HOME

TheInfoList




In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a fractal is a subset of
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
with a
fractal dimension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

fractal dimension
that strictly exceeds its
topological dimension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. Fractals appear the same at different scales, as illustrated in successive magnifications of the
Mandelbrot set The Mandelbrot set () is the set of complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation ...

Mandelbrot set
. Fractals often exhibit similar patterns at increasingly smaller scales, a property called
self-similarity __NOTOC__ has an infinitely repeating self-similarity when it is magnified. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra ...
, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the
Menger spongeImage:Menger-Schwamm-farbig.png, upright=1.4, An illustration of ''M''4, the sponge after four iterations of the construction process In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpi ...
, it is called affine self-similar. Fractal geometry lies within the mathematical branch of
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
. One way that fractals are different from finite
geometric figures Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ...
is how they
scale Scale or scales may refer to: Mathematics * Scale (descriptive set theory)In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set (mathematics), set of point (mathematics), points in some Poli ...
. Doubling the edge lengths of a
polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

polygon
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 dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

volume
scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). 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 (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
. This power is called the
fractal dimension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

fractal dimension
of the fractal, and it usually exceeds the fractal's
topological dimension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. Analytically, most fractals are nowhere
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
. An infinite
fractal curve A fractal curve is, loosely, a mathematical curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may b ...
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 also resembles a surface. 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 Linguistics is the scientific study of language, meaning tha ...

recursion
, 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 ga ...
but not
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
functions in the 19th century by the seminal work of
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian A Bohemian () is a resident of Bohemia Bohemia ( ; cs, Čechy ; ; hsb, Čěska; szl, Czechy) is the westernmost a ...

Bernard Bolzano
,
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
, and
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

Karl Weierstrass
, and on to the coining of the word ''
fractal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
'' 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a shape made of parts similar to the whole in some way." Still later, Mandelbrot proposed "to use ''fractal'' without a pedantic definition, to use ''
fractal dimension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

fractal dimension
'' 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 having fractal dimensions, 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, exam ...
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 natural, physical, material world or universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter an ...
,
technology Technology ("science of craft", from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. I ...
,
art Art is a diverse range of (products of) human activities Humans (''Homo sapiens'') are the most populous and widespread species of primates, characterized by bipedality, opposable thumbs, hairlessness, and intelligence allowing the use ...
, architectureOstwald, 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, architecture of buildings. It was written by Michael J. Ostwald and Josephine Vaughan, both of whom ...
''. Birhauser, Basel. .
and
law Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its bounda ...
. Fractals are of particular relevance in the field of
chaos theory Chaos theory is an interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from several other fields ...
because the graphs of most chaotic processes are fractals. Many real and model networks have been found to have fractal features such as self similarity.


Etymology

The term "fractal" was coined by the mathematician 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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

dimensions
to geometric
patterns in nature Patterns in nature are visible regularities of form found in the natural world. These patterns A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable man ...
.


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 upA piece generated in Apophysis Fractal art is a form of algorithmic art created by calculating fractal In mathematics, a fractal is a subset of Euclidean space with a Hausdorff dimension, fractal dimension that strictly exceeds its to ...
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 becaus ...

infinite regress
in parallel mirrors or the
homunculus A homunculus ( , , ; "little person") is a representation of a small human being. Popularized in sixteenth-century alchemy File:Aurora consurgens zurich 044 f-21v-44 dragon-pot.jpg, Depiction of Ouroboros from the alchemical treatise ''Aurora ...
, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

fractal dimension
greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric
shapes A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. ...

shapes
are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is rep-tiled 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 fractal In mathematics, a fractal is a subset of Euclidean space with a Hausdorff dimension, fractal dimension that st ...

Koch curve
. 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!). The fact that the Koch curve has a fractal dimension differing from its ''conventionally understood'' dimension (that is, its topological dimension) is what makes it a fractal. This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
". 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. The scope and application of measurement are dependent on the context and discipline. In natural science Natu ...
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 dea ...

computer graphics
, with several notable people contributing canonical fractal forms along the way. A common theme in traditional African architecture 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 ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, " ...
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 Linguistics is the science, scientific study of language. It e ...

recursive
self-similarity __NOTOC__ has an infinitely repeating self-similarity when it is magnified. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra ...
(although he made the mistake of thinking that only the
straight line In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics Mathematics (from Greek: ) includes the study of such topics as numbe ...

straight line
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

Karl Weierstrass
presented the first definition of a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

function
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 discret ...

graph
that would today be considered a fractal, having the non-
intuitive Intuition is the ability to acquire knowledge Knowledge is a familiarity, awareness, or understanding of someone or something, such as facts ( descriptive knowledge), skills (procedural knowledge), or objects (Knowledge by acquaintance, acqu ...
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 ga ...
but nowhere differentiable 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 created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
, who attended lectures by Weierstrass, published examples of
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
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 remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 188 ...

Cantor set
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 group ...
and
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the s ...
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 Sweden, 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 no ...

Helge von Koch
, 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 ...

Koch snowflake
. Another milestone came a decade later in 1915, when
Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish people, Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of fu ...

Wacław Sierpiński
constructed his famous
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

triangle
then, one year later, his
carpet A carpet is a textile A textile is a flexible material made by creating an interlocking bundle of yarn Yarn is a long continuous length of interlocked fibres, suitable for use in the production of textiles, sewing, crocheting, kni ...

carpet
. By 1918, two French mathematicians,
Pierre Fatou Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth ...

