__NOTOC__
![Standard self-similarity](https://upload.wikimedia.org/wikipedia/commons/7/7c/Standard_self-similarity.png)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
coastline
The coast, also known as the coastline or seashore, is defined as the area where land meets the ocean, or as a line that forms the boundary between the land and the coastline. The Earth has around of coastline. Coasts are important zones in ...
s, are statistically self-similar: parts of them show the same statistical properties at many scales.
[PDF]
/ref> Self-similarity is a typical property of fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
s. Scale invariance
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.
The technical term ...
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
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...
is both symmetrical
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
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
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
, whereas any portion of a straight line
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
may resemble the whole, further detail is not revealed.
A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity
measured at different times are different but the corresponding dimensionless quantity at given value of remain invariant. It happens if the quantity exhibits dynamic scaling Dynamic scaling (sometimes known as Family-Vicsek scaling) is a litmus test that shows whether an evolving system exhibits self-similarity. In general a function is said to exhibit dynamic scaling if it satisfies:
:f(x,t)\sim t^\theta \varphi \left ...
. The idea is just an extension of the idea of similarity of two triangles. Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide.
Peitgen ''et al.'' explain the concept as such:
Since mathematically, a fractal may show self-similarity under indefinite magnification, it is impossible to recreate this physically. Peitgen ''et al.'' suggest studying self-similarity using approximations:
This vocabulary was introduced by Benoit Mandelbrot in 1964.
Self-affinity
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, self-affinity is a feature of a fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
whose pieces are scaled by different amounts in the x- and y-directions. This means that to appreciate the self similarity of these fractal objects, they have to be rescaled using an anisotropic
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
.
Definition
A compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
''X'' is self-similar if there exists a finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. Th ...
''S'' indexing a set of non-surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
s for which
:
If , we call ''X'' self-similar if it is the only non-empty
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
. When the set ''S'' has only two elements, the monoid is known as the dyadic monoid
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractiona ...
. The dyadic monoid can be visualized as an infinite binary tree
In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary t ...
; more generally, if the set ''S'' has ''p'' elements, then the monoid may be represented as a p-adic
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...
tree.
The automorphisms of the dyadic monoid is the modular group; the automorphisms can be pictured as hyperbolic rotation
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping.
For a fixed positive real number , t ...
s of the binary tree.
A more general notion than self-similarity is Self-affinity
__NOTOC__
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 se ...
.
Examples
The Mandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
is also self-similar around Misiurewicz point
In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical point (mathematics), critical point is strictly p ...
s.
Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in teletraffic engineering
Teletraffic engineering, telecommunications traffic engineering, or just traffic engineering when in context, is the application of transportation traffic engineering theory to telecommunications. Teletraffic engineers use their knowledge of stat ...
, packet switched
In telecommunications, packet switching is a method of grouping data into '' packets'' that are transmitted over a digital network. Packets are made of a header and a payload. Data in the header is used by networking hardware to direct the pack ...
data traffic patterns seem to be statistically self-similar. This property means that simple models using a Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
are inaccurate, and networks designed without taking self-similarity into account are likely to function in unexpected ways.
Similarly, stock market
A stock market, equity market, or share market is the aggregation of buyers and sellers of stocks (also called shares), which represent ownership claims on businesses; these may include ''securities'' listed on a public stock exchange, as ...
movements are described as displaying self-affinity
__NOTOC__
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 se ...
, i.e. they appear self-similar when transformed via an appropriate affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
for the level of detail being shown. Andrew Lo
Andrew Wen-Chuan Lo () (born 1960) is the Charles E. and Susan T. Harris Professor of Finance at the MIT Sloan School of Management. Lo is the author of many academic articles in finance and financial economics. He founded AlphaSimplex Group in ...
describes stock market log return self-similarity in econometrics
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
.
Finite subdivision rules
In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeati ...
are a powerful technique for building self-similar sets, including the Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Thr ...
and the Sierpinski triangle.
In cybernetics
The viable system model
The viable system model (VSM) is a model of the organizational structure of any autonomous system capable of producing itself. A viable system is any system organised in such a way as to meet the demands of surviving in the changing environment. On ...
of Stafford Beer
Anthony Stafford Beer (25 September 1926 – 23 August 2002) was a British theorist, consultant and professor at the Manchester Business School. He is best known for his work in the fields of operational research and management cybernetics.
...
is an organizational model with an affine self-similar hierarchy, where a given viable system is one element of the System One of a viable system one recursive level higher up, and for whom the elements of its System One are viable systems one recursive level lower down.
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
A fern (Polypodiopsida or Polypodiophyta ) is a member of a group of vascular plants (plants with xylem and phloem) that reproduce via spores and have neither seeds nor flowers. The polypodiophytes include all living pteridophytes except t ...
, which bears a marked resemblance to natural ferns. Other plants, such as Romanesco broccoli
Romanesco broccoli (also known as Roman cauliflower, Broccolo Romanesco, Romanesque cauliflower, Romanesco or broccoflower) is an edible flower bud of the species ''Brassica oleracea''. It is chartreuse in color, and has a form naturally approx ...
, exhibit strong self-similarity.
In music
* Strict canons display various types and amounts of self-similarity, as do sections of fugues
In music, a fugue () is a contrapuntal compositional technique in two or more voices, built on a subject (a musical theme) that is introduced at the beginning in imitation (repetition at different pitches) and which recurs frequently in the c ...
.
* A Shepard tone
A Shepard tone, named after Roger Shepard, is a sound consisting of a superposition of sine waves separated by octaves. When played with the bass pitch of the tone moving upward or downward, it is referred to as the ''Shepard scale''. This cr ...
is self-similar in the frequency or wavelength domains.
* The Danish
Danish may refer to:
* Something of, from, or related to the country of Denmark
People
* A national or citizen of Denmark, also called a "Dane," see Demographics of Denmark
* Culture of Denmark
* Danish people or Danes, people with a Danish a ...
composer
A composer is a person who writes music. The term is especially used to indicate composers of Western classical music, or those who are composers by occupation. Many composers are, or were, also skilled performers of music.
Etymology and Defi ...
Per Nørgård
Per Nørgård (; born 13 July 1932) is a Danish composer and music theorist. Though his style has varied considerably throughout his career, his music has often included repeatedly evolving melodies—such as the infinity series—in the vein o ...
has made use of a self-similar integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.
An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...
named the 'infinity series' in much of his music.
* In the research field of music information retrieval
Music information retrieval (MIR) is the interdisciplinary science of retrieving information from music. MIR is a small but growing field of research with many real-world applications. Those involved in MIR may have a background in academic musicol ...
, self-similarity commonly refers to the fact that music often consists of parts that are repeated in time. In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.[ (Also se]
Google Books
See also
References
External links
"Copperplate Chevrons"
— a self-similar fractal zoom movie
— New articles about Self-Similarity. Waltz Algorithm
Self-affinity
*
*
*
{{DEFAULTSORT:Self-Similarity
Fractals
Scaling symmetries
Homeomorphisms
Self-reference