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.