Substitution tiling

Remarks

Example: in $\R^2$, let $T_1 = [0,1]^2$, $\lambda = 2$, and let $\omega(T_1)$ be the four unit squares $T_1 + a$ with $a \in \set{0,1}^2$. Then $\omega^k(T_1)$ is the $2^k \times 2^k$ block of unit squares. The grid of all unit squares $T_1 + a$, $a \in \Z^2$, is a substitution tiling, since each finite patch of it lies in a translate of some block. Non-example: for the same rule, a tiling of $\R^2$ by rows of unit squares in which the rows are shifted by $1/2$ against each other is not a substitution tiling. Its patches across two rows are not translates of patches of the grid. The Penrose tilings, cut into Robinson triangles, are substitution tilings with expansion factor the golden ratio. Their prototiles are the finitely many rotated and reflected copies of the two triangles. More generally, $\lambda$ may be replaced by an expanding linear map of $\R^d$.