العربية
中国
Français
Deutsch
Ελληνική
हिन्दी
Italiano
日本語
Português
Русский
EspañolSELF-SIMILARITY
(Redirected from Self-similar)
In mathematics, a 'self-similar' object is exactly or approximately similar to a part of itself, e.g., 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[1]. 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.
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.
Self-similarity also 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[2]. 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.
Mandelbrot set is self-similar around Misiurewicz points.
★ Droste effect
★ Self-reference
★ Zipfs law
1. Benoît Mandelbrot, ''How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension''
2. Leland ''et al.'' "On the self-similar nature of Ethernet traffic", ''IEEE/ACM Transactions on Networking'', Volume '2', Issue 1 (February 1994)
★ "Copperplate Chevrons" - a self-similar fractal zoom movie
A Koch curve has an infinitely repeating self-similarity when it is magnified.
In mathematics, a 'self-similar' object is exactly or approximately similar to a part of itself, e.g., 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[1]. 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.
| Contents |
| Definition |
| Examples |
| See also |
| References |
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.
Examples
An image of a fern which exhibits affine self-similarity.
Self-similarity also 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[2]. 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.
Mandelbrot set is self-similar around Misiurewicz points.
See also
★ Droste effect
★ Self-reference
★ Zipfs law
References
1. Benoît Mandelbrot, ''How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension''
2. Leland ''et al.'' "On the self-similar nature of Ethernet traffic", ''IEEE/ACM Transactions on Networking'', Volume '2', Issue 1 (February 1994)
★ "Copperplate Chevrons" - a self-similar fractal zoom movie
This article provided by Wikipedia. To edit the contents of this article, click here for original source.
psst.. try this: add to faves
