Substitution Tiling - Mathematical Definition

Mathematical Definition

We will consider regions in that are well-behaved, in the sense that a region is a nonempty compact subset that is the closure of its interior.

We take a set of regions as prototiles. A placement of a prototile is a pair where is an isometry of . The image is called the placement's region. A tiling T is a set of prototile placements whose regions have pairwise disjoint interiors. We say that the tiling T is a tiling of W where W is the union of the regions of the placements in T.

A tile substitution is often loosely defined in the literature. A precise definition is as follows.

A tile substitution with respect to the prototiles P is a pair, where is a linear map, all of whose eigenvalues are larger than one in modulus, together with a substitution rule that maps each to a tiling of . The tile substitution induces a map from any tiling T of a region W to a tiling of, defined by

Note, that the prototiles can be deduced from the tile substitution. Therefore it is not necessary to include them in the tile substitution .

Every tiling of, where any finite part of it is congruent to a subset of some is called a substitution tiling (for the tile substitution ).