# Self-similarity

In mathematics, a **self-similar** object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales.^{[2]} 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. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.

## 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

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.^{[3]} 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.^{[4]} Andrew Lo describes stock market log return self-similarity in econometrics.^{[5]}

Finite subdivision rules are a powerful technique for building self-similar sets, including the Cantor set and the Sierpinski triangle.

### In nature

Self-similarity can be found in nature, as well. To the right is a mathematically generated, perfectly self-similar image of a fern, which bears a marked resemblance to natural ferns. Other plants, such as Romanesco broccoli, exhibit strong self-similarity.

### In music

- A Shepard tone is self-similar in the frequency or wavelength domains.
- The Danish composer Per Nørgård has made use of a self-similar integer sequence named the 'infinity series' in much of his music.

## See also

- Droste effect
- Long-range dependency
- Non-well-founded set theory
- Recursion
- Self-affinity
- Self-dissimilarity
- Self-reference
- Self-replication
- Tweedie distributions
- Zipf's law

## References

- ↑ Mandelbrot, Benoit B. (1982).
*The Fractal Geometry of Nature*, p.44. ISBN 978-0716711865. - ↑ Mandelbrot, Benoit B. (5 May 1967). "How long is the coast of Britain? Statistical self-similarity and fractional dimension". New Series.
**156**(3775).*Science*: 636–638. doi:10.1126/science.156.3775.636. Retrieved 11 January 2016. PDF - ↑ Leland
*et al.*"On the self-similar nature of Ethernet traffic",*IEEE/ACM Transactions on Networking*, Volume**2**, Issue 1 (February 1994) - ↑ Benoit Mandelbrot (February 1999). "How Fractals Can Explain What's Wrong with Wall Street". Scientific American.
- ↑ Campbell, Lo and MacKinlay (1991) "Econometrics of Financial Markets ", Princeton University Press! ISBN 978-0691043012

## External links

- "Copperplate Chevrons" — a self-similar fractal zoom movie
- "Self-Similarity" — New articles about Self-Similarity. Waltz Algorithm