@HolomorphicShoes the lens idea is interesting; I haven't thought about it that way before, but it makes sense. I'm sure someone has figured out formulas for how different shaped lenses distort light, but I don't know anything about it. I wonder what lens shape would make an arctan distortion!
@HolomorphicShoes This doesn't distort much when r is small, so the "inner" fibers look more like circles. But the "outer" circles get distorted a lot.
@HolomorphicShoes oh, I think I understand your question: in that Hopf fibration picture the loops are not actually circles because what I used isn't exactly stereographic projection. Stereographic projection of S^3 fills all of R^3, but I wanted something that fits in a ball of radius 1. So I compressed it where a point in R^3 with spherical coordinates (r, theta, phi) was sent to (arctan(r), theta, phi).
@HolomorphicShoes Maybe you mean one could choose a different projection to R^3? A fact about stereographic projection is that it always sends circles to circles, but you could choose a different type of projection maybe?
@HolomorphicShoes wow thanks for checking! I was waiting for you to ask a question, but I totally missed it above!
I'm not sure I understand it though. The Hood fibration is a specific function from S^3 to S^2, and it's fibers are definitely circles in S^3. So I'm not sure what you mean by "a matter of perspective".
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@11011110 yeah, same for me; I should have been either more specific, or more vague. Good luck to the people working on it!
Working in topology and category theory;
Assoc. Prof. at Ohio State, Newark
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