Comments on: 4-D fractal film
http://www.metafilter.com/42813/4D-fractal-film/
Comments on MetaFilter post 4-D fractal filmThu, 16 Jun 2005 13:24:35 -0800Thu, 16 Jun 2005 13:24:35 -0800en-ushttp://blogs.law.harvard.edu/tech/rss604-D fractal film
http://www.metafilter.com/42813/4D-fractal-film
Video: <a href="http://www.fleischfilm.com/html/gestalt.htm">4-dimensional quaternions</a> (group of fractals) are visualized by projecting them into three-dimensional space. (x[n+1]=x[n]^p, baby.post:www.metafilter.com,2005:site.42813Thu, 16 Jun 2005 13:21:07 -0800signalfractalsquaternions4dBy: signal
http://www.metafilter.com/42813/4D-fractal-film#958665
That should, of course, have read: "x[n+1]=x[n]^p, baby."comment:www.metafilter.com,2005:site.42813-958665Thu, 16 Jun 2005 13:24:35 -0800signalBy: signal
http://www.metafilter.com/42813/4D-fractal-film#958670
<a href="http://www.grau1001.de/">This </a>is pretty cool, too, if you into this sort of thing. Look for the links in the bottom right div.comment:www.metafilter.com,2005:site.42813-958670Thu, 16 Jun 2005 13:26:09 -0800signalBy: teece
http://www.metafilter.com/42813/4D-fractal-film#958671
I'd like to state for the record that quaternions are cool. They are like complex numbers, but rather than being of the form <i>a+bi</i>, they are of the form <i>a+bi+cj+dk</i>. Thus there are three 'imaginary' numbers rather than one in a quaternion.
Develepod by Hamilton. The <a href="http://mathworld.wolfram.com/Quaternion.html">MathWorld</a> article is cool, if you are so inclined.comment:www.metafilter.com,2005:site.42813-958671Thu, 16 Jun 2005 13:26:56 -0800teeceBy: gnutron
http://www.metafilter.com/42813/4D-fractal-film#958700
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.comment:www.metafilter.com,2005:site.42813-958700Thu, 16 Jun 2005 13:49:42 -0800gnutronBy: Kickstart70
http://www.metafilter.com/42813/4D-fractal-film#958707
)
God, without that bracket you nearly killed us all!! Fractal edges are infinitely sharp!comment:www.metafilter.com,2005:site.42813-958707Thu, 16 Jun 2005 13:52:43 -0800Kickstart70By: escabeche
http://www.metafilter.com/42813/4D-fractal-film#958716
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?comment:www.metafilter.com,2005:site.42813-958716Thu, 16 Jun 2005 14:05:32 -0800escabecheBy: delmoi
http://www.metafilter.com/42813/4D-fractal-film#958725
How can you have a 4-dimensional anything with only one variable?comment:www.metafilter.com,2005:site.42813-958725Thu, 16 Jun 2005 14:12:43 -0800delmoiBy: teece
http://www.metafilter.com/42813/4D-fractal-film#958730
delmoi: I'm assuming that <i>n</i> is a step in an iteration, <i>p</i> and <i>c</i> are parameters, and <i>x</i> 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.comment:www.metafilter.com,2005:site.42813-958730Thu, 16 Jun 2005 14:18:11 -0800teeceBy: delmoi
http://www.metafilter.com/42813/4D-fractal-film#958737
<i>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.</i>
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.comment:www.metafilter.com,2005:site.42813-958737Thu, 16 Jun 2005 14:23:34 -0800delmoiBy: ikalliom
http://www.metafilter.com/42813/4D-fractal-film#958746
These are <a href="http://www.chaospro.de/documentation/html/fractaltypes/quaternions/theory.htm">four-dimensional fractals</a>, 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.comment:www.metafilter.com,2005:site.42813-958746Thu, 16 Jun 2005 14:30:55 -0800ikalliomBy: vernondalhart
http://www.metafilter.com/42813/4D-fractal-film#958749
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<sub>n+1</sub> = Z<sub>n</sub><sup>2</sup> + c.
