The Hat Tiling
Though I haven’t blogged about them before, I’m a fan of aperiodic tilings. I even have a few up for sale at Spoonflower. What usually passes for excitement in the tiling world is Penrose suing someone for using his tilings, but this time there’s real news: a strong aperiodic tiling of the plane with one shape. There’s a thread about it at Hacker News (linked to a paywalled article), or if you’re into tilings, there’s always the 90-page preprint.
The long-awaited hat “monotile” feels like a bit of a cheat because its reflection is also used in the tiling. (See the darker blue tiles below.) Other tilings don’t use asymmetric tiles, and even when there’s an asymmetric variant of the tiles (to enforce the matching rules by shape alone), it doesn’t include reflections.
There is a whole family of hats that are continuous perturbations of the central hat: 
© 2023 M. C. DeMarco
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