I appreciate the posts that people are writing! But I don't like writing my own!! Anyway, I'll try.

I'm a research mathematician, interested in social media for both professional and recreational reasons. If you follow me, you'll see a mix of math and non-math.

Here are four pictures of things I've been involved in. I'll add a little more in a comment below.

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The first two are pictures of Wikipedia pages, featuring images that I made. I'm proud of them (a) because they help visualize the math involved and (b) other people seem to appreciate them too.

The second two are books I've coauthored, with Donald Yau. I'm proud of those because they explain some complicated stuff that I know about. Both books can be downloaded from my website. Links below :)

@nilesjohnson I happen to have a student currently working on one of those four things (associahedra)! Probably from a different point of view though. Will post more when preprint becomes available, likely in a couple months.

@11011110 cool! I was surprised when I learned how many different places those structures showed up. The particular configuration in my diagrams comes from one of their original uses, in the thesis of Tamari. (I forget the rest of those details now though!)

@nilesjohnson I had no idea that you made that Hopf fibration image! I've seen it everywhere!

@nilesjohnson your Hopf fibration work is epic!

I read a lot of the materials you shared before realizing you are in haha.

I have found answers to most of my Hopf-fibration-related questions, but one question remains.

Would you be okay with me asking here?

@nilesjohnson Thanks so much! In this image, I'm showing different views of a sphere and a plane in the center and right-hand columns. The only difference is the viewpoint. The right-hand side is distorted because we are viewing S2 from the outside haha. So, a transformation is the act of 'steering the sphere' haha.

How do we control the literal viewpoint for Hopf fibrations to see D-shaped loops vs. circles? Most of materials treat the fibers as symmetric, which is a matter of perspective.

@nilesjohnson You can ignore that the contours are cosine here and not the exponential function haha

@nilesjohnson hi, does this question make any sense? If not, I can rephrase. Thanks again!

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

@nilesjohnson Hi! Yes, circles to circles :) I meant when you look through a circular fibers on a transparent sphere, we could see a circle, an ellipse, or a distorted circle. So, how do we control the (apparent) distortion, such as what we see here? Do you use the camera viewpoint as well as the positions shown on the right? Cosine just happened to be what I had handy (apparent distortion of ellipses instead of apparent distortion of circles). Thanks so much, and apologies for the wordiness!

@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 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.

@nilesjohnson Oh my gosh, thank you so much! Arctan! That's super interesting and exactly what I wanted to know! Can we think about the distortion as a lens of sorts that 'corrects' focal length of an image in the center of the image surface but bends the edges? I took a picture of a dotted surface with my cell phone camera resting on a spherical lens as a simple example to check my understanding. I'm playing with some parametric plots to visualize the precise effect, but alas, conference prep!

@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!

@nilesjohnson Oh yay! I am glad that idea does not sound too off the rails haha. I will look for papers, after I have a best guess :) Here is some simple doodling in the complex plane haha. Maybe oscillating states of soap film knot/link complements? Mainly because I happen to have a spot light w/ concentric circular 'fibers' haha. So many fun, low-key quirky things to explore! Oh, and Lie algebras! Definitely Lie algebras, although probably just to teach myself more of the basics. Thanks again!

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