![Mandelbrot sequence new](https://upload.wikimedia.org/wikipedia/commons/a/a4/Mandelbrot_sequence_new.gif)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a 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 meas ...
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
topological invariant, topologically invariant way.
Informal discussion
F ...
. 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 simil ...
.
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 toge ...
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 ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
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). The de ...
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 language ...
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 meas ...
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
topological invariant, topologically invariant way.
Informal discussion
F ...
).
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 rec ...
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.
![LineSegment selfSimilar svg](https://upload.wikimedia.org/wikipedia/commons/1/10/LineSegment_selfSimilar_svg.svg)
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 mathematics ...
, 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 ...
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 liber ...
,
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 rig ...
, 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 illu ...
'' 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
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 is zero, of a ...
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
topological invariant, topologically invariant way.
Informal discussion
F ...
." 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 on ...
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 meas ...
'' as a generic term applicable to ''all'' the variants".
The consensus among mathematicians is that theoretical fractals are infinitely self-similar
iterated
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
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, ex ...
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 physics, physical world or universe. "Nature" can refer to the phenomenon, 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. ...
,
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, science, ...
,
art
Art is a diverse range of human activity, and resulting product, that involves creative or imaginative talent expressive of technical proficiency, beauty, emotional power, or conceptual ideas.
There is no generally agreed definition of wha ...
,
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 building ...
[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 have co ...
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 coordina ...
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, foa ...
.
Introduction
![FractalTree](https://upload.wikimedia.org/wikipedia/commons/3/30/FractalTree.gif)
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
Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still digital images, animations, and media. Fractal art developed from the mid-1980s onwards. It is a genre of computer ...
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 beca ...
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 meas ...
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 on ...
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 Mar ...
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 3
2 = 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).
![3D Computer Generated Fractal](https://upload.wikimedia.org/wikipedia/commons/1/1c/3D_Computer_Generated_Fractal.png)
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 t ...
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
![Von Koch curve](https://upload.wikimedia.org/wikipedia/commons/f/fd/Von_Koch_curve.gif)
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 i ...
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 space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
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
Intuition is the ability to acquire knowledge without recourse to conscious reasoning. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledge; unconscious cognition; ...
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 ...
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 its ...
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 of ...
, who attended lectures by Weierstrass,
published examples of
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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.
Thr ...
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, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
introduced a category of fractal that has come to be called "self-inverse" fractals.
![Fractal tree](https://upload.wikimedia.org/wikipedia/commons/a/a9/Fractal_tree.gif)
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 Edge (geometry), edges and three Vertex (geometry), 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, an ...
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 hav ...
. 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
P ...
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 a ...
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, and ...
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
In mathematics, the Lévy C curve is a self-similar fractal curve that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and Georg Faber in 1910, but now bears the name of French mathematician Pa ...
.
![60 degrees 2x recursive IFS](https://upload.wikimedia.org/wikipedia/commons/e/ef/60_degrees_2x_recursive_IFS.jpg)
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 phy ...
started writing about self-similarity in papers such as ''
'', 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 si ...
.
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 on ...
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 meas ...
s, but whereas these numbers quantify
complexity
Complexity characterises the behaviour of a system or model whose components interaction, interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.
The term is generall ...
(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
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 is zero, of a ...
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
topological invariant, topologically invariant way.
Informal discussion
F ...
.
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, space ...
s such as the
Hilbert curve
The Hilbert curve (also known as the Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling Peano curves discovered by Giuseppe ...
.
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
In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole.
Emergence ...
(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 mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
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
![KarperienFractalBranch](https://upload.wikimedia.org/wikipedia/commons/b/b7/KarperienFractalBranch.jpg)
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
Unpredictable may refer to:
Film and TV
* "Unpredictable" (''Eureka''), an episode of ''Eureka''
Music Albums
* ''Unpredictable'' (Classified album), 2000
* ''Unpredictable'' (Jamie Foxx album), 2005
* ''Unpredictable'' (Malik B. album), 2015 ...
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 fractal ...
(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.
Thr ...
, 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 injective. ...
,
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 sho ...
,
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 attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
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 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 popular ...
)
* ''
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 some ...
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 attribute ...
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, spac ...
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 betwee ...
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
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
); 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
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wit ...
