1.4. CONTRUCTING INTERESTING TILINGS 15

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

Rn,

restricting it to the neighbor-

hood of a k-plane in

Rn

(that’s the “cut”), and then projecting the periodic

structure onto that k-plane. For generic embeddings of the k-plane in

Rn,

the result will be an aperiodic tiling of the k-plane. By considering all

translates (in

Rn)

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

R2.

Pick an irrational number α 0 and a point