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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