selfaffine

Self-affine is a term used in fractal geometry to describe sets or objects that are invariant under a family of affine contractions. An affine map has the form f(x) = A x + b, where A is a linear transformation and b a translation. A set F in the plane or higher dimensions is called self-affine if it is the attractor of an iterated function system consisting of contractive affine maps {f_i}. In other words, F = ∪ f_i(F). Self-affine means the scaling is anisotropic: different directions may be scaled by different factors, in contrast to self-similar sets where all directions scale uniformly.

A classic example is the Bedford–McMullen carpet, defined by subdividing a square into a grid of rectangles and applying a collection of affine maps that shrink differently along each axis. More generally, many self-affine fractals are generated by iterated function systems in which the linear parts A_i are diagonal or otherwise nonuniform. The family of Gatzouras–Lalley and Lalley–Gatzouras carpets provides a broad range of self-affine examples with rich dimensional properties.

Dimension theory for self-affine sets is more subtle than for self-similar sets. The Hausdorff and box-counting dimensions can depend on the full set of contraction ratios and orientations, and there is no universal simple formula like the similarity dimension. Results such as Falconer’s work on random self-affine sets give conditions under which the dimensions can be determined or estimated, though exact values are known only in special cases. Self-affine geometry thus occupies a central place in the study of anisotropic fractals and their applications in mathematics and modeling.