Figure 1.13. Starting from a corner seed, we only tile a
quarter plane.
There are several natural extensions of the notions of self-similarity and
of substitution tilings. If T is a tiling and T is locally derivable from a
stretched version of T , then T is called “pseudo-self-similar”. This is some-
what weaker than self-similarity, since the tiles of T may not group exactly
into the tiles of λT . However, Priebe and Solomyak [PS] have proven that
every pseudo-self-similar tiling of the plane is MLD to a self-similar tiling, al-
beit one whose tiles may have fractal boundaries. In dimensions greater than
one, we sometimes consider expansive linear maps instead of just rescaling.
If L is an expansive linear transformation and L(T ) can be subdivided into
T , then we say L is “self-affine”, and if T is locally derivable from L(T ) we
say that T is “pseudo-self-affine”. In the literature, the term “substitution
tiling” is sometimes used for each of these generalizations, and “substitution
tiling space” is sometimes used for the hulls of such tilings. In the bulk of
this book, however, substitutions will be limited to rescalings by a factor
λ 1 followed by subdivision into geometric pieces.
Several famous planar substitutions are shown in Figure 1.14.
Cut-and-project tilings. A k-dimensional cut-and-project tiling is
obtained by taking a periodic structure in
restricting it to the neighbor-
hood of a k-plane in
(that’s the “cut”), and then projecting the periodic
structure onto that k-plane. For generic embeddings of the k-plane in
the result will be an aperiodic tiling of the k-plane. By considering all
translates (in
of the periodic structure, we can construct a space of cut-
and-project tilings. The topological and dynamical properties of this space
are closely related to those of the n-torus and to the method of cutting.
We illustrate the method with a family of one-dimensional tilings ob-
tained by projecting from
Pick an irrational number α 0 and a point
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