digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

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]

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.
Further information: patterns in nature

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.

References

1. ^ Mandelbrot, Benoit B. (1982). The Fractal Geometry of Nature, p.44. ISBN 978-0716711865.
2. ^ Benoit Mandelbrot (May 1967). "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". Science Magazine.
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
 48703 videos foundNext >
 M3-32/Psychedelic Trance {self-similarity} - NEW SUPER AURA 2013I'm not dead, I just pop in every now and then and check how these guys are doing. Aesthetics continues to be my game. --------------------------------------... Self Similarity and FractalsThis is an explanation of self similarity and fractals. More Info on Self Similarity and Fractals: http://www.youtube.com/watch?v=kJ4EsAhBh1g http://www.phys... Self-Similarity in Fractal GeometryCreated on July 1, 2010 using FlipShare. Self similaritySierpinski gasket and Koch curve. Tutorial: Self-Similarity and ThinkingThinkBlocks ( http://www.thinkandthrive.com ) teach advanced thinking skills to children and adults. Northern Structures - Self Similarityhttp://www.discogs.com/Northern-Structures-Self-Similarity-EP/release/2988333. Self-Similarity Exhibition Art VideoCarmen Hathaway/Abenaki Artworks new media original This art video was screened in new media artist Carmen Hathaway's solo exhibition Self-Similarity at the ... [cubegrams] NEW SUPER AURA #05 - {self-similarity}Track #05 title: {self-similarity} album: NEW SUPER AURA circle: cubegrams composition: Camellia [DOWNFORCE] source: original album info: http://www.cubegram... Cycles of Fifths in Lines of Self Similarity (minimalist music for percussion ensemble)"Cycles of Fifths in Lines of Self-Similarity (Horizontal Color Forms 9)" by Steve Kornicki The version for percussion ensemble is performed here by the Univ... Steve Kornicki: Cycles of Fifths in Lines of Self Similarity (for guitar quintet with score)The piece uses a process of music composition wherein five twelve-tone rows constructed from perfect fifths are extended across the five parts for the 6' 30'...
 48703 videos foundNext >
 7 news items
 Mesoscopic States of Possibility Granular Scales of Mesosaurs in Nannofossils EIN News (press release) Sun, 31 Aug 2014 03:06:30 -0700 ... approach[A Reyes-Coronado,et.al:"Self-organzatin approach for THz polaritonic metamaterials",manuscript, Optical Society of America,2012] whereas :".mai properties of fractal-based system are self-similarity, self-organization, goal-orientation ... A simple model clarifies the complicated relationships of complex networks Nature.com Wed, 27 Aug 2014 02:09:26 -0700 According to the definition of self-similarity (i.e., when an entire object is exactly or approximately similar to a part of itself), scale-free networks can be regarded as self-similar with respect to the probability of the degree or can exhibit a ... Simple Fractal Branching Explains the Ediacaran Rangeomorphs Discovery Institute Thu, 14 Aug 2014 05:56:15 -0700 ... can be reconstructed by applying approximately self-similar branch production rules (as described above) at increasingly fine scales, although these may be modified by different parameter values to give subtle variations in the level of self ... AppAdvice How to spend a \$25 iTunes gift card for Aug. 22, 2014 AppAdvice Fri, 22 Aug 2014 08:45:00 -0700 Frax is powered by fractal geometry, the same principle that creates patterns of self-similarity in our natural world. From spiral galaxies to hurricanes, from tree limbs to branching blood vessels, the universe is filled with endless variation and ... Independent Online Exploring an African palace Independent Online Sun, 10 Aug 2014 23:11:15 -0700 Another way of describing this is the use of so-called “self-similarity”, in which the individual parts follow the same form as the whole. At Ammazulu, it may seem that indigenous people simply built the structure using local materials, but the ... Fate Of A Misguided Scholar 3 Vibe Ghana Fri, 01 Aug 2014 13:48:45 -0700 ... who, though not through any faults of his, fails to recognize the facial fingerprints of “truth” in The African Sun when he is confronted with them, for, once again, his miscomprehension of the concept of logarithmic spiral, of self-similarity, an ... Presentan académicos de CETYS proyectos de investigación Uniradio Informa Wed, 06 Aug 2014 17:11:15 -0700 ... Allocation in Computational Grids with Uncertainty: Self-Similarity and Heavy Tails por el Dr. Adán Hirales Carbajal y Propuesta teórica-experimental de un sistema de criptografía óptica cuántica de segunda generación del Dr. Josué Aarón López Leyva.
 Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }

Oops, we seem to be having trouble contacting Twitter

You can talk about Self-similarity with people all over the world in our discussions.