John Conway disovered a technique using infinite, finitely presented groups that in a number of interesting cases resolves the question of whether a region in the plane can be tessellated by given tiles. The idea is that the tiles can be interpreted as describing relators in a group, in such a way that the plane region can be tiled, only if the group element which describes the boundary of the region is the trivial element 1.

@article{ConwaysTilingGroupsonJSTOR,
title = {Conway's Tiling Groups on JSTOR},
abstract = {John Conway disovered a technique using infinite, finitely presented groups that in a number of interesting cases resolves the question of whether a region in the plane can be tessellated by given tiles. The idea is that the tiles can be interpreted as describing relators in a group, in such a way that the plane region can be tiled, only if the group element which describes the boundary of the region is the trivial element 1.},
url = {https://www.jstor.org/stable/2324578},
year = 1990,
author = {William P. Thurston},
comment = {},
urldate = {2020-04-15},
collections = {Fun maths facts,Geometry,The groups group}
}