Tiling With Polyominoes And Combinatorial Group Theory
01 January 1990
When can a given finite region consisting of cells in a regular lattice (triangular, square, or hexagonal) in R sup 2 be perfectly tiled by tiles drawn from a finite set of tile shapes? This paper gives necessary conditions for the existence of such tilings using boundary invariants, which are combinatorial group-theoretic invariants associated to the boundaries of the tile shapes and the regions to be tiled. Boundary invariants are used to solve problems concerning the tiling of triangular-shaped regions of hexagons in the hexagonal lattice with contain tiles consisting of three hexagons.