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In this video, we cover an exciting new result in mathematics - the Empty Hexagon Problem has finally been solved, bringing a “happy ending” to a 100-year-old mathematical journey! It started with a simple idea from the 1930s, and evolved into a rich history of geometric questions, culminating in a spectacular and lucky solution in 2024 that makes use of the newest developments in SAT solvers, a computerized proof method, filling in the last piece of the puzzle.
Special thanks to Professor Marijn Heule and Carnegie Mellon University for sponsoring this video, to Joe Horton, Bernardo Subercaseaux, and Cayden Codel for their help with the project.
If you or your institution is interested in sponsoring the topic of a future PurpleMind video, please contact me via [email protected]!
Relevant Links:
Heule Paper: https://arxiv.org/pdf/2403.00737
Subercaseaux et al. Paper: https://arxiv.org/abs/2403.17370
Horton Paper: https://tinyurl.com/md48uf3u
Overmars Paper: https://link.springer.com/content/pdf/10.1007/s00454-002-2829-x.pdf
Nicolás Paper: https://link.springer.com/article/10.1007/s00454-007-1343-6#preview
Gerken Paper: https://link.springer.com/article/10.1007/s00454-007-9018-x
Happy Ending Problem: https://en.wikipedia.org/wiki/Happy_ending_problem
Point Realizer Game: https://w-realizer.surge.sh/ (special shoutout to Bernardo Subercaseaux for making this!) You can upload your own constraints for this game as a text file on the website. See this example for reference (it’s the unrealizable set of constraints I used as an example in the video): https://pastebin.com/BvWqcd1d
Another game from Bernardo where you can try to place points without making convex/empty polygons: https://k-hole-tester.surge.sh/
Math animations are made using Manim, by 3Blue1Brown.
Character animation done by Divyesh Patel.
Discord Server: https://discord.gg/smuxnzZ5Zf. Feel free to join!
Business Inquiries: [email protected]