Somebody in the office saw examples of the Hopf fibration, or Hopf Map, and challenged me to it. Now, I hadn't heard of it before and it's quite an abstract thing that I don't understand much, but here is what I think is going on, and the result of which you can see above:

As far as I understand this is a continous mapping from a 3-sphere to a 2-sphere but it does so by going via bundles of circles (1-sphere).

For each point on S3 we construct a C2 complex number (out of two, originally build from the point's coordinates). The space R4 in which S3 sits is represented as 2 complex numbers in C2. A function maps this to a single complex number. So, that's the mapping.

Then, to visualize this we do a higher-dimensional stereographic projection where we identify each point in R3 with a point on S3. You can do that in 2d for example where each point on a 2d plane would be projected onto a 2d sphere and you effectively fit the entire infinite space, bar one point, and display it. It's the same with 3 dimensions. We can map the 3-sphere to R3 space and luckily for us we can see and understand 3d space!

Another way to look at this is to see the embedding space R4 of the S3 sphere as a the set of quaternions. Then a point, say Q, on S3 can be represented as a quaternion and since a quaternion can be used as a rotation in R3 we can represent Q as a point, say P0, in 3d. However, if we run that backwards we see that many points in R3 can be traced back to Q. The collection of those 'many' points has a structure: a circle. And the collection of those circles also has a special structure and that's what we see in the animations above.

Each of these circles is a 'fiber' and together they are a bundle of fibers. They are each interlinked. I like to think of fiber bundles as a sort of multi-dimensional copy stamping. Imagine you stamp lines on a sphere. Then the sphere is the base space and the lines form the fiber bundle. Each line can then be projected back to the sphere, and in reverse each point on the sphere can be backtraced to a collection of points (the line).

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