I feel like SAT solvers and the like are getting a lot more attention on HN recently (for example https://news.ycombinator.com/item?id=44259476) - justifiably so! I think that they're a great tool that's often criminally underused in industry for a whole subset of problems.

Terrific article by Nicolas Hatcher! Aperiodic tilings are fun to make from paper, wood, and ceramics. I've cut tiles from ceramic field tiles.

No surprise that concave cuts in ceramics are a high stress, so Kite and Dart tiles don't work very well (the dart is likely to crack). Same is true for the Turtle, Hat, and Spectre.

Rhombus tiles are everywhere convex, and the P3 Rhombus tiles are easy to cut in a diamond saw (or even a snap-cutter). With a diamond band-saw, it's possible to make Penrose rhombs with curved (parabolic) edges.

But cutting tiles from stock field tiles produces sharp surface edges -- you don't want these as bathroom floor tiles. Also, you waste a lot of the field tile as scrap. To get "friendly" tile shoulders, I'm experimenting with making Penrose tiles directly from high-fired porcelain clay.

Interesting article! I'm stuck on the following claim about tiling Hats though:

> In each center of an hexagon you can have any of the 12 possibilities:

> Any of the 6 rotations of the Hat

> Any of the 6 rotations of the anti-Hat

For this claim to hold, it must be the case that a Hat (or anti-Hat) occupies the same area as a hexagon. But they don't: a hexagon is made of 6 kites, while a Hat is made of 8. So, some hexagons must contain no corresponding (anti-)Hat -- specifically, for every 8 hexagons, there must be 6 (anti-)Hats.

This seems to complicate the SAT formulation. But could the fix be as simple as adding a 13th possibility, "No hat at this hexagon centre occupies more than half of its kites"? Or are additional constraints needed?

[dead]