General Introduction Of the many examples of aperiodic tilings or aperiodic point sets in Euclidean space found in recent years, two classes stand out as particularly interesting and aesthetically pleasing. These are the sub- stitution tilings, tilings which are self-similar in a rather strong sense described in [GS] [Rl] [SI] [AP], and the tilings and patterns ob- tained by the method of cut and projection from higher dimensional periodic sets described in [dBl] [KrNe] [KD]. In this memoir we consider the second class. However, some of the best studied and most physically useful examples of aperiodic tilings, for example the Penrose tiling [Pe] and the octagonal tiling (see [Soc]), can be ap- proached as examples of either class. Therefore we study specially those tilings which are in the overlap of these two classes, and exam- ine some of their necessary properties. Tilings and patterns in Euclidean space can be compared by vari- ous degrees of equivalence, drawn from considerations of geometry and topology [GS]. Two tilings can be related by simple geometric tranformations (shears or rotations), topological distortions (bending edges), or by more radical adaptation (cutting tiles in half, joining adjacent pairs etc). Moreover, point patterns can be obtained from tilings in locally defined ways (say, by selecting the centroids or the vertices of the tiles) and vice versa (say, by the well-known Voronoi construction). Which definition of equivalence is chosen is determined by the problem in hand. In this paper, we adopt definitions of equivalence (pointed con- jugacy and topological conjugacy, 1.4.5) which allow us to look, without loss of generality, at sets of uniformly isolated points (point patterns) in Euclidean space. In fact these patterns will typically have the Meyer property [Lai] (see 1.4.5). Therefore in this introduction, and often throughout the text, we formulate our ideas and results in terms of point patterns and keep classical tilings in mind as an implicit example. The current rapid growth of interest in projection method pat- terns started with the discovery of material quasicrystals in 1984 [SBGC], although these patterns had been studied before then. Quasicrystaline material surprised the physical world by showing sharp Bragg peaks under X-ray scattering, a phenomenon usually associated only with periodic crystals. Projection method patterns Received by the editor October 24, 2000. 1

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