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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 fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
strictly exceeding the
topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. Informal discussion For ordinary Euclidean ...
. Many fractals appear similar at various scales, as illustrated in successive magnifications of 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 ...
. This exhibition of similar patterns at increasingly smaller scales is called
self-similarity __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 ...
, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the
Menger sponge In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Si ...
, the shape is called
affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...
self-similar. Fractal geometry lies within the mathematical branch of
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
. One way that fractals are different from finite
geometric figures Lists of shapes cover different types of geometric shape and related topics. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools. Mathematics * List of mathematical shapes * List of two- ...
is how they scale. Doubling the edge lengths of a filled
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the
radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
of a filled sphere is doubled, its
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
and is in general greater than its conventional dimension. This power is called the
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
of the geometric object, to distinguish it from the conventional dimension (which is formally called the
topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. Informal discussion For ordinary Euclidean ...
). Analytically, many fractals are nowhere
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
. An infinite
fractal curve A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectif ...
can be conceived of as winding through space differently from an ordinary line – although it is still topologically 1-dimensional, its fractal dimension indicates that it locally fills space more efficiently than an ordinary line. Starting in the 17th century with notions of
recursion Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematic ...
, fractals have moved through increasingly rigorous mathematical treatment to the study of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
but not
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
functions in the 19th century by the seminal work of
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his li ...
,
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
, and
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, and on to the coining of the word ''
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 ill ...
'' in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot defined ''fractal'' as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the
topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. Informal discussion For ordinary Euclidean ...
." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or fragmented
geometric shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...
that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole." Still later, Mandelbrot proposed "to use ''fractal'' without a pedantic definition, to use ''
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
'' as a generic term applicable to ''all'' the variants". The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many
examples Example may refer to: * '' exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, e ...
have been formulated and studied. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media and found in
nature Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
,
technology Technology is the application of knowledge to reach practical goals in a specifiable and reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in medicine, scien ...
, art,
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing buildings ...
Ostwald, Michael J., and Vaughan, Josephine (2016) ''
The Fractal Dimension of Architecture ''The Fractal Dimension of Architecture'' is a book that applies the mathematical concept of fractal dimension to the analysis of the architecture of buildings. It was written by Michael J. Ostwald and Josephine Vaughan, both of whom are architec ...
''. Birhauser, Basel. .
and
law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vario ...
. Fractals are of particular relevance in the field of
chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to hav ...
because they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction).


Etymology

The term "fractal" was coined by the mathematician
Benoît Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
in 1975. Mandelbrot based it on the Latin , meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional
dimensions In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
to geometric
patterns in nature Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, ...
.


Introduction

The word "fractal" often has different connotations for the lay public as opposed to mathematicians, where the public is more likely to be familiar with fractal art than the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background. The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in the
infinite regress An infinite regress is an infinite series of entities governed by a recursive principle that determines how each entity in the series depends on or is produced by its predecessor. In the epistemic regress, for example, a belief is justified bec ...
in parallel mirrors or the
homunculus A homunculus ( , , ; "little person") is a representation of a small human being, originally depicted as small statues made out of clay. Popularized in sixteenth-century alchemy and nineteenth-century fiction, it has historically referred to the ...
, the little man inside the head of the little man inside the head ...). The difference for fractals is that the pattern reproduced must be detailed. This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric
shapes A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie o ...
are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is
rep-tile In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by ...
d into pieces each 1/3 the length of the original, then there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled into pieces each scaled down by a factor of 1/3 in both dimensions, there are a total of 32 = 9 pieces. We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/''r'', there are a total of ''r''''n'' pieces. Now, consider the
Koch curve 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 ...
. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So, strictly by analogy, we can consider the "dimension" of the Koch curve as being the unique real number ''D'' that satisfies 3''D'' = 4. This number is what mathematicians call the ''fractal dimension'' of the Koch curve; it is certainly ''not'' what is conventionally perceived as the dimension of a curve (this number is not even an integer!). In general, a key property of fractals is that the fractal dimension differs from the ''conventionally understood'' dimension (formally called the topological dimension). This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
". In a concrete sense, this means fractals cannot be measured in traditional ways. To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of
measuring Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e. the snowflake has an infinite perimeter.


