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.

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