selfsimilarity

Self-similarity is a property in which a structure or pattern looks similar at different scales. In mathematical contexts it is often formalized as invariance under a family of scale transformations. There are several related notions, including exact self-similarity, statistical self-similarity, and self-affinity.

Geometric exact self-similarity arises when a set S can be decomposed into scaled copies of itself. If S equals the union of fi(S) for a finite collection of similitudes fi(x) = ri R_i x + ti with contraction ratios 0 < ri < 1, then S is exactly self-similar. The similarity dimension s is the unique solution to the equation sum_i ri^s = 1. Under the open set condition, the Hausdorff and box-counting dimensions of S coincide and equal s. Classic examples include the Cantor set and the Sierpinski triangle.

In probability and stochastic processes, self-similarity can be statistical or distributional. A process X(t) is self-similar with exponent H if, for all a > 0, {X(at)} has the same finite-dimensional distributions as {a^H X(t)}. Fractional Brownian motion is a prominent example with H in (0,1). Such processes often exhibit long-range dependence and scale-invariant behavior.

Real-world phenomena frequently display approximate self-similarity over several orders of magnitude, such as coastlines, river networks, cloud fields, and rough surfaces. Distinctions are made between exact self-similarity (strict invariance under scaling) and self-similarity in a statistical sense (invariance in distribution).

Applications span fractal geometry, computer graphics, material science, turbulence, and financial modeling, where multiscale structure is a central feature. See also fractal geometry, Hurst exponent, and iterated function systems.