History

The history of fractals traces a path from chiefly theoretical studies to modern applications in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, with several notable people contributing canonical fractal forms along the way. A common theme in traditional
African architecture Like other aspects of the culture of Africa, the architecture of Africa is exceptionally diverse. Throughout the history of Africa, Africans have developed their own local architectural traditions. In some cases, broader regional styles can be ...
is the use of fractal scaling, whereby small parts of the structure tend to look similar to larger parts, such as a circular village made of circular houses. According to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
pondered
recursive Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
self-similarity __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 ...
(although he made the mistake of thinking that only the
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
was self-similar in this sense). In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them. Indeed, according to various historical accounts, after that point few mathematicians tackled the issues and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters". Thus, it was not until two centuries had passed that on July 18, 1872
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
presented the first definition of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
with a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
that would today be considered a fractal, having the non- intuitive property of being everywhere
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
but
nowhere differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in ...
at the Royal Prussian Academy of Sciences. In addition, the quotient difference becomes arbitrarily large as the summation index increases. Not long after that, in 1883,
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
, who attended lectures by Weierstrass, published examples of
subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset of ...
s of the real line known as
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. T ...
s, which had unusual properties and are now recognized as fractals. Also in the last part of that century,
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
and
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
introduced a category of fractal that has come to be called "self-inverse" fractals. One of the next milestones came in 1904, when
Helge von Koch Niels Fabian Helge von Koch (25 January 1870 – 11 March 1924) was a Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake, one of the earliest fractal curves to be described. He was born to Swedish nobility ...
, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand-drawn images of a similar function, which is now called 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 ...
. Another milestone came a decade later in 1915, when
Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and to ...
constructed his famous
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
then, one year later, his
carpet A carpet is a textile floor covering typically consisting of an upper layer of pile attached to a backing. The pile was traditionally made from wool, but since the 20th century synthetic fibers such as polypropylene, nylon, or polyester ...
. By 1918, two French mathematicians,
Pierre Fatou Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him. Biography ...
and
Gaston Julia Gaston Maurice Julia (3 February 1893 – 19 March 1978) was a French Algerian mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are cl ...
, though working independently, arrived essentially simultaneously at results describing what is now seen as fractal behaviour associated with mapping
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
and iterative functions and leading to further ideas about attractors and repellors (i.e., points that attract or repel other points), which have become very important in the study of fractals. Very shortly after that work was submitted, by March 1918,
Felix Hausdorff Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, an ...
expanded the definition of "dimension", significantly for the evolution of the definition of fractals, to allow for sets to have non-integer dimensions. The idea of self-similar curves was taken further by Paul Lévy, who, in his 1938 paper ''Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole'', described a new fractal curve, the Lévy C curve. Different researchers have postulated that without the aid of modern computer graphics, early investigators were limited to what they could depict in manual drawings, so lacked the means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered (the Julia set, for instance, could only be visualized through a few iterations as very simple drawings). That changed, however, in the 1960s, when
Benoit Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of p ...
started writing about self-similarity in papers such as ''
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoit Mandelbrot, first published in ''Science'' on 5 May 1967. In this paper, Mandelbrot discusses self-similar curves that ...
'', which built on earlier work by
Lewis Fry Richardson Lewis Fry Richardson, FRS (11 October 1881 – 30 September 1953) was an English mathematician, physicist, meteorologist, psychologist, and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of s ...
. In 1975 Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical
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 ...
, captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal". In 1980,
Loren Carpenter Loren C. Carpenter (born February 7, 1947) is a computer graphics researcher and developer. Biography He was a co-founder and chief scientist of Pixar Animation Studios. He is the co-inventor of the Reyes rendering algorithm and is one of the ...
gave a presentation at the
SIGGRAPH SIGGRAPH (Special Interest Group on Computer Graphics and Interactive Techniques) is an annual conference on computer graphics (CG) organized by the ACM SIGGRAPH, starting in 1974. The main conference is held in North America; SIGGRAPH Asia ...
where he introduced his software for generating and rendering fractally generated landscapes.