,
Burning Ship fractal
The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function:
:z_ = (, \operatorname \left(z_n\right), +i, \operatorname \left(z_n\right), )^2 + c, \quad z_0= ...
,
Nova fractal
The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial or transcendental function. It is the Julia set of the meromorphic function which is given by Newton's method. ...
and
Lyapunov fractal
In mathematics, Lyapunov fractals (also known as Markus–Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, ''r'', periodically switches between ...
. 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
A Lévy flight is a random walk in which the step-lengths have a Lévy distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random direct ...
,
percolation clusters,
self avoiding walks,
fractal landscapes
A fractal landscape is a surface that is generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the result of the procedure is not a deterministic fractal surface, ...
, trajectories of
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
and the
Brownian tree
In probability theory, the Brownian tree, or Aldous tree, or Continuum Random Tree (CRT) is a special case from random real trees which may be defined from a Brownian excursion. The Brownian tree was defined and studied by David Aldous in three a ...
(i.e., dendritic fractals generated by modeling
diffusion-limited aggregation
Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by T.A. Witten Jr. and L.M. Sander in 1981, is app ...
or reaction-limited aggregation clusters).
![Finite subdivision of a radial link](https://upload.wikimedia.org/wikipedia/commons/d/d6/Finite_subdivision_of_a_radial_link.png)
*''
Finite subdivision rule
In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeati ...
s'' – use a recursive
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
algorithm for refining tilings
[J. 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
Cell division is the process by which a parent cell (biology), cell divides into two daughter cells. Cell division usually occurs as part of a larger cell cycle in which the cell grows and replicates its chromosome(s) before dividing. In eukar ...
.
[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.
Thr ...
and the
Sierpinski carpet are examples of finite subdivision rules, as is
barycentric subdivision
In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension on simplicial complexes is a canonical method to refine them. Therefore, the barycentric subdivision is an important tool i ...
.
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
natural phenomena with fractal features. The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for
fractal analysis
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical dataset, or a pattern or signal extracted from p ...
. Some specific applications of fractals to technology are listed
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
artifacts which are not characteristics of true fractals.
Modeled fractals may be sounds,
digital images, electrochemical patterns,
circadian rhythm
A circadian rhythm (), or circadian cycle, is a natural, internal process that regulates the sleep–wake cycle and repeats roughly every 24 hours. It can refer to any process that originates within an organism (i.e., Endogeny (biology), endogeno ...
s, etc.
Fractal patterns have been reconstructed in physical 3-dimensional space
and virtually, often called "
in silico
In biology and other experimental sciences, an ''in silico'' experiment is one performed on computer or via computer simulation. The phrase is pseudo-Latin for 'in silicon' (correct la, in silicio), referring to silicon in computer chips. It ...
" modeling.
Models of fractals are generally created using
fractal-generating software
Fractal-generating software is any type of graphics software that generates images of fractals. There are many fractal generating programs available, both free and commercial. Mobile apps are available to play or tinker with fractals. Some programm ...
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, foa ...
can be modeled on a computer by using recursive
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
s and
L-systems
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 some ...
techniques.
The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a
frond
A frond is a large, divided leaf. In both common usage and botanical nomenclature, the leaves of ferns are referred to as fronds and some botanists restrict the term to this group. Other botanists allow the term frond to also apply to the lar ...
from a
fern
A fern (Polypodiopsida or Polypodiophyta ) is a member of a group of vascular plants (plants with xylem and phloem) that reproduce via spores and have neither seeds nor flowers. The polypodiophytes include all living pteridophytes except t ...
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
Microfilaments, also called actin filaments, are protein filaments in the cytoplasm of eukaryotic cells that form part of the cytoskeleton. They are primarily composed of polymers of actin, but are modified by and interact with numerous other pr ...
*
Alga
Algae (; singular alga ) is an informal term for a large and diverse group of photosynthetic eukaryotic organisms. It is a polyphyletic grouping that includes species from multiple distinct clades. Included organisms range from unicellular mi ...
e
*
Animal coloration
Animal coloration is the general appearance of an animal resulting from the reflection or emission of light from its surfaces. Some animals are brightly coloured, while others are hard to see. In some species, such as the peafowl, the male h ...
patterns
*
Blood vessel
The blood vessels are the components of the circulatory system that transport blood throughout the human body. These vessels transport blood cells, nutrients, and oxygen to the tissues of the body. They also take waste and carbon dioxide away ...
s and
pulmonary vessels
* Brownian motion (generated by a one-dimensional
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
).