He seems to use on the webpage a very analagous formula, x<sub>n+1</sub> = x<sub>n</sub><sup>p</sup> - 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<sup>-1</sup>(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.comment:www.metafilter.com,2005:site.42813-958749Thu, 16 Jun 2005 14:31:35 -0800vernondalhartBy: Rubbstone
http://www.metafilter.com/42813/4D-fractal-film#958750
creepy.
compelling,stark, beautiful and inspiring
but mostly creepycomment:www.metafilter.com,2005:site.42813-958750Thu, 16 Jun 2005 14:31:46 -0800RubbstoneBy: vernondalhart
http://www.metafilter.com/42813/4D-fractal-film#958761
<a href="http://www.metafilter.com/mefi/42813#958725">delmoi:</a> <i>How can you have a 4-dimensional anything with only one variable?</i>
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.comment:www.metafilter.com,2005:site.42813-958761Thu, 16 Jun 2005 14:39:45 -0800vernondalhartBy: teece
http://www.metafilter.com/42813/4D-fractal-film#958773
delmoi, sorry, <i>c</i> is in the linked page, but not in the post.
But you misunderstood me. Neither <i>n</i> nor <i>p</i> nor <i>c</i> are variables, they are parameters. The 4 variables that make it a 4-D equation are all contained in <i><b>x</b></i>=(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) [just picking some random vairables to represent the 4 dimensions].
It's the same with vector equations. <b><i>F</i></b>=<i>m<b>a</b></i> is a 3 dimensional problem in Newtonian physics, because <b><i>a</i></b>=(<i>x, <i>y</i>, <i>z</i>)</i>.comment:www.metafilter.com,2005:site.42813-958773Thu, 16 Jun 2005 14:50:56 -0800teeceBy: Mikey-San
http://www.metafilter.com/42813/4D-fractal-film#958783
THIS THREAD MAKES MY HEAD ASPLODE.
But damn, fascinating stuff.comment:www.metafilter.com,2005:site.42813-958783Thu, 16 Jun 2005 14:57:42 -0800Mikey-SanBy: erisfree
http://www.metafilter.com/42813/4D-fractal-film#958845
Oh. My. Wow.comment:www.metafilter.com,2005:site.42813-958845Thu, 16 Jun 2005 16:04:24 -0800erisfreeBy: Chuckles
http://www.metafilter.com/42813/4D-fractal-film#958923
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)
<img src="http://www.control.toronto.edu/~lidstone/slide16.gif" alt="Gimbal Lock">
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 <a href="http://www.control.toronto.edu/~lidstone/research.htm">slide 17 and slide 18 links here</a> 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!comment:www.metafilter.com,2005:site.42813-958923Thu, 16 Jun 2005 17:27:43 -0800ChucklesBy: recursive
http://www.metafilter.com/42813/4D-fractal-film#958929
Also, the soundtrack is totally frickin rad baby.comment:www.metafilter.com,2005:site.42813-958929Thu, 16 Jun 2005 17:41:29 -0800recursiveBy: pompomtom
http://www.metafilter.com/42813/4D-fractal-film#959042
Mikey-San: The trick is to not get <i>any</i> of it. Safe as houses.comment:www.metafilter.com,2005:site.42813-959042Thu, 16 Jun 2005 20:11:26 -0800pompomtomBy: DeepFriedTwinkies
http://www.metafilter.com/42813/4D-fractal-film#959082
Man, I have no idea what any of you guys (except Mikey-San) are talking about. But they shore is purty to look at.comment:www.metafilter.com,2005:site.42813-959082Thu, 16 Jun 2005 21:20:25 -0800DeepFriedTwinkiesBy: BlackLeotardFront
http://www.metafilter.com/42813/4D-fractal-film#960181
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?!comment:www.metafilter.com,2005:site.42813-960181Fri, 17 Jun 2005 18:26:10 -0800BlackLeotardFront