Pierre Fatou
and
Gaston Julia Gaston Maurice Julia (3 February 1893 – 19 March 1978) was a French mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are closely relat ...
, though working independently, arrived essentially simultaneously at results describing what is now seen as fractal behaviour associated with mapping
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) include ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) inc ...
'', which built on earlier work by
Lewis Fry Richardson Lewis Fry Richardson, FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family ...

Lewis Fry Richardson
. 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 number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation ...

Mandelbrot set
, 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. He was a co-founder and chief scientist of Pixar, Pixar Animation Studios. He is the co-inventor of the Reyes rendering algorithm and is one of the authors ...

Loren Carpenter
gave a presentation at the
SIGGRAPH SIGGRAPH (Special Interest Group on Computer Graphics and Interactive Techniques) is an annual Meeting, conference on computer graphics (CG) organized by the ACM SIGGRAPH, starting in 1974. The main conference is held in North America; SIGGRA ...
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 is the form of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, texture, or material type. Classification of simple shapes Some simple shapes can ...

geometric shape
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

fractal dimension
s, but whereas these numbers quantify
complexity Complexity characterises the behaviour of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environme ...
(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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. However, this requirement is not met by
space-filling curve In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathemat ...
s such as the
Hilbert curve The Hilbert curve (also known as the Hilbert space-filling curve) is a continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probab ...

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, in addition to being nowhere differentiable and able to have a
fractal dimension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

fractal dimension
, be only generally characterized by a
gestalt ''Gestalt'', a German word for form or shape, may refer to: * Holism, the idea that natural systems and their properties should be viewed as wholes, not as loose collections of parts Psychology * Gestalt psychology (also known as "Gestalt theory" ...
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 number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation ...

Mandelbrot set
'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 In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wi ...
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 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 In mathematics, a fractal is a subset of Euclidean space with a Hausdorff dimension, fractal dimension that strictly exceeds its topological dimension. Fractals appear the same at diffe ...
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 In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, ...
(related to the next criterion in this list). * Irregularity locally and globally that is not easily described in traditional Euclidean geometric language. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls". * Simple and "perhaps
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 Linguistics is the science, scientific study of language. It e ...

recursive
" definitions; ''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, is easily described in Euclidean language, has the same
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point (geometry), ...
as
topological dimension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, and is fully defined 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 In chaos theory Chaos theory is an interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from ...

butterfly effect
, a small change in a single variable can have an unpredictable outcome. * ''
Iterated function systems In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981. IFS fractals, a ...
(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 ...

Koch snowflake
,
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 188 ...

Cantor set
, Haferman carpet, , Sierpinski gasket,
Peano curve 400px, Three iterations of a Peano curve construction, whose limit is a space-filling curve. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arit ...

Peano curve
, ,
T-square A T-square is a technical drawing Drawing is a form of visual art The visual arts are art forms such as painting Painting is the practice of applying paint, pigment, color or other medium to a solid surface (called the "matrix" o ...
,
Menger spongeImage:Menger-Schwamm-farbig.png, upright=1.4, An illustration of ''M''4, the sponge after four iterations of the construction process In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpi ...
* ''
Strange attractor In the mathematical field of dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples ...
s'' – 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 In mathematics, a fractal is a subset of Euclidean space with a Hausdorff dimension, fractal dimension that strictly exceeds its topological dimension. Fractals appear the same at diffe ...
image, or the
logistic map The logistic map is a polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only t ...

logistic map
) * ''
L-system An L-system or Lindenmayer system is a wikt:parallel, parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make string (computer science), strings, a collection of Production ...

L-system
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 Vector graphics are computer graphics images that are defined in terms of points on a Cartesian plane, which are connected by lines and curves to form polygons and other shapes. V ...
patterns such as space-filling curves and tilings * ''Escape-time fractals'' – use a formula or recurrence relation 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 number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation ...

Mandelbrot set
, 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 remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 188 ...

Cantor set
and the are examples of finite subdivision rules, as is barycentric subdivision.


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 A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable man ...
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 * Clouds and rainfall areas * Coastlines * Impact crater, Craters * Crystals * DNA * Earthquakes * Fault lines * Geometrical optics * Heart rates * Heart sounds *Lake shorelines and areas * Lightning bolts * Mountain goat horns * Networks *Polymers * Percolation * Mountain, Mountain ranges * Wind wave, Ocean waves * Pineapple * Psychological subjective perception * Proteins * Rings of Saturn * river, River networks * Romanesco broccoli * Snowflakes * Soil pores *Surfaces in Turbulence, turbulent flows * Trees * Dust grains * Brownian motion (generated by a one-dimensional Wiener process). 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 Protein, proteins, to Organelle, organelles, to whole cells.


In creative works

Since 1999, more than 10 scientific groups have performed fractal analysis on over 50 of Jackson Pollock's (1912–1956) paintings which were created by pouring paint directly onto his 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 A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environme ...
* 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.


Ion propulsion

When two-dimensional fractals are iterated many times, the perimeter of the fractal increases up to infinity, but the area may never exceed a certain value. A fractal in three-dimensional space is similar; such a fractal may have an infinite surface area, but never exceed a certain volume. This can be utilized to maximize the efficiency of ion propulsion when choosing electron emitter construction and material. If done correctly, the efficiency of the emission process can be maximized.


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

Santo Banerjee, M. K. Hassan, Sayan Mukherjee and A. Gowrisankar, Fractal Patterns in Nonlinear Dynamics and Applications. (CRC Press, Taylor & Francis Group, 2019)


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 number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation ...

Mandelbrot set
.) * 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

*
Scaling and Fractals
presented by Shlomo Havlin, Bar-Ilan University
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