Everything about Self-similarity totally explained
In
mathematics, a
self-similar object is exactly or approximately
similar to a part of itself (for example 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. Self-similarity is a typical property of
fractals.
Scale invariance is an exact form of self-similarity where at any magnification there's a smaller piece of the object that's
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.
Definition
A
compact topological space X is self-similar if there exists a
finite set S indexing a set of non-
surjective homeomorphisms
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
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.
Further Information
Get more info on 'Self-similarity'.
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