Definition and characteristics

One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented
geometric shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...
that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"; this is generally helpful but limited. Authors disagree on the exact definition of ''fractal'', but most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in. One point agreed on is that fractal patterns are characterized by
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
s, but whereas these numbers quantify
complexity Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence. The term is generally used to ch ...
(i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns. In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose Hausdorff–Besicovitch dimension is greater than its
topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. Informal discussion For ordinary Euclidean ...
. However, this requirement is not met by
space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, ...
s such as the Hilbert curve. Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According to Falconer, fractals should be only generally characterized by a
gestalt Gestalt may refer to: Psychology * Gestalt psychology, a school of psychology * Gestalt therapy, a form of psychotherapy * Bender Visual-Motor Gestalt Test, an assessment of development disorders * Gestalt Practice, a practice of self-exploration ...
of the following features; * Self-similarity, which may include: :* Exact self-similarity: identical at all scales, such as 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 ...
:* Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., 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 ...
's satellites are approximations of the entire set, but not exact copies. :* Statistical self-similarity: repeats a pattern
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
ally so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals like the well-known example of the
coastline of Britain The coastline of the United Kingdom is formed by a variety of natural features including islands, bays, headlands and peninsulas. It consists of the coastline of the island of Great Britain and the north-east coast of the island of Ireland, as w ...
for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines fractals like the Koch snowflake. :* Qualitative self-similarity: as in a time series :*
Multifractal A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed ...
scaling: characterized by more than one fractal dimension or scaling rule * Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may have emergent properties (related to the next criterion in this list). * Irregularity locally and globally that cannot easily be described in the language of traditional
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
other than as the limit of a
recursively Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
defined sequence of stages. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls";''see Common techniques for generating fractals''. As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks detail, and is easily described in Euclidean language without a need for recursion.


Common techniques for generating fractals

Images of fractals can be created by fractal generating programs. Because of the butterfly effect, a small change in a single variable can have an unpredictable outcome. * '' Iterated function systems (IFS)'' – use fixed geometric replacement rules; may be stochastic or deterministic; e.g.,
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 ...
,
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. T ...
, Haferman carpet, Sierpinski carpet, Sierpinski gasket,
Peano curve In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not in ...
, Harter-Heighway dragon curve,
T-square A T-square is a technical drawing instrument used by draftsmen primarily as a guide for drawing horizontal lines on a drafting table. The instrument is named after its resemblance to the letter T, with a long shaft called the "blade" and a sh ...
,
Menger sponge In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Si ...
* '' Strange attractors'' – use iterations of a map or solutions of a system of initial-value differential or difference equations that exhibit chaos (e.g., see
multifractal A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed ...
image, or the
logistic map The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popula ...
) * ''
L-system An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into som ...
s'' – use string rewriting; may resemble branching patterns, such as in plants, biological cells (e.g., neurons and immune system cells), blood vessels, pulmonary structure, etc. or
turtle graphics In computer graphics, turtle graphics are vector graphics using a relative cursor (the "turtle") upon a Cartesian plane (x and y axis). Turtle graphics is a key feature of the Logo programming language. Overview The turtle has three attribut ...
patterns such as
space-filling curves In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, spa ...
and tilings * ''Escape-time fractals'' – use a
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
or
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
at each point in a space (such as the complex plane); usually quasi-self-similar; also known as "orbit" fractals; e.g., 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 ...
, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly. * ''Random fractals'' – use stochastic rules; e.g., Lévy flight, Percolation theory, percolation clusters, Self-avoiding walk, self avoiding walks, fractal landscapes, trajectories of Brownian motion and the Brownian tree (i.e., dendritic fractals generated by modeling diffusion-limited aggregation or reaction-limited aggregation clusters). *''Finite subdivision rules'' – use a recursive topological algorithm for refining tilingsJ. W. Cannon, W. J. Floyd, W. R. Parry. ''Finite subdivision rules''. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196. and they are similar to the process of cell division.J. W. Cannon, W. Floyd and W. Parry
''Crystal growth, biological cell growth and geometry''.
Pattern Formation in Biology, Vision and Dynamics, pp. 65–82. World Scientific, 2000. , .
The iterative processes used in creating 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. T ...
and the Sierpinski carpet are examples of finite subdivision rules, as is barycentric subdivision.


Applications


Simulated fractals

Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to the practical limits of physical time and space. Models may simulate theoretical fractals or #fractals in nature, natural phenomena with fractal features. The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for fractal analysis. Some specific applications of fractals to technology are listed #fractals in technology, elsewhere. Images and other outputs of modelling are normally referred to as being "fractals" even if they do not have strictly fractal characteristics, such as when it is possible to zoom into a region of the fractal image that does not exhibit any fractal properties. Also, these may include calculation or display Artifact (error), artifacts which are not characteristics of true fractals. Modeled fractals may be sounds, digital images, electrochemical patterns, circadian rhythms, etc. Fractal patterns have been reconstructed in physical 3-dimensional space and virtually, often called "in silico" modeling. Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above. As one illustration, trees, ferns, cells of the nervous system, blood and lung vasculature, and other branching
patterns in nature Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, ...
can be modeled on a computer by using recursive algorithms and L-systems techniques. The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects. A limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithms.


