Thread by @AlexKontorovich

Awesome @MoMath1 presentation on the discovery of the Hat! A summary 🧵:

This is Dave Smith, a mathematical artist. He spends *a lot* of time just messing around, seeing what shapes he can tile in usual ways.

Nov 20, 2022, he emails @cs_kaplan to say: he can't figure out...

2/ how to get this shape to tile periodically. [By the way, Craig, I'd love to know more of the history predating this email -- how did he stumble onto it?]

4 days later: "now wouldn't that be a thing?" !!!

There are two problems: (i) does it really tile the whole plane? And...

3/ (ii) assuming so, it really *impossible* to find some *other* tiling which is periodic.

Same day, Craig replies that he's put it into his software, and indeed it seems to be ever expandable, no sign of running into trouble. (It could've been that 10 layers fit but not 11!)

4/ They play around and eventually find the meta-structure.

Jan 2023, Chaim Goodman-Strauss and Joseph Samuel Myers join the team. By the end of the month, aperiodicity is proved!

The key idea is that you can recover global structure from local! If you look at all possible ...

5/ neighboring patterns at various depths, you learn that you're *forced* into the meta-structure of their aperiodic tiling. So there's no way to do it periodically!

This proof technique reminds me quite a lot of Doron Zeilberger and (his computer) Shalosh B. Ekad's proof of...

6/ Conway's "lost" Cosmological theorem in the Look-And-Say (audioactive decay) sequence.

Anyway, backing up to Dec 6, Dave had yet another discovery. Early Feb: Joseph shows this one is also an einstein, and in fact there's an infinite continuous family of such! This leads to..

7/ yet another, much more conceptual, idea to prove aperiodicity! If both the "hat" and "chevron" have periodic tilings, then the triangular lattice can be translated to the same but scaled by sqrt2 - impossible!

Anyway, amazing work, and congrats again to all involved!

PS Looks like the video link has been posted online; enjoy!