Welcome to everyone who has come here from Rudy Rucker’s blog, from Vimeo, or from my LKL talk. I have attempted here to give a a bit more explanation of my recent 4D rotation animations, in as non-technical a way as possible.
I have done some experiments with 3D inversion and this developed out of that. Now if you dont know it there is this fantastic video by Douglas Arnold and Jonathan Rogness called Moebius transformations revealed. Its beautifully done and I really recommend you watch it.
The key point for the following discussion is at 0:35
I was very impressed by how clear and intuitive this made the notion of inversion in 2D. So I got to thinking about something similar for my inversions in 3D.
For this I’ve used something called the 3-sphere.
(Now its important to note that what in everyday language is called a sphere is referred to by mathematicians as a 2-sphere. A 1-sphere is a circle and a 3-sphere (or hypersphere) is an object which lives in 4-dimensional space just as an everyday 2-sphere lives in 3D space.)
So I wrote some code which uses an inverse stereographic projection (often described in 3D, but it generalises naturally to higher dimensions) to project Euclidean 3-space (ie normal, everyday 3D space) onto the 3-Sphere (A space of non-Euclidean geometry, a curved space like those of Einstein’s relativity).
I then performed a 4-dimensional rotation on this 3-Sphere while stereographically projecting back down to Euclidean 3-space. Just as the ‘Moebius transformations revealed’ video projects rotation in 3D to the 2D plane, my animations project rotation in 4D to a 3D space so that we can see it.
Continue to page 2 of 4-Dimensional Rotations
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December 11, 2008 at 7:14 pm
[...] Space Symmetry Structure [...]
January 13, 2009 at 6:39 pm
was this made in vvvv? If so care to share the code? If not mybe when I’m not lazy I’ll reuse what you’ve presented and try to code my own… cool stuff
January 13, 2009 at 10:48 pm
Thanks. This is Rhinoscript, not vvvv. Let me know if you do code a version, I’d be happy to help
January 19, 2009 at 4:36 am
[...] Space Symmetry Structure [...]
February 6, 2009 at 5:33 pm
[...] one then has lines and circles about which the space revolves. I began to explore this a bit in my post here, using stereographic projection in 4D. Things rapidly become rather confusing, with the [...]
March 3, 2009 at 12:15 am
[...] 4-Dimensional Rotations « Space Symmetry Structure (tags: inspiration animation fun visualization math) [...]
March 11, 2009 at 12:05 pm
[...] finally, 4-dimensional [...]
April 5, 2009 at 12:26 am
Perhaps if the atomic structures of the material that forms some hypothetical newly evolved cortex of the brain – if this atomic material were to contain an axis lying in a fourth direction; let’s say a .0000006 seconds into the future and .0000005 seconds into the past, and with particles at these distances, then information of four space might be able to be processed into perceptual interpretation mediated through the 4-D neurons of this cortex. -scriAlphi
April 5, 2009 at 1:55 pm
schriAlphi – have you read Penrose’s The Emperor’s New Mind ? I think the ideas there are somewhat similar to what you describe.
April 5, 2009 at 3:08 pm
Daniel, tks for the reply. As a matter of fact I have the book, but haven’t read it in it’s entirety. I submitted this idea as a result of having read a discription of a science fiction pseudochemical as described by Isaac Asimov called thiotimeline or something like that. This possessed those very properties i described but not as part of any cortex. But were reactively chained to extend this miniscule penetration into the future to up to 2 seconds. I’ll serch for the article or any reference to thiotimoline (or something like that) in google. -schriAlphi
May 23, 2009 at 7:29 pm
[...] on rheotomic surfaces Those coming from BoingBoing (!!!) in search of Gnarl might also enjoy my 4D rotation animations or experimants with Cellular Automata. Possibly related posts: (automatically generated)Rheotomic [...]
June 9, 2009 at 10:25 pm
Great work Daniel, I first saw the youtube inversion and wondered about the 3D. Can you share the Rihnoscript for us that want to code in… say OPenGL?
June 10, 2009 at 5:49 pm
Thanks Alex,
Yea, will post the code soon(ish), and maybe a grasshopper version.
I’d love to see this transferred to some other languages, and I’m interested in learning some OpenGL myself
October 4, 2009 at 7:33 pm
[...] Daniel Piker has some very interesting demonstrations of 4-dimensional rotations: [...]