Self-similarity

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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

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

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.

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]

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

Andrew Lo describes Stock Market log return self-similarity in Econometrics.[5]

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.