It turns out that in this case, the 2 circles I have been describing as the poles of the 4D rotation are not unique at all. In fact, absolutely every point in 3D space lies on such a circle. Where a 2D rotation has one pole, and a 3D rotation of a 2-sphere has 2 poles, an isoclinic 4D rotation of the 3-sphere has nothing but poles, in fact it has an infinite number of them. The 3-sphere, and hence its stereographic projection to 3-space, is entirely filled by circles. Under one full turn of such a rotation every point in the space loops exactly once through every one of these circles. The circles are arranged in a stunningly beautiful way, so that each circle links with every other circle, and no 2 circles ever intersect. Sadly we cannot see the full perfection of this arrangement in our Euclidean 3-space because when they are projected down, these circles are of different sizes, and there will always be one of them which projects to a straight line (a circle of infinite radius). But in the 3-sphere all the circles are of the same size, each one is related to the whole in exactly the same way, and the true regularity and smoothness of the arrangement is revealed.
Such an arrangement of curves which covers a whole space without crossing itself is known as a fibration of that space, and the one I just described is called the Hopf fibration after its discoverer Heinz Hopf. No analogous fibration exists for the 2-sphere, a fact sometimes described as the hairy-ball theorem. (If you have a ball covered in fur, there exists no possible way of combing that fur so that it all lays flat – you must always have one or more tufts, points where there is a sudden change of direction, like the poles of the 2-sphere described earlier). The fact that such a perfectly smooth ‘combing’ of the 3-sphere exists has profound implications in maths and physics. It is a central part of Roger Penrose’s Twistor theory (which aims to ‘provide an adequate formalism for the union of quantum theory and general relativity’ and is noted for its great mathematical elegance), appearing as a ‘Robinson Congruence’ – a ‘twisting shear-free congruence of rays’.
Well that’s about enough for today, but I hope to keep producing animations to better show the 3-sphere and related topological marvels in a way that is intuitively understandable.
I also intend to write a post on the actual workings of the code – the matrices, coordinates and trigonometry involved. I didn’t want to do that here, as I’m trying to make this post as accessible as possible. Also I’ve used a fairly intuitive, hacking and guessing sort of approach, so it will be hard to explain clearly, but I’ll try soon, and hopefully someone out there can help me express it in terms of Quaternions/Clifford algebra, as Im sure that would be much more elegant and powerful.
When learning about these ideas I found the following sources very helpful:
The Flat Torus in the Three-Sphere by Thomas Banchoff
Also, I havent studied the work of Blaine Lawson enough to fully understand it yet, but I think what I am doing must relate closely to some of his work, as my script generates what appears to be the sudanese mobius band and lawson klein bottle which he was the first to describe. I first found out about these from the animations of George Francis.
Finally, I would like to point out that I am not formally educated in any of this, so its highly possible that I have made mistakes. I would be very grateful for any corrections or clarifications from real mathematicians.