The boundary of the Mandelbrot set is a famous example of a fractal.

Another view of the Mandelbrot set.

A fractal is generally "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole,"^[1] a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured". Download high resolution version (1024x768, 189 KB)This is a file from the Wikimedia Commons, a repository of free content hosted by the Wikimedia Foundation. ... Download high resolution version (1024x768, 189 KB)This is a file from the Wikimedia Commons, a repository of free content hosted by the Wikimedia Foundation. ... Initial image of a Mandelbrot set zoom sequence with continuously coloured environment The Mandelbrot set is a set of points in the complex plane that forms a fractal. ... Image File history File links Download high-resolution version (1280x960, 380 KB) Mandelbrot set. ... Image File history File links Download high-resolution version (1280x960, 380 KB) Mandelbrot set. ... Look up shape in Wiktionary, the free dictionary. ... A self-similar object is exactly or approximately similar to a part of itself. ... Benoît B. Mandelbrot, PhD, (born November 20, 1924) is a Franco-American mathematician, best known as the father of fractal geometry. Benoît Mandelbrot was born in Poland, but his family moved to France when he was a child; he is a dual French and American citizen and was... For other uses, see Latin (disambiguation). ...

A fractal often has the following features:

• It has a fine structure at arbitrarily small scales.
• It is too irregular to be easily described in traditional Euclidean geometric language.
• It is self-similar (at least approximately or stochastically).
• It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
• It has a simple and recursive definition.^[2]

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics. A self-similar object is exactly or approximately similar to a part of itself. ... Stochastic, from the Greek stochos or goal, means of, relating to, or characterized by conjecture; conjectural; random. ... In mathematics, the Hausdorff dimension is an extended non-negative real number associated to any metric space. ... In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement in which no point is included in more than n+1 elements. ... Hilbert curve, first order Hilbert curves, first and second order Hilbert curves, first to third order A Hilbert curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891. ... A recursive definition is one that defines something in terms of itself, albeit in a useful way. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...

History

To create a Koch snowflake, start with an equilateral triangle and replace the middle third of every line segment with a pair of line segments that form an equilateral "bump." Then perform the same replacement on every line segment of the resulting shape, ad infinitum. With every iteration, the perimeter of this shape grows by 4/3rds. The Koch snowflake is the result of an infinite number of these iterations, and has an infinite length, while its area remains finite. For this reason, the Koch snowflake and similar constructions were sometimes called "monster curves."

The mathematics behind fractals began to take shape in the 17th century when philosopher Leibniz considered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense). Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The first four iterations of the Koch snowflake. ... The word iteration is sometimes used in everyday English with a meaning virtually identical to repetition. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Leibniz redirects here. ... This article is about the concept of recursion. ...

It took until 1872 before a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable. In 1904, Helge von Koch, dissatisfied with Weierstrass's very abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch snowflake. In 1915, Waclaw Sierpinski constructed his triangle and, one year later, his carpet. Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions. The idea of self-similar curves was taken further by Paul Pierre 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. Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... In BIOLOGY, the SUMMER VACATION function was the first example found of a Chumba wumbafunction with the property that it is continuous everywhere but differentiable nowhere. ... Intuition is an unconscious form of knowledge. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Weierstrass function may also refer to the Weierstrass elliptic function () or the Weierstrass sigma, zeta, or eta functions. ... Niels Fabian Helge von Koch (January 25, 1870 - March 11, 1924) was a Swedish mathematician, who gave his name to the famous fractal known as the Koch curve, which was one of the earliest fractal curves to have been described. ... The first four iterations of the Koch snowflake. ... Wacław Franciszek Sierpiński, was born on March 14, 1882 in Warsaw and died on October 21, 1969 in Warsaw. ... Sierpinski triangle The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after WacÅ‚aw SierpiÅ„ski who described it in 1916. ... The Sierpinski carpet is a plane fractal first described by WacÅ‚aw SierpiÅ„ski. ... Paul Pierre Lévy (September 15, 1886 - December 15, 1971) was a French mathematician who was active especially in probability theory, introduced martingales and Lévy flights. ... In mathematics, the Lévy C curve is a self similar fractal that was first described and whose differentiability properties were analysed by E.Cesaro in 1906 and G. Farber in 1910, but now bears the name of French mathematician Paul Lévy, who was the first to describe its...

Georg Cantor also gave examples of subsets of the real line with unusual properties—these Cantor sets are also now recognized as fractals. Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ... “Superset” redirects here. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

Iterated functions in the complex plane were investigated in the late 19th and early 20th centuries by Henri Poincaré, Felix Klein, Pierre Fatou and Gaston Julia. However, without the aid of modern computer graphics, they lacked the means to visualize the beauty of many of the objects that they had discovered. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Jules TuPac Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Felix Christian Klein (April 25, 1849, Düsseldorf, Germany – June 22, 1925, Göttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... Pierre Fatou was the first to define the Mandelbrot set. ... Gaston Maurice Julia (February 3, 1893 – March 19, 1978) was a French mathematician who devised the formula for the Julia set. ...