Natural phenomena with fractal features

Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees. Phenomena known to have fractal features include: * Actin cytoskeleton * Algae * Animal coloration patterns * Blood vessels and pulmonary vessels * Brownian motion (generated by a one-dimensional Wiener process). * Clouds and rainfall areas * Coastlines * Impact crater, Craters * Crystals * DNA * Dust grains * Earthquakes * Fault lines * Geometrical optics * Heart rates * Heart sounds *Lake shorelines and areas * Lightning bolts * Mountain goat horns * Polymers * Percolation * Mountain, Mountain ranges * Wind wave, Ocean waves * Pineapple * Proteins * Psychedelic Experience *Purkinje cell, Purkinje cells * Rings of Saturn * river, River networks * Romanesco broccoli * Snowflakes * Soil pores *Surfaces in Turbulence, turbulent flows * Trees File:Frost patterns 2.jpg, Frost crystals occurring naturally on cold glass form fractal patterns File:Optical Billiard Spheres dsweet.jpeg, Fractal basin boundary in a geometrical optical system File:Glue1 800x600.jpg, A fractal is formed when pulling apart two glue-covered Acryloyl group, acrylic sheets File:Square1.jpg, High-voltage breakdown within a block of acrylic glass creates a fractal Lichtenberg figure File:Romanesco broccoli (Brassica oleracea).jpg, Romanesco broccoli, showing self-similar form approximating a natural fractal File:Fractal defrosting patterns on Mars.jpg, Fractal defrosting patterns, polar Mars. The patterns are formed by sublimation of frozen CO2. Width of image is about a kilometer. File:Brefeldia maxima plasmodium on wood.jpg, Slime mold ''Brefeldia maxima'' growing fractally on wood


Fractals in cell biology

Fractals often appear in the realm of living organisms where they arise through branching processes and other complex pattern formation. Ian Wong and co-workers have shown that migrating cells can form fractals by clustering and Branching process, branching. Neuron, Nerve cells function through processes at the cell surface, with phenomena that are enhanced by largely increasing the surface to volume ratio. As a consequence nerve cells often are found to form into fractal patterns. These processes are crucial in cell physiology and different Pathology, pathologies. Multiple subcellular structures also are found to assemble into fractals. Diego Krapf has shown that through branching processes the actin filaments in human cells assemble into fractal patterns. Similarly Matthias Weiss showed that the endoplasmic reticulum displays fractal features. The current understanding is that fractals are ubiquitous in cell biology, from proteins, to organelles, to whole cells.


In creative works

Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created by Jackson Pollock by pouring paint directly onto horizontal canvasses. Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks. Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals. Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns. It involves pressing paint between two surfaces and pulling them apart. Cyberneticist Ron Eglash has suggested that fractal geometry and mathematics are prevalent in African art, games, divination, trade, and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles. Hokky Situngkir also suggested the similar properties in Indonesian traditional art, batik, and ornament (art), ornaments found in traditional houses. Ethnomathematician Ron Eglash has discussed the planned layout of Benin city using fractals as the basis, not only in the city itself and the villages but even in the rooms of houses. He commented that "When Europeans first came to Africa, they considered the architecture very disorganised and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics that they hadn’t even discovered yet." In a 1996 interview with Michael Silverblatt, David Foster Wallace admitted that the structure of the first draft of ''Infinite Jest'' he gave to his editor Michael Pietsch was inspired by fractals, specifically the Sierpinski triangle (a.k.a. Sierpinski gasket), but that the edited novel is "more like a lopsided Sierpinsky Gasket". Some works by the Dutch artist M. C. Escher, such as Circle Limit III, contain shapes repeated to infinity that become smaller and smaller as they get near to the edges, in a pattern that would always look the same if zoomed in. File:Animated fractal mountain.gif, A fractal that models the surface of a mountain (animation) File:FRACTAL-3d-FLOWER.jpg, 3D recursive image File:Fractal-BUTTERFLY.jpg, Recursive fractal butterfly image File:Apophysis-100303-104.jpg, A fractal flame


Physiological responses

Humans appear to be especially well-adapted to processing fractal patterns with D values between 1.3 and 1.5. When humans view fractal patterns with D values between 1.3 and 1.5, this tends to reduce physiological stress.


