Self-similarity
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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
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![](/w/images/thumb/4/4b/Fractal_fern_explained.png/200px-Fractal_fern_explained.png)
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
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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.
- ↑ Lua error in package.lua at line 80: module 'strict' not found. PDF
- ↑ Leland et al. "On the self-similar nature of Ethernet traffic", IEEE/ACM Transactions on Networking, Volume 2, Issue 1 (February 1994)
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 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