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.