18 1. BASIC NOTIONS
line removed and then glued in twice — once as the limit of one side and
once as the limit of the other.
The space Ωα can be constructed for any irrational α. It is true (but
not obvious!) that when α is a quadratic irrational number, then Ωα can
also be constructed a substitution. In particular, the space of tilings
with α = τ = (1 +
5)/2 is none other than the Fibonacci tiling space.
There are many ways to generalize this construction. The strip does not
have to come from a unit square, but can take the form L × E, where the
“window” E is any (reasonable) set in the orthogonal complement to L. The
source space does not have to be
but instead can be any locally compact
Abelian group. For instance, the Penrose tiling is most naturally described
as a projection from
× Z5 to
For a more detailed description of
cut-and-project tilings, their generalizations, and the topology of the tiling
spaces they generate, see [Moo].
Topologically, a space of cut-and-project tilings is always a torus in the
higher-dimensional space, minus a lower-dimensional piece where an integer
point lies on the boundary of the strip S, plus limits as those singular points
are approached from various directions. Forest, Hunton and Kellendonk
have developed powerful methods for computing topological invariants of
such a space in terms of the geometry of the window E, and have written an
excellent book on the subject [FHK]. These techniques have a very different
flavor from the other content of this book, and will not be discussed further.
Local matching rules. The constructions of substitution tilings and
cut-and-project tilings are highly non-local. For substitution tilings, we
apply a condition to patches of arbitrarily large size. For cut-and-project
tilings, we invoke a higher dimension and project globally. At first glance,
neither would seem to model actual materials, where atoms and molecules
are held together by local forces.
A different approach is to take a set S of tiles together with rules about
how two tiles can fit together. Usually these rules can be implemented
geometrically, by adding bumps and notches to the tiles, as in Figure 1.16.
Let ΩS be the set of all tilings of the plane (or
by translates of the tiles
in S. If ΩS is non-empty and if every tiling in ΩS is aperiodic, we say that
S is an aperiodic set.
Tiling theory largely arose from a decidability question: given a set of
tiles, is it possible to determine algorithmically whether or not they tile the
plane? It was generally believed that any set that tiled the plane could tile
the plane periodically, and it was possible to determine whether a set of tiles
could produce a periodic tiling. However, it turned out that some sets of
tiles do tile the plane, but only non-periodically! The first aperiodic set, by
Berger [Ber], had thousands of tiles. Simpler sets were later constructed by
Robinson [Rob], by Kari and Culik [Kar, Cul], and by others. The most
famous aperiodic set contains the Penrose kites and darts of Figure 1.16:
two shapes in ten orientations, or twenty tiles in all.