Applications in technology

* Fractal antennas *Fractal transistor * Fractal heat exchangers * Digital imaging * Architecture * Urban growth * Categorisation, Classification of histopathology slides * Fractal landscape or Coastline
complexity Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence. The term is generally used to ch ...
* Detecting 'life as we don't know it' by fractal analysis * Enzymes (Michaelis-Menten kinetics) * Algorithmic composition, Generation of new music * Signal (information theory), Signal and Fractal compression, image compression * Creation of digital photographic enlargements * Fractal in soil mechanics * Game Design, Computer and video game design * Computer Graphics * Life, Organic environments * Procedural generation * Fractography and fracture mechanics * SAXS, Small angle scattering theory of fractally rough systems * T-shirts and other fashion * Generation of patterns for camouflage, such as MARPAT * Digital sundial * Technical analysis of price series * Fractal dimension on networks, Fractals in networks * Medicine * Neuroscience * Diagnostic Imaging * Pathology * Geology *Geography * Archaeology * Soil mechanics * Seismology * Search and rescue * Technical analysis * Morton order#Applications, Morton order space filling curves for GPU cache coherency in texture mapping, rasterisation and indexing of turbulence data.


See also

*Banach fixed point theorem *Bifurcation theory *Box counting *Cymatics *Determinism *Diamond-square algorithm *Droste effect *Feigenbaum function *Form constant *Fractal cosmology *Fractal derivative *Fractalgrid *Fractal string *Fracton *Graftal *Greeble *Infinite regress *Lacunarity *List of fractals by Hausdorff dimension *Mandelbulb *Mandelbox *Macrocosm and microcosm *Matryoshka doll *Menger Sponge *Multifractal system *Newton fractal *Percolation *Power law *List of important publications in mathematics#Fractal geometry, Publications in fractal geometry *Random walk *Self-reference *Self-similarity *Systems theory *Strange loop *Turbulence *Wiener process


Notes


References


Further reading

* Barnsley, Michael F.; and Rising, Hawley; ''Fractals Everywhere''. Boston: Academic Press Professional, 1993. * Duarte, German A.; ''Fractal Narrative. About the Relationship Between Geometries and Technology and Its Impact on Narrative Spaces''. Bielefeld: Transcript, 2014. * Falconer, Kenneth; ''Techniques in Fractal Geometry''. John Wiley and Sons, 1997. * Jürgens, Hartmut; Heinz-Otto Peitgen, Peitgen, Heinz-Otto; and Saupe, Dietmar; ''Chaos and Fractals: New Frontiers of Science''. New York: Springer-Verlag, 1992. *Benoit Mandelbrot, Mandelbrot, Benoit B.; ''The Fractal Geometry of Nature''. New York: W. H. Freeman and Co., 1982. * Peitgen, Heinz-Otto; and Saupe, Dietmar; eds.; ''The Science of Fractal Images''. New York: Springer-Verlag, 1988. * Clifford A. Pickover, Pickover, Clifford A.; ed.; ''Chaos and Fractals: A Computer Graphical Journey – A 10 Year Compilation of Advanced Research''. Elsevier, 1998. * Jones, Jesse; ''Fractals for the Macintosh'', Waite Group Press, Corte Madera, CA, 1993. . * Lauwerier, Hans; ''Fractals: Endlessly Repeated Geometrical Figures'', Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991. , cloth. paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix. * * Wahl, Bernt; Van Roy, Peter; Larsen, Michael; and Kampman, Eric
''Exploring Fractals on the Macintosh''
Addison Wesley, 1995. * Lesmoir-Gordon, Nigel; ''The Colours of Infinity: The Beauty, The Power and the Sense of Fractals''. 2004. (The book comes with a related DVD of the Arthur C. Clarke documentary introduction to the fractal concept and 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 ...
.) * Liu, Huajie; ''Fractal Art'', Changsha: Hunan Science and Technology Press, 1997, . * Gouyet, Jean-François; ''Physics and Fractal Structures'' (Foreword by B. Mandelbrot); Masson, 1996. , and New York: Springer-Verlag, 1996. . Out-of-print. Available in PDF version at. *


External links

*
Hunting the Hidden Dimension
PBS ''Nova (American TV series), NOVA'', first aired August 24, 2011
Benoit Mandelbrot: Fractals and the Art of Roughness
, TED (conference), TED, February 2010
Technical Library on Fractals for controlling fluid

Equations of self-similar fractal measure based on the fractional-order calculus
��2007) {{Authority control Fractals, Mathematical structures Topology Computational fields of study