# Self-Affine Tiles

There exist tiles that are self-affine, but not self-similar.

If `{ T`_{i} } is an IFS
generating a self-similar tile, and `A` is an
affine transformation, then the attractor of `{ A`^{-1}·T_{i}.A } is the image
of `{ T`_{i} } under `A`, and is also a figure with a similarity dimension
of 2. In some cases (e.g. the rep-4 square) the attractor remains self-similar,
but usually it is self-affine, but not self-similar. If the unit cell for the
attractor of `{ T`_{i} }
consists of a single copy of the attractor, or all the copies have the same
orientation (± rotation by 180°), or `A` is also
a similarity transform then the attractor is also a tile. Otherwise the
attractors is one of a set of co-tiles, as transformation of the plane by `A` changes the shape of differently orientated copies
differently.

There are also sets of interconvertible self-affine tiles which do not
contain any self-similar members. Simple examples can be found among the
tilings of a square by rectangles, and among the tilings of triangles by
triangles.

© 2016 Stewart R. Hinsley