Why are these simple polygons so hard to make sense of?

Support me on Patreon! https://www.patreon.com/PurpleMindCS Tier levels include special roles in my Discord server, seeing your name on screen at the end of videos, and getting preview of upcoming content. I will be making frequent posts on Patreon about my next video, which is about a recent research result from a team at Carnegie Mellon University. The first sneak peek is live right now -- head over there to check it out. If you'd like to aid the success of this channel, this is the best way to do it! Every contribution is sincerely, greatly, appreciated. This is episode "3B" in a series called "Brainstorm." The topics for these videos are puzzle-style problems, broken into two parts each, a couple months apart. -- In the first part, I introduce the problem, and you (the viewers!!!) try to solve the problem yourself, or at least come up with some initial approaches, first steps, or partial solutions. -- In the second part (this video), I follow up with animated discussions of the solutions that were sent to me, mixed in with my own thoughts. The problem is about a beautiful geometric construct called a Voronoi Diagram (https://en.wikipedia.org/wiki/Voronoi_diagram), formed from a set of "site" locations (points) in 2D space and a corresponding set of "cells" -- polygons which are the regions closest to the sites inside of them. The central question of the video is about finding the average perimeter of a random cell in a diagram with n uniformly random sites. As a community, we find that mathematically proving the answer to this question is extremely difficult, but experimental data suggest a very simple formula as a great approximation. Along the way, the journey leads us to rediscover a fundamental geometric quantity, ask questions about how circles overlap with squares, and find an approximation for a quite strange-looking five-dimensional integral. Thanks so much to everyone who submitted something for this problem, and to everyone, in general, for watching the video! I’m planning on continuing this series, so stay tuned :) Link to Meijering paper referenced in the video: https://pearl-hifi.com/06_Lit_Archive/02_PEARL_Arch/Vol_16/Sec_53/Philips_Rsrch_Reports_1946_thru_1977/Philips%20Research%20Reports-08-1953.pdf Brainstorm Episode 1A: https://youtu.be/w6xl0IWrZYc Brainstorm Episode 1B: https://youtu.be/ZMXC00TUlOI Brainstorm Episode 2A: https://youtu.be/p0J8kIoQF6I Brainstorm Episode 2B: https://youtu.be/pkZ-qUUBYzw Brainstorm Episode 3A (for this video): https://youtu.be/57N5h3E3Raw Feel free to join my Discord server: https://discord.gg/smuxnzZ5Zf Business Inquiries: [email protected] These animations are made using Manim, by 3Blue1Brown.