Difference between revisions of "Self-similarity"
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| − | In [[mathematics]], a '''self-similar''' [[ | + | In [[mathematics]], a '''self-similar''' [[Mathematical object|object]] is exactly or approximately [[Similarity (geometry)|similar]] to a part of itself (i.e. the whole has the same shape as one or more of the parts). |
== Description == | == Description == | ||
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Self-similarity is a typical property of [[Fractal|fractals]]. | Self-similarity is a typical property of [[Fractal|fractals]]. | ||
| − | == Scale invariance | + | == Scale invariance == |
[[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. | [[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. | ||
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* [[Chaos theory]] | * [[Chaos theory]] | ||
* [[Fractal]] | * [[Fractal]] | ||
| + | * [[Mathematical object]] | ||
* [[Mathematics]] | * [[Mathematics]] | ||
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* [[Scale invariance]] | * [[Scale invariance]] | ||
| + | * [[Self-reference]] | ||
* [[Similarity (geometry)]] | * [[Similarity (geometry)]] | ||
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* [https://en.wikipedia.org/wiki/Self-similarity Self-similarity] @ Wikipedia | * [https://en.wikipedia.org/wiki/Self-similarity Self-similarity] @ Wikipedia | ||
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| + | [[Category:Language]] | ||
| + | [[Category:Logic]] | ||
| + | [[Category:Mathematics]] | ||
| + | [[Category:Patterns]] | ||
| + | [[Category:Recursion]] | ||
| + | [[Category:Structures]] | ||
Latest revision as of 18:00, 27 April 2016
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).
Contents
Description
Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales.
Fractals
Self-similarity is a typical property of fractals.
Scale invariance
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.
Characteristics
The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales.
Counterexample
As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.
See also
- Chaos theory
- Fractal
- Mathematical object
- Mathematics
- Scale invariance
- Self-reference
- Similarity (geometry)
External links
- Self-similarity @ Wikipedia
