Symmetry of Tilings of the Plane

We discuss two new results on tilings of the plane. In the first, we give sufficient conditions for the tilings associated with an inflation rule to be uniquely ergodic under translations, the conditions holding for the pinwheel inflation rule. In the second result we prove there are matching rules for the pinwheel inflation rule, making the system the first known to have complete rotational symmetry. We consider tilings of the Euclidean plane, E2, by (orientation-preserving) congruent copies of a fixed finite set of prototiles. Prototiles are topological disks in the plane satisfying some mild restrictions on their shapes, as detailed below. Congruent copies of prototiles are called tiles, and a tiling is simply an unordered collection of tiles whose union is the plane and in which each pair of tiles has disjoint interiors. We are concerned here with two constructions associated with a fixed finite set S = {Pj} of prototiles, the most important of which is the set X(S) of all tilings by tiles from S. In particular, we are interested in understanding the purest cases, in which all the tilings in X(S) are "essentially the same"; we will define this precisely further on. Two examples are exhibited in Figures 1 and 2 on the next page, both with two prototiles; in Figure 1, S = Sk produces only a checkerboard-like tiling (and all congruences), and in Figure 2, S = Sp produces the well-known tilings of Penrose [3, 4, 6]. Tilings like those of Penrose are not usually invariant under any congruence of the plane (other than the identity), so to analyze their symmetries we introduce some elementary ergodic theory and another basic construction which can sometimes be associated with a prototile set S, the set XF{S) of tilings defined by an "inflation function" F. An "inflation rule" for S, if it exists, consists of a dilation DF of E2 by some factor kF < 1 and a finite set {C,*} of congruences of E2, such that for each Pj e S we have (1) Pj = {JCjkDFPnk k where the elements of each union have pairwise disjoint interiors. The inflation function F associated with such a rule is defined on tiles (and then sets of tiles), with sets of tiles as values, as follows. If the tile P is "of tile-type j ", that is, P = CPj where Pj e S and C is a congruence, then (2) F:P^F(P) = {D-lCCjkDFPnk}. (Intuitively, P can be replaced by a set of "small-size tiles" by (1), which are Received by the editors February 12, 1993. 1991 Mathematics Subject Classification. Primary 52C20, 58F11, 47A35. Research supported in part by Texas ARP Grant 003658-113. ©1993 American Mathematical Society 0273-0979/93 $1.00+ $.25 per page 213