A *tiling* `$T$`

in `$R^d$`

is a countable set of tiles, which is a covering as well as a packing of `$R^d$`

. I.e., the union of all tiles in `$T$`

is `$R^d$`

, and the intersection of the interior of two different tiles in `$T$`

is empty.
If `$T$`

contains only finitely many congruence classes of tiles (resp. translation classes), then a family `${T_1, T_2, \ldots ,T_m}$`

of representants of each class is called the family of prototiles. With the help of these prototiles, `$T$`

can be written as `$T=\{ f_i (T_{j(i)}) \mid f_i \textrm{ isometries }, i=1,2,3,\ldots \}$`

. In the case of finitely many prototiles up to translation one has an even simpler representation, namely `$T=\{ T_{j(i)} + x_i \mid i=1,2,3,\ldots \}$`

.