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.