Matching Rules and Substitution Tilings

Chaim Goodman-Strauss
1998 Annals of Mathematics  
A substitution tiling is a certain globally de ned hierarchical structure in a geometric space; we show that for any substitution tiling in 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, in nite collections of forced aperiodic tilings are constructed. The theorem covers all known examples of hierarchical aperiodic tilings. Figure 1: A substitution tiling On the left in gure 1, L-shaped
more » ... in gure 1, L-shaped tiles are repeatedly in ated and subdivided". We de ne our terms more precisely in Section 1 As this process is iterated, larger and larger regions of the plane are tiled with L-tiles hierarchically arranged into larger and larger images of in ated and subdivided L-tiles, as at right in gure 1 the thicker lines have been added to emphasize the hierarchy. We can then de ne a global structure the substitution tiling" induced by the in ation and division of the tiles. But L-tiles can tile the plane in myriad ways. Is there a set of local conditions| matching rules" | that, if satis ed everywhere, force the hierarchical structure of the substitution tiling to emerge? One can show that no such rules exist for unmarked L-tiles. However, we can nd a set of marked L-tiles, and matching 1 This paper is dedicated to Raphael Robinson and Hao Wang.
doi:10.2307/120988 fatcat:eu7by3no6zdt5kdjk3orvnfvza