = x_n^p - c which is a bit different, but in the same vein.

Another point of interest about Quaternions is that you can use them to model rotations in three dimensions - Much as complex numbers can be used to model rotations in two dimensions

Basically, if you look at the purely imaginary parts of quaternions, that is the parts of the form xi + yj + zk, you end up with a three dimensional space. Then if you conjugate by any quaternion of norm 1 (ie if w is such a quaternion, you compute w^-1(xi + yj + zk)w) you get in effect a rotation in three dimensions. This was what Hamilton was looking for when he came up with these as I recall; he hit the jackpot with the idea when he realized that his imaginary components need not commute (specifically, ij = -ji = k). He figured this out while walking around in Dublin, and then proceeded to vandalize a bridge by carving it into it.
posted by vernondalhart at 2:31 PM on June 16, 2005

creepy.
compelling,stark, beautiful and inspiring
but mostly creepy
posted by Rubbstone at 2:31 PM on June 16, 2005

delmoi: How can you have a 4-dimensional anything with only one variable?

Well, by having a four dimensional variable. If you consider, for example, complex functions (that is, functions from the complex plane to itself) those are functions of one variable - but since the complex plane is "the same" as two dimensional real space, you can also see it as a function of a two dimensional real variable. This is the same, since the quaternions are "the same" as four dimensional real space.
posted by vernondalhart at 2:39 PM on June 16, 2005

delmoi, sorry, c is in the linked page, but not in the post.

But you misunderstood me. Neither n nor p nor c are variables, they are parameters. The 4 variables that make it a 4-D equation are all contained in x=(x, y, z, t) [just picking some random vairables to represent the 4 dimensions].

It's the same with vector equations. F=ma is a 3 dimensional problem in Newtonian physics, because a=(x, y, z).
posted by teece at 2:50 PM on June 16, 2005

THIS THREAD MAKES MY HEAD ASPLODE.

But damn, fascinating stuff.
posted by secret about box at 2:57 PM on June 16, 2005

Oh. My. Wow.
posted by erisfree at 4:04 PM on June 16, 2005

Well, the really cool thing about quaternions is that you can interpolate rotations in three space. That, and there is no gimbal lock singularity...

Interpolate rotations meaning that if you have the start and end of a rotation but you want to know what the half way point is, with quaternions it is easy to calculate.

Gimbal lock being when two of the three axes of rotation become aligned because the middle axis is rotated 90deg.
(This really is an animation, but I guess you have to refresh the page if you miss it)

Note that the Yaw and the Roll axes are aligned when the stand rotates 90deg about the Pitch axis. Traditional methods for representing rotation (like Z-Y-Z Euler angles) exhibit a singularity at this point, but quaternions don't.

The slide 17 and slide 18 links here might also be interesting. They show what happens to a gimbal near gimbal lock, it isn't pretty (in terms of the usefulness of my thesis, the animations are pretty enough).

Self link, woo-hoo!
posted by Chuckles at 5:27 PM on June 16, 2005

Also, the soundtrack is totally frickin rad baby.
posted by recursive at 5:41 PM on June 16, 2005

Mikey-San: The trick is to not get any of it. Safe as houses.
posted by pompomtom at 8:11 PM on June 16, 2005

Man, I have no idea what any of you guys (except Mikey-San) are talking about. But they shore is purty to look at.
posted by DeepFriedTwinkies at 9:20 PM on June 16, 2005

That was motherfreaking insane! The background music sounds like Wolf Eyes remixed by Oval! The decision to do that in black and white was wise - too many fractal animations look corny b.c. of weak color schemes. That was like a glimpse into the lowest circles of hell. What the hell was that egg?!
posted by BlackLeotardFront at 6:26 PM on June 17, 2005