Self-similarity

A Koch curve has an infinitely repeating self-similarity when it is magnified.

Standard (trivial) self-similarity.^[1]

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 $\{ f_s : s\in S \}$ for which

$X=\bigcup_{s\in S} f_s(X)$

If $X\subset Y$, we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for $\{ f_s : s\in S \}$. We call

$\mathfrak{L}=(X,S,\{ f_s : s\in S \} )$

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

Self-similarity in the Mandelbrot set shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)

An image of a fern which exhibits affine self-similarity

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.

A triangle subdivided repeatedly using barycentric subdivision. The complement of the large circles is becoming a Sierpinski carpet

In nature

Close-up of a Romanesco broccoli.

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

1. Mandelbrot, Benoit B. (1982). The Fractal Geometry of Nature, p.44. ISBN 978-0716711865.
2. 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
3. Leland et al. "On the self-similar nature of Ethernet traffic", IEEE/ACM Transactions on Networking, Volume 2, Issue 1 (February 1994)
4. Benoit Mandelbrot (February 1999). "How Fractals Can Explain What's Wrong with Wall Street". Scientific American.
5. 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