Aperiodic Monotiles

Until recently, it was unknown whether there was a geometric shape that would tile a plane infinitely without repeating patterns. I certainly would have assumed that this was impossible, yet there are examples of pairs of shapes which can achieve this (though having rotational symmetry), like the Penrose tiling.

More generally, this was referred to as the Einstein problem.

In November 2022, David Smith discovered a family of shapes that seemed to solve the Einstein problem. Craig S. Kaplan, Joseph Samuel Myers, and Chaim_Goodman-Strauss collaborated on a rigorous proof.

I made 3D-printed versions of the hat and spectre tiles to play around with assembling patches myself. While it is not possible to assemble a repeating set of tiles, there is no guarantee that an arbitrary patch will be infinitely extendable. This makes it an interesting puzzle, since it seems to become harder to add tiles with increasing size.

I used the polygonal version of the spectre because I prefer the look of the straight edges compared to the curved edges of the more strict version. The modified edges prevent the shape from tiling periodically when some are flipped.