Tiling Atlas

Hello, I'd like to showcase some new results in tilings. This summer, I was contacted by a student named Alessandro Longo who tried to enumerate k-Euclidean tilings for his thesis. He eventually found online the algorithm I made for this problem several years ago. He used his thesis to rigorously prove what I had only believed, i.e. that the algorithm is truly capable of enumerate all tilings that satisfy set conditions. With that in hand, he built this website: thetilingatlas.com I am really humbled that my amateur algorithm could get this far! We have tilings of all kinds, but mostly hyperbolic ones (they are just easier to find), and now we even have a stand-alone search program. Not very user-friendly yet, I'm afraid. I wanted to share some results. They are from recent searches, so not uploaded yet. preview.redd.it/zgzx0dju5yzh1.png This is a very complex hyperbolic tiling, but you need more slices to see it. So if we going straight up, we get preview.redd.it/3p0vpnh06yzh1.png preview.redd.it/g83g2h526yzh1.png And moving aside to better capture the apeirogons, preview.redd.it/m9nixkq56yzh1.png preview.redd.it/l02xu8ka6yzh1.png It contains 8 kinds of regular polygons with 4 different edge lengths. For Euclidean tilings, I have tried to look at 14-Archimedean ones. The first one, Amazing Amalgam, was found using my algorithm in 2022, I think: preview.redd.it/z5xf3jvp6yzh1.png This one is 24-uniform, but the search program is now fast enough to go to k=40 in reasonable amount of time, so here's a 40-uniform result: preview.redd.it/8ob4p73v6yzh1.png We hope that enyone who's interested in tilings will give the site a look :)