These are definitely interesting. They make me wonder what the underlying continuous function is, and if it's the same function for all the pictures. These mostly look like aliasing to me. Which makes sense, perhaps is nearly obvious, because that's what you get when you plot the mod of a multiply on a grid. It's pretty easy to reproduce something close to the large image (prime 9973), by just plotting the continuous…
There’s not really a reasonable continuous analog, IMO (unless you just want to see a single solid blob, like in your picture, but that hides all of the interesting part of the structure). But there are moiré-like patterns. Check out https://www.youtube.com/watch?v=qhbuKbxJsk8 for some insight based on a related type of diagram (related to a single row at a time from the OP’s diagrams).
Re: Beauty in Mathematics: Modular Multiplication Tables
#21That was awesome thanks for sharing. For the first minute I wasn't expecting much, but the reflections part was so cool! Plus, this connects directly to the cyclic curves in the complex plane, e.g., https://mathlesstraveled.com/2015/06/04/random-cyclic-curves... The curves formed by the envelope of intersecting lines in the circular multiplication table are a slice of cyclic curve space. I animated this after reading the above blog post: https://youtu.be/myGXoLCnx-A You can see some of the curves in the lower left corner are the same ones from Mathologer.