Niles Johnson @nilesjohnson@mathstodon.xyz

@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 dang it, that's supposed to say Hopf fibration, not Hood :/

@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".

@HolomorphicShoes sure, what's on your mind?

I've just relearned the following lesson: If you're going to make something by generators and relations, you should check and prove whatever coherence it satisfies. You're going to need it sooner or later if you're doing anything nontrivial with your construction.

@11011110 and now it's back! I knew complaining would help ;)

@11011110 yeah, same for me; I should have been either more specific, or more vague. Good luck to the people working on it!

arxiv seems to be down right now, and I'm trying not to freak out

Going through some Beast Academy level 1 with my 7yo this summer, looking forward to seeing how they cover category theory.

@amy @wilbowma @maxsnew It's grueling! A supportive coauthor helps a lot. And, like many things, it's something that becomes more possible as one gains more experience with smaller projects.

@wilbowma @maxsnew Accepting that I will write a bad textbook was the only way I could make any progress at all.