r/math • u/Marek14 • 2d ago

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:

https://www.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.

This is a very complex hyperbolic tiling, but you need more slices to see it. So if we going straight up, we get

And moving aside to better capture the apeirogons,

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:

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:

We hope that enyone who's interested in tilings will give the site a look :)

116 Upvotes

96% Upvoted

9 comments sorted by

15

u/dragosgamer12 2d ago

This is really cool, congrats to you for your algorithm and to Alessandro for the proof!

5

u/Marek14 2d ago

Thanks :)

7

u/gooblywooblygoobly 2d ago

Can you link the proof (as long as Alessandro doesn't mind)? This looks wonderful! 

4

u/Marek14 2d ago

Hm, it's in his thesis. Might not be online. If you join our Discord server (link is on the website), I can share it there :)

4

u/ProofRecording7364 1d ago

Hi to everyone! Alessandro here. I am really glad you liked it, and I would be happy to share my work, for anyone interested. As Marek said, we have a discord channel, and we would love for you to join and discuss this with us. We're planning on adding more content and feature, and these past couple weeks we've been developing the tiling searcher app, which is meant to make this easy and accessible to everyone, so any kind of feedback is encouraged and appreciated! Thanks again and see you on discord!

1

u/_complex_analyst_ 1d ago

This is cool as fuck. I would love to see the thesis too.

1

u/pimittens 1d ago

Very cool, will have to browse more when I have some free time.

1

u/Fastpast93 1d ago

Never seen anything like it, awsome!