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 ).
Read more about this topic: Substitution Tiling
Famous quotes containing the words mathematical and/or definition:
“As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.”
—Blaise Pascal (1623–1662)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)