* Clouds and rainfall areas
* Coastlines
*
Craters
* Crystals
*
DNA
* Dust grains
*
Earthquakes
An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, from ...
*
Fault line
In geology, a fault is a planar fracture or discontinuity in a volume of rock across which there has been significant displacement as a result of rock-mass movements. Large faults within Earth's crust result from the action of plate tectonic ...
s
* Geometrical optics
* Heart rates
*
Heart sounds
Heart sounds are the noises generated by the beating heart and the resultant flow of blood through it. Specifically, the sounds reflect the turbulence created when the heart valves snap shut. In cardiac auscultation, an examiner may use a stetho ...
*
Lake
A lake is an area filled with water, localized in a basin, surrounded by land, and distinct from any river or other outlet that serves to feed or drain the lake. Lakes lie on land and are not part of the ocean, although, like the much large ...
shorelines and areas
*
Lightning
Lightning is a naturally occurring electrostatic discharge during which two electric charge, electrically charged regions, both in the atmosphere or with one on the land, ground, temporarily neutralize themselves, causing the instantaneous ...
bolts
* Mountain goat horns
* Polymers
* Percolation
*
Mountain ranges
A mountain range or hill range is a series of mountains or hills arranged in a line and connected by high ground. A mountain system or mountain belt is a group of mountain ranges with similarity in form, structure, and alignment that have arise ...
*
Ocean waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of t ...
* Pineapple
*
Proteins
Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, respo ...
*
Psychedelic Experience
A psychedelic experience (known colloquially as a trip) is a temporary altered state of consciousness induced by the consumption of a psychedelic substance (most commonly LSD, mescaline, psilocybin mushrooms, or DMT). For example, an acid tr ...
*
Purkinje cells
Purkinje cells, or Purkinje neurons, are a class of GABAergic inhibitory neurons located in the cerebellum. They are named after their discoverer, Czech anatomist Jan Evangelista Purkyně, who characterized the cells in 1839.
Structure
The ...
*
Rings of Saturn
The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirel ...
*
River networks
*
Romanesco broccoli
Romanesco broccoli (also known as Roman cauliflower, Broccolo Romanesco, Romanesque cauliflower, Romanesco or broccoflower) is an edible flower bud of the species ''Brassica oleracea''. It is chartreuse in color, and has a form naturally approx ...
* Snowflakes
* Soil pores
*Surfaces in
turbulent
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between t ...
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 acrylic
Acrylic may refer to:
Chemicals and materials
* Acrylic acid, the simplest acrylic compound
* Acrylate polymer, a group of polymers (plastics) noted for transparency and elasticity
* Acrylic resin, a group of related thermoplastic or thermosett ...
sheets
File:Square1.jpg, High-voltage breakdown within a block of acrylic glass creates a fractal Lichtenberg figure
A Lichtenberg figure (German ''Lichtenberg-Figuren''), or Lichtenberg dust figure, is a branching electric discharge that sometimes appears on the surface or in the interior of insulating materials. Lichtenberg figures are often associated wit ...
File:Romanesco broccoli (Brassica oleracea).jpg, Romanesco broccoli
Romanesco broccoli (also known as Roman cauliflower, Broccolo Romanesco, Romanesque cauliflower, Romanesco or broccoflower) is an edible flower bud of the species ''Brassica oleracea''. It is chartreuse in color, and has a form naturally approx ...
, showing self-similar
__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 ...
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
Slime mold or slime mould is an informal name given to several kinds of unrelated eukaryotic organisms with a life cycle that includes a free-living single-celled stage and the formation of spores. Spores are often produced in macroscopic mu ...
''Brefeldia maxima
''Brefeldia maxima'' is a species of non-parasitic plasmodial slime mold, and a member of the class Myxogastria, Myxomycetes. It is common name, commonly known as the tapioca slime mold because of its peculiar pure white, tapioca pudding-like app ...
'' 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.
Nerve cells
A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa. No ...
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
Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemical ...
and different
pathologies
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in t ...
