fractalsselfsimilarity

Fractal self-similarity is a central idea in fractal geometry describing patterns that replicate themselves at different scales. An object is exactly self-similar when it can be decomposed into parts that are scaled copies of the whole. In mathematical constructions, this property is often produced by iterated function systems, a finite set of contraction mappings whose fixed point is a self-similar set. The level of scaling is captured by the fractal or similarity dimension, which tends to be non-integer and reflects how detail scales with magnification.

Exactly self-similar structures include classic examples generated by simple rules. The Cantor set consists of two copies of itself scaled by a factor of 1/3; the Sierpinski triangle, the Koch snowflake, and the Menger sponge are well-known fractals that exhibit precise self-similarity across scales.

Not all self-similarity is exact. Many natural objects display statistical or finite self-similarity, meaning parts resemble the whole only approximately or in a probabilistic sense. Coastlines, clouds, and terrain often show approximate scale invariance over a range of magnifications. Random or stochastic fractals, such as fractional Brownian motion, exhibit self-similarity in a statistical sense, characterized by descriptors like the Hurst exponent that govern roughness and scaling behavior.

Dimensions and measurement techniques, such as box-counting and Hausdorff dimension, quantify self-similarity and complexity. Fractal self-similarity informs diverse applications, including computer graphics, texture synthesis, signal processing, and the modeling of natural phenomena, where scale-invariant patterns provide compact descriptors of complex structures.