Self-Similarity

An object is said to be self-similar if it looks "roughly" the same on any scale. Fractals are a particularly interesting class of self-similar objects. Self-similar objects with parameters  and  are described by a power law such as

where

is the "dimension" of the scaling law, known as the Hausdorff dimension.

REFERENCES:

Harris, J. W. and Stocker, H. "Scaling Invariance and Self-Similarity" and "Construction of Self-Similar Objects." §4.11.1-4.11.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 113, 1998.

Hutchinson, J. "Fractals and Self-Similarity." Indiana Univ. J. Math. 30, 713-747, 1981.