.
Multiple subcellular structures also are found to assemble into fractals.
Diego Krapf
Diego Krapf (born November 21, 1973) is an Argentine-Israeli-American physicist known for his work on anomalous diffusion and ergodicity breaking. He currently is a professor in the Department of Electrical and Computer Engineering at Colorado St ...
has shown that through branching processes the
actin
Actin is a family of globular multi-functional proteins that form microfilaments in the cytoskeleton, and the thin filaments in muscle fibrils. It is found in essentially all eukaryotic cells, where it may be present at a concentration of over ...
filaments in human cells assemble into fractal patterns.
Similarly Matthias Weiss showed that the
endoplasmic reticulum
The endoplasmic reticulum (ER) is, in essence, the transportation system of the eukaryotic cell, and has many other important functions such as protein folding. It is a type of organelle made up of two subunits – rough endoplasmic reticulum ( ...
displays fractal features. The current understanding is that fractals are ubiquitous in cell biology, from
protein
Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, respo ...
s, to
organelle
In cell biology, an organelle is a specialized subunit, usually within a cell, that has a specific function. The name ''organelle'' comes from the idea that these structures are parts of cells, as organs are to the body, hence ''organelle,'' the ...
s, to whole cells.
In creative works
Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created by
Jackson Pollock
Paul Jackson Pollock (; January 28, 1912August 11, 1956) was an American painter and a major figure in the abstract expressionist movement. He was widely noticed for his " drip technique" of pouring or splashing liquid household paint onto a hor ...
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
Decalcomania (from french: décalcomanie) is a decorative technique by which engravings and prints may be transferred to pottery or other materials.
A shortened version of the term is used for a mass-produced commodity art transfer or product l ...
, a technique used by artists such as
Max Ernst
Max Ernst (2 April 1891 – 1 April 1976) was a German (naturalised American in 1948 and French in 1958) painter, sculptor, printmaker, graphic artist, and poet. A prolific artist, Ernst was a primary pioneer of the Dada movement and Surrealism ...
, can produce fractal-like patterns. It involves pressing paint between two surfaces and pulling them apart.
Cyberneticist
Ron Eglash
Ron Eglash (born December 25, 1958 in Chestertown, Maryland) is an American who works in cybernetics, professor in the School of Information at the University of Michigan with a secondary appointment in the School of Design, and an author widely ...
has suggested that fractal geometry and mathematics are prevalent in
African art
African art describes the modern and historical paintings, sculptures, installations, and other visual culture from native or indigenous Africans and the African continent. The definition may also include the art of the African diasporas, su ...
, games,
divination
Divination (from Latin ''divinare'', 'to foresee, to foretell, to predict, to prophesy') is the attempt to gain insight into a question or situation by way of an occultic, standardized process or ritual. Used in various forms throughout histor ...
, 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
Hokky Situngkir (born February 7, 1978) is an Indonesian scientist who researches complexity theory at Surya University. He is the founder of the Bandung Fe Institute, a research institute for social complexity research. His academic activities ...
also suggested the similar properties in Indonesian traditional art,
batik
Batik is an Indonesian technique of wax-resist dyeing applied to the whole cloth. This technique originated from the island of Java, Indonesia. Batik is made either by drawing dots and lines of the resist with a spouted tool called a ''ca ...
, and
ornaments
An ornament is something used for decoration.
Ornament may also refer to:
Decoration
*Ornament (art), any purely decorative element in architecture and the decorative arts
*Biological ornament, a characteristic of animals that appear to serve on ...
found in traditional houses.
Ethnomathematician Ron Eglash has discussed the planned layout of
Benin city
Benin City is the capital and largest city of Edo State, Edo State, Nigeria. It is the fourth-largest city in Nigeria according to the 2006 census, after Lagos, Kano (city), Kano, and Ibadan, with a population estimate of about 3,500,000 as of ...
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
Michael Silverblatt (born August 6, 1952) is a literary critic and American broadcaster who hosted ''Bookworm'', a nationally syndicated radio program focusing on books and literature, from 1989 to 2022.
''Bookworm'' is broadcast by Los Angeles ...
