Good writeup of 'Combinatorial coordinates for the aperiodic Spectre tiling' from Simon Tatham here : https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi...
Next frontier: aperiodic tilings with irrational angles (meant, tiles having angles of x*2pi were x is irrational). Or are these proven to be impossible?
Because both the hats and spectres are basically subset of triangular grid. Penrose tilings are subset of regular grid, too. Can we get rid of these underlaying regular grids.
Interestingly this was found by a “hobbyist tiler”, David Smith, who is the first author. He was interviewed on how he found it in this YouTube video: https://youtu.be/4HHUGnHcDQw?si=VsHLqVUdw6ihERg2
Something that is unclear to me: are hat reflections allowed? I think they are, but it would be good to have confirmation. In short, if you allow reflections, are the tilings still guaranteed to be aperiodic?
Does anyone of if there are any consequences of the existence of monotiles in algebra or number theory?
(2023)
After publication of Spectres, I don't know if there much interest anymore on Hats. Spectres are like Hats, but eliminate the need of reflections for tiling.
https://cs.uwaterloo.ca/~csk/spectre/