Self-similarity

Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole.

For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape.

The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales.

A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity

[3][4][5] Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide.

[6]Since mathematically, a fractal may show self-similarity under indefinite magnification, it is impossible to recreate this physically.

Peitgen et al. suggest studying self-similarity using approximations:In order to give an operational meaning to the property of self-similarity, we are necessarily restricted to dealing with finite approximations of the limit figure.

This is done using the method which we will call box self-similarity where measurements are made on finite stages of the figure using grids of various sizes.

[8] In mathematics, self-affinity is a feature of a fractal whose pieces are scaled by different amounts in the x- and y-directions.

This means that to appreciate the self similarity of these fractal objects, they have to be rescaled using an anisotropic affine transformation.

For example, in teletraffic engineering, packet switched data traffic patterns seem to be statistically self-similar.

[9] This property means that simple models using a Poisson distribution are inaccurate, and networks designed without taking self-similarity into account are likely to function in unexpected ways.

[10] Andrew Lo describes stock market log return self-similarity in econometrics.