,
David Foster Wallace
David Foster Wallace (February 21, 1962 – September 12, 2008) was an American novelist, short story writer, essayist, and university professor of English and creative writing. Wallace is widely known for his 1996 novel '' Infinite Jest'', whi ...
admitted that the structure of the first draft of ''
Infinite Jest
''Infinite Jest'' is a 1996 novel by American writer David Foster Wallace. Categorized as an encyclopedic novel, ''Infinite Jest'' is featured in ''TIME'' magazine's list of the 100 best English-language novels published between 1923 and 2005.
...
'' 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
Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints.
Despite wide popular interest, Escher was for most of his life neglected in t ...
, such as
Circle Limit III
''Circle Limit III'' is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and then "fall back again whence they came".Escher, as quoted by .
It is one of a series of fo ...
, 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
Fractal flames are a member of the iterated function system class of fractals created by Scott Draves in 1992. Draves' open-source code was later ported into Adobe After Effects graphics softwareChris Gehman and Steve Reinke (2005). ''The Sharp ...
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 antenna
A fractal antenna is an antenna that uses a fractal, self-similar design to maximize the effective length, or increase the perimeter (on inside sections or the outer structure), of material that can receive or transmit electromagnetic radiation ...
s
*Fractal transistor
* Fractal heat exchangers
* Digital imaging
* Architecture
* Urban growth
*
Classification Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood.
Classification is the grouping of related facts into classes.
It may also refer to:
Business, organizat ...
of
histopathology
Histopathology (compound of three Greek words: ''histos'' "tissue", πάθος ''pathos'' "suffering", and -λογία '' -logia'' "study of") refers to the microscopic examination of tissue in order to study the manifestations of disease. Spe ...
slides
*
Fractal landscape
A fractal landscape is a surface that is generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the result of the procedure is not a deterministic fractal surface, ...
or
Coast
The coast, also known as the coastline or seashore, is defined as the area where land meets the ocean, or as a line that forms the boundary between the land and the coastline. The Earth has around of coastline. Coasts are important zones in n ...
line
complexity
Complexity characterises the behaviour of a system or model whose components interaction, interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.
The term is generall ...
* Detecting 'life as we don't know it' by fractal analysis
* Enzymes (
Michaelis-Menten kinetics)
*
Generation of new music
*
Signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
and
image compression
Image compression is a type of data compression applied to digital images, to reduce their cost for storage or transmission. Algorithms may take advantage of visual perception and the statistical properties of image data to provide superior r ...
* Creation of digital photographic enlargements
*
Fractal in soil mechanics
A fractal is an irregular geometric object with an infinite nesting of structure at all scales. It is mainly applicable in soil chromatography and soil micromorphology (Anderson, 1997). Internal structure, pore size distribution and pore geometry ...
*
Computer and video game design
*
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 ...
*
Organic
Organic may refer to:
* Organic, of or relating to an organism, a living entity
* Organic, of or relating to an anatomical organ
Chemistry
* Organic matter, matter that has come from a once-living organism, is capable of decay or is the product ...
environments
*
Procedural generation
In computing, procedural generation is a method of creating data algorithmically as opposed to manually, typically through a combination of human-generated assets and algorithms coupled with computer-generated randomness and processing power. In ...
*
Fractography
Fractography is the study of the fracture surfaces of materials. Fractographic methods are routinely used to determine the cause of failure in engineering structures, especially in product failure and the practice of forensic engineering or fa ...
and
fracture mechanics
Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics t ...
*
Small angle scattering theory of fractally rough systems
*
T-shirt
A T-shirt (also spelled tee shirt), or tee, is a style of fabric shirt named after the T shape of its body and sleeves. Traditionally, it has short sleeves and a round neckline, known as a ''crew neck'', which lacks a collar. T-shirts are general ...
s and other fashion
* Generation of patterns for camouflage, such as
MARPAT
MARPAT (short for Marine pattern) is a multi-scale camouflage pattern in use with the United States Marine Corps, designed in 2001 and introduced from late 2002 to early 2005 with the Marine Corps Combat Utility Uniform (MCCUU), which replaced ...
*
Digital sundial
A digital sundial is a clock that indicates the current time with numerals formed by the sunlight striking it. Like a classical sundial, the device contains no moving parts. It uses no electricity nor other manufactured sources of energy. The di ...
