Self-similarity


Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|A Koch snowflake has an infinitely repeating self-similarity when it is magnified.
In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. 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.
Peitgen et al. explain the concept as such:
Since mathematically, a fractal may show self-similarity under arbitrary magnification, it is impossible to recreate this physically. Peitgen et al. suggest studying self-similarity using approximations:
This vocabulary was introduced by Benoit Mandelbrot in 1964.

Self-affinity

Image:Self-affine set.png|thumb|right|A self-affine fractal with Hausdorff dimension = 1.8272
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.

Definition

A compact topological space X is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms for which
If, we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for. We call.
a self-similar structure. The homeomorphisms may be iterated, resulting in an iterated function system. The composition of functions creates the algebraic structure of a monoid. When the set S has only two elements, the monoid is known as the dyadic monoid. The dyadic monoid can be visualized as an infinite binary tree; more generally, if the set S has p elements, then the monoid may be represented as a p-adic tree.
The automorphisms of the dyadic monoid is the modular group; the automorphisms can be pictured as hyperbolic rotations of the binary tree.
A more general notion than self-similarity is self-affinity.

Examples

Image:Feigenbaumzoom.gif|left|thumb|201px|Self-similarity in the Mandelbrot set shown by zooming in on the Feigenbaum point at
Image:Fractal fern explained.png|thumb|right|300px|An image of the Barnsley fern which exhibits affine self-similarity
The Cantor discontinuum is self-similar since any of its closed subsets is a continuous image of the discontinuum.
The Mandelbrot set is also self-similar around Misiurewicz points.
Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in teletraffic engineering, packet switched data traffic patterns seem to be statistically self-similar. 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.
Similarly, stock market movements are described as displaying self-affinity, i.e. they appear self-similar when transformed via an appropriate affine transformation for the level of detail being shown. Andrew Lo describes stock market log return self-similarity in econometrics.
Finite subdivision rules are a powerful technique for building self-similar sets, including the Cantor set and the Sierpinski triangle.
Some space filling curves, such as the Peano curve and Moore curve, also feature properties of self-similarity.

In cybernetics

The viable system model of Stafford Beer is an organizational model with an affine self-similar hierarchy, where a given viable system is one element of the System One of a viable system one recursive level higher up, and for whom the elements of its System One are viable systems one recursive level lower down.

In nature

Self-similarity can be found in nature, as well. Plants, such as Romanesco broccoli, exhibit strong self-similarity.

In music

  • Strict canons display various types and amounts of self-similarity, as do sections of fugues.
  • A Shepard tone is self-similar in the frequency or wavelength domains.
  • The Danish composer Per Nørgård made use of a self-similar integer sequence named the infinity series in much of his music.
  • In the research field of music information retrieval, self-similarity commonly refers to the fact that music often consists of parts that are repeated in time. In other words, music is self-similar under temporal translation, rather than under scaling.