@MISC{Goodman-strauss00openquestions, author = {Chaim Goodman-strauss}, title = {Open Questions in Tiling}, year = {2000} }

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Abstract

In the following survey, weconsider connections between several open questions regarding tilings in general settings. Along the way, we support a careful revision of the de nition of aperiodicity, and pose several new conjectures. We also give several new examples. Over the years a numberofinteresting questions have been asked about the combinatorial complexity of tilings in the plane or other spaces. Here we consider several of these questions, and their connections, in a very general setting. Our most general question is simply: \How complex can the behaviour of a given protoset be?" Figure 1: A monohedral tiling A protoset may behave inmanyways: First, a protoset might admit no tilings whatsoever. In this case, as long as the setting isn't too pathological, there is some upper bound on the size of con gurations that this protoset can form (for if the protoset admits arbitratrily large con gurations, one can produce a tiling of the entire space; cf. Theorem 3.8.1 in [16]). One measure of this bound is the Heesch number of the protoset, de ned below. Or if a protoset does admit tilings, it might admit \strongly periodic " tilings | that is tilings