* Technical analysis of price series
*
Fractals in networks
* Medicine
*
Neuroscience
Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, development ...
*
Diagnostic Imaging
Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to rev ...
*
Pathology
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in ...
* Geology
*
Geography
Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and ...
*
Archaeology
Archaeology or archeology is the scientific study of human activity through the recovery and analysis of material culture. The archaeological record consists of artifacts, architecture, biofacts or ecofacts, sites, and cultural landscap ...
*
Soil mechanics
Soil mechanics is a branch of soil physics and applied mechanics that describes the behavior of soils. It differs from fluid mechanics and solid mechanics in the sense that soils consist of a heterogeneous mixture of fluids (usually air and wat ...
*
Seismology
Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
*
Search and rescue
Search and rescue (SAR) is the search for and provision of aid to people who are in distress or imminent danger. The general field of search and rescue includes many specialty sub-fields, typically determined by the type of terrain the search ...
*
Technical analysis
In finance, technical analysis is an analysis methodology for analysing and forecasting the direction of prices through the study of past market data, primarily price and volume. Behavioral economics and quantitative analysis use many of the sam ...
*
Morton order
In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve, Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points. It i ...
space filling curves for
GPU
A graphics processing unit (GPU) is a specialized electronic circuit designed to manipulate and alter memory to accelerate the creation of images in a frame buffer intended for output to a display device. GPUs are used in embedded systems, mobil ...
cache coherency
In computer architecture, cache coherence is the uniformity of shared resource data that ends up stored in multiple local caches. When clients in a system maintain caches of a common memory resource, problems may arise with incoherent data, whi ...
in
texture mapping
Texture mapping is a method for mapping a texture on a computer-generated graphic. Texture here can be high frequency detail, surface texture, or color.
History
The original technique was pioneered by Edwin Catmull in 1974.
Texture mapping ...
,
rasterisation
In computer graphics, rasterisation (British English) or rasterization (American English) is the task of taking an image described in a vector graphics format (shapes) and converting it into a raster image (a series of pixels, dots or lines, whic ...
and indexing of turbulence data.
See also
*
Banach fixed point theorem
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certa ...
*
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Mo ...
*
Box counting
Box counting is a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. The essence of the pro ...
*
Cymatics
Cymatics (from grc, κῦμα, translit=kyma, translation=wave) is a subset of modal vibrational phenomena. The term was coined by Hans Jenny (1904-1972), a Swiss follower of the philosophical school known as anthroposophy. Typically the surf ...
*
Determinism
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
*
Diamond-square algorithm
The diamond-square algorithm is a method for generating heightmaps for computer graphics. It is a slightly better algorithm than the three-dimensional implementation of the midpoint displacement algorithm, which produces two-dimensional landscap ...
*
Droste effect
The Droste effect (), known in art as an example of ''mise en abyme'', is the effect of a picture recursively appearing within itself, in a place where a similar picture would realistically be expected to appear. This produces a loop which in ...
*
Feigenbaum function In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum:
* the solution to the Feigenbaum-Cvitanović functional equation; and
* the scaling fun ...
*
Form constant
A form constant is one of several geometric patterns which are recurringly observed during hypnagogia, hallucinations and altered states of consciousness.
History
In 1926, Heinrich Klüver systematically studied the effects of mescaline (peyote ...
*
Fractal cosmology
In physical cosmology, fractal cosmology is a set of minority cosmological theories which state that the distribution of matter in the Universe, or the structure of the universe itself, is a fractal across a wide range of scales (see also: multif ...
*
Fractal derivative
In applied mathematics and mathematical analysis, the fractal derivative or Hausdorff derivative is a non-Newtonian generalization of the derivative dealing with the measurement of fractals, defined in fractal geometry. Fractal derivatives were ...
*
Fractalgrid In electric power distribution, a fractalgrid is a system-of-systems architecture of distributed energy resources or DERs.
In a fractalgrid topology, multiple microgrids are strategically arranged to follow a fractal or recursive pattern. Fractals, ...
*
Fractal string
An ordinary fractal string \Omega is a bounded, open subset of the real number line. Such a subset can be written as an at-most-countable union of connected open intervals with associated lengths \mathcal=\ written in non-increasing order; we ...
*
Fracton
A fracton is a collective quantized vibration on a substrate with a fractal structure..
