In response to this CompSciTheory (cstheory) question, A simple problem whose decidability is not known, I posted that:
It is unknown whether or not it is decidable to determine if a given shape can tile the plane,
referring to an earlier cstheory question. This is even open for polyomino tiles.
[![Tiling][1]][1]
(Image from [Wikipedia](https://en.wikipedia.org/wiki/Polyomino#Tiling_the_plane_with_copies_of_a_single_polyomino).)