Isomorphism of Hierarchical Structures

We consider hierarchical structures such as Fibonacci sequences and Penrose tilings, and examine the consequences of different choices for the definition of isomorphism. In particular we discuss the role such a choice plays with regard to matching rules for such structures. May 1998 * Research supported in part by NSF Grant No. DMS-9531584 and Texas ARP Grant 003658-152 ** Research supported in part by NSF Grant No. DMS-9626698 and Texas ARP Grant 003658-152 0. Introduction This paper concerns certain hierarchical tilings of Euclidean space, of which a well known example is a "kite & dart" tiling of the plane [Gar]. Such tilings are noted for two properties: their unusual rotational symmetries ("ten-fold symmetry "), and the fact that their global structures derive from local ("matching") rules. We will examine the consequences of different choices for the definition of isomorphism between such structures. To be more specific, we assume that any tiling of interest is embedded in a c..