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
Previous Page Next Page