Tiling

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 $T=\{ T_{j(i)} + x_i \mid i=1,2,3,\ldots \}$.