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 around 1:55
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.