> Distorting the shapes of the tile edges a tiny bit is enough to prevent the unwanted extra tilings, and produce a tile which rules out all but single-handedness non-periodic tilings of the plane by its shape alone.
> The distortion can be anything you like: the precise shapes don’t matter, as long as they match.
My favourite way to do this is to make each of the original edges the hypotenuse of a 1-by-2 right-angled triangle. This results in a 27-sided shape with only two different edge lengths and lots of rectangular 'tabs' and 'slots' to help fit the tiles together. Image: https://mathstodon.xyz/@OscarCunningham/110488184680656303
> The distortion can be anything you like: the precise shapes don’t matter, as long as they match.
My favourite way to do this is to make each of the original edges the hypotenuse of a 1-by-2 right-angled triangle. This results in a 27-sided shape with only two different edge lengths and lots of rectangular 'tabs' and 'slots' to help fit the tiles together. Image: https://mathstodon.xyz/@OscarCunningham/110488184680656303