## Tile

For our purposes, a tile in $\mathbb{R}^d$ is defined as a nonempty compact subset of $\mathbb{R}^d$ which is the closure of its interior. Sometimes a tile is required to be connected, or to be homeomorphic to a closed ball. For a substitution one needs a finite set of tiles which serve as building blocks for the substitution tilings. These are called prototiles. Sometimes the term is used in a slightly different sense. If it is stated that ‘$T$ is a tile’, then this can mean that there is a tiling $T$ such that $T$ is the single prototile of $T$. For instance, in this sense each square is a tile, but a regular pentagon is not a tile.