Matching rules and substitution tilings


A substitution tiling is a certain globally defined hierarchical structure in a geometric space; we show that for any substitution tiling in $\mathbb{E}^d$, $d>1$, subject to relatively mild conditions, one can construct local rules that force the desired global structure to emerge. As an immediate corollary, infinite collections of forced aperiodic tilings are constructed. The theorem covers all known examples of hierarchical aperiodic tilings.


Chaim Goodman-Strauss