I wish I could buy ceramic Penrose P3 tiles to put on my floor or wall. They 2 different tiles instead of 1, but they're simple diamond shapes, and they tile aperiodically.
The discovery of the aperiodic monotile was what finally pushed me over the line to sign up for a ceramics beginners course funnily enough! Give me about a year...
This feels like a niche market someone needs to go after. I would love to have a bathroom floor or a kitchen backsplash tiled in 'hat' in 4 colours (ensuring no two same-colour tiles touched of course).
Could the chromatic number be less than 4 (for some or all hat/turtle/spectre tilings)? Note that for the lattice tilings by squares and equilateral triangles the chromatic number is 2, while for the tiling by regular hexagons it is 3.
ZeroRogue has drawn dual graphs of hat tilings https://twitter.com/ZenoRogue/status/1639644061823819777 so you can just try to color them.
Which raises the question: how many colors of the inverted hat tile would you need? I don’t believe the inverted hats ever need to touch one another…
I suspect, given the 1:1 correspondence to a hex grid (where some of the hexagons map to an inverted/non inverted pair of hats) that they describe in the original paper that it would be possible to tile with just three colors of noninverted hat, and one color of inverted hat.