Fractons are the fractal analog of phonons. Phonons are the result of applying translational symmetry to the potential in a Schrödinger equation. Fractal s ...
*
Graftal
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 some ...
*
Greeble
A greeble ( ), or "nurnies", is a part harvested from plastic modeling kits to be applied to an original model as a detail element. The practice of using parts in this manner is called "kitbashing".
Etymology
The term "greeblies" was first use ...
*
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 beca ...
*
Lacunarity
Lacunarity, from the Latin lacuna, meaning "gap" or "lake", is a specialized term in geometry referring to a measure of how patterns, especially fractals, fill space, where patterns having more or larger gaps generally have higher lacunarity. B ...
*
List of fractals by Hausdorff dimension
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff dimension, Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."
Presented here is a list of fractals, ordered by increasing Hausdorff ...
*
Mandelbulb
The Mandelbulb is a three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis and in 2009 further developed by Daniel White and Paul Nylander using spherical coordinates.
A canonical 3-dimensional Mandelbrot set does not e ...
*
Mandelbox
In mathematics, the mandelbox is a fractal with a boxlike shape found by Tom Lowe in 2010. It is defined in a similar way to the famous Mandelbrot set as the values of a parameter such that the origin does not escape to infinity under iteration ...
*
Macrocosm and microcosm
*
Matryoshka doll
Matryoshka dolls ( ; rus, матрёшка, p=mɐˈtrʲɵʂkə, a=Ru-матрёшка.ogg), also known as stacking dolls, nesting dolls, Russian tea dolls, or Russian dolls, are a set of wooden dolls of decreasing size placed one inside ano ...
*
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 ...
*
Multifractal system
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. ...
*
Newton fractal
The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial or transcendental function. It is the Julia set of the meromorphic function which is given by Newton's method. ...
*
Percolation
Percolation (from Latin ''percolare'', "to filter" or "trickle through"), in physics, chemistry and materials science, refers to the movement and filtering of fluids through porous materials.
It is described by Darcy's law.
Broader applicatio ...
*
Power law
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
*
Publications in fractal geometry
*
Random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
*
Self-reference
Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding. In philoso ...
*
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 ...
*
Systems theory
Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structu ...
*
Strange loop
A strange loop is a cyclic structure that goes through several levels in a hierarchical system. It arises when, by moving only upwards or downwards through the system, one finds oneself back where one started.
Strange loops may involve self-refe ...
*
Turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
*
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
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;
Peitgen, Heinz-Otto; and Saupe, Dietmar; ''Chaos and Fractals: New Frontiers of Science''. New York: Springer-Verlag, 1992.
*
Mandelbrot, Benoit B.; ''
The Fractal Geometry of Nature
''The Fractal Geometry of Nature'' is a 1982 book by the Franco-American mathematician Benoît Mandelbrot.
Overview
''The Fractal Geometry of Nature'' is a revised and enlarged version of his 1977 book entitled ''Fractals: Form, Chance and Dimen ...
''. 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.
*
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
Sir Arthur Charles Clarke (16 December 191719 March 2008) was an English science-fiction writer, science writer, futurist, inventor, undersea explorer, and television series host.
He co-wrote the screenplay for the 1968 film '' 2001: A Spac ...
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
The Public Broadcasting Service (PBS) is an American public broadcasting, public broadcaster and Non-commercial activity, non-commercial, Terrestrial television, free-to-air television network based in Arlington, Virginia. PBS is a publicly fu ...
''
NOVA
A nova (plural novae or novas) is a transient astronomical event that causes the sudden appearance of a bright, apparently "new" star (hence the name "nova", which is Latin for "new") that slowly fades over weeks or months. Causes of the dramati ...
'', first aired August 24, 2011
Benoit Mandelbrot: Fractals and the Art of Roughness,
TED
TED may refer to:
Economics and finance
* TED spread between U.S. Treasuries and Eurodollar
Education
* ''Türk Eğitim Derneği'', the Turkish Education Association
** TED Ankara College Foundation Schools, Turkey
** Transvaal Education Depa ...
, February 2010
Technical Library on Fractals for controlling fluidEquations of self-similar fractal measure based on the fractional-order calculus2007)
{{Authority control
Mathematical structures
Topology
Computational fields of study