In the 1960s, Benoît Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which built on earlier work by Lewis Fry Richardson. Finally, in 1975 Mandelbrot coined the word "fractal" to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal". Benoît B. Mandelbrot, PhD, (born November 20, 1924) is a Franco-American mathematician, best known as the father of fractal geometry. Benoît Mandelbrot was born in Poland, but his family moved to France when he was a child; he is a dual French and American citizen and was... How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension is a paper by mathematician Benoît Mandelbrot, first published in Science in 1967. ... For the Hollyoaks character, see Lewis Richardson (Hollyoaks) Lewis Fry Richardson (October 11, 1881 - September 30, 1953) was an innovative mathematician, physicist and psychologist. ... In mathematics, the Lebesgue covering dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement with no point included in more than n+1 elements. ...

Examples

A Julia set, a fractal related to the Mandelbrot set

A relatively simple class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, and Koch curve. Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. Fractals can be deterministic (all the above) or stochastic (that is, non-deterministic). For example, the trajectories of the Brownian motion in the plane have a Hausdorff dimension of 2. Download high resolution version (1024x768, 564 KB)A julia set with seed coordinates (-0. ... Download high resolution version (1024x768, 564 KB)A julia set with seed coordinates (-0. ... In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-time behavior under repeated iteration of can change drastically under arbitrarily small perturbations. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... Sierpinski triangle The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Wacław Sierpiński who described it in 1916. ... The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński. ... In mathematics, the Menger sponge is a fractal curve. ... A dragon curve is the generic name for any member of a family of self similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. ... Space-filling curves or Peano curves are curves, first described by Giuseppe Peano, whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube). ... The first four iterations of the Koch snowflake. ... Standard Lyapunov logistic fractal with iteration sequence AB Generalized Lyapunov logistic fractal with iteration sequence AABAB 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... In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i. ... The term deterministic may refer to: the more general notion of determinism from philosophy, see determinism a type of algorithm as discussed in computer science, see deterministic algorithm scientific determinism as used by Karl Popper and Stephen Hawking deterministic system in mathematics deterministic system in philosophy deterministic finite state machine... Stochastic, from the Greek stochos or goal, means of, relating to, or characterized by conjecture; conjectural; random. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ...

Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals (see attractor). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so it has a Hausdorff dimension equal to its topological dimension of 2—but what is truly surprising is that the boundary of the Mandelbrot set also has a Hausdorff dimension of 2 (while the topological dimension of 1), a result proved by Mitsuhiro Shishikura in 1991. A closely related fractal is the Julia set. For other uses, see Chaos Theory (disambiguation). ... Phase space of a dynamical system with focal stability. ... The Lorenz attractor is an example of a non-linear dynamical system. ... In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ... In statistics one can study the distribution of a random variable. ... Initial image of a Mandelbrot set zoom sequence with continuously coloured environment The Mandelbrot set is a set of points in the complex plane that forms a fractal. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of... Mitsuhiro Shishikura (Japanese: 宍倉 光広 Shishikura Mitsuhiro) is a Japanese mathematician working in the field of complex dynamics. ... In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-time behavior under repeated iteration of can change drastically under arbitrarily small perturbations. ...

Even simple smooth curves can exhibit the fractal property of self-similarity. For example the power-law curve (also known as a Pareto distribution) produces similar shapes at various magnifications. A power law is any polynomial relationship that exhibits the property of scale invariance. ... The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ...

Generating fractals

Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set.

Three common techniques for generating fractals are: Image File history File links Mandelbrot-similar-x1. ... Image File history File links Mandelbrot-similar-x6. ... Image File history File links Mandelbrot-similar-x100. ... Image File history File links Mandelbrot-similar-x2000. ...

• Escape-time fractals — These are defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal and the Lyapunov fractal.
• Iterated function systems — These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals.
• Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters.

In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Initial image of a Mandelbrot set zoom sequence with continuously coloured environment The Mandelbrot set is a set of points in the complex plane that forms a fractal. ... In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-time behavior under repeated iteration of can change drastically under arbitrarily small perturbations. ... Main body of the Burning Ship fractal; window-coodinates: -1. ... Standard Lyapunov logistic fractal with iteration sequence AB Generalized Lyapunov logistic fractal with iteration sequence AABAB 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... Menger sponge, created by using IFS. Iterated function systems or IFSs, are a kind of fractal which were conceived in their present form by John Hutchinson in 1981 and popularized by Michael Barnsleys book Fractals Everywhere. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński. ... The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Waclaw Sierpinski. ... Intuitively, a continuous curve in the 2-dimensional plane or in the 3-dimensional space can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the... The first four iterations of the Koch snowflake. ... A dragon curve is the generic name for any member of a family of self similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. ... The T-Square is a fractal curve of infinite length inside finite area. ... In mathematics, the Menger sponge is a fractal curve. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... A Lévy flight, named after the French mathematician Paul Pierre Lévy, is a type of random walk in which the increments are distributed according to a heavy tail distribution. ... A fractal landscape is essentially a two-dimensional form of the fractal coastline, which can be considered a stochastic generalization of the Koch curve. ... A Brownian tree example A Brownian tree, whose name is derived from Robert Brown via Brownian motion, is a form of computer art that was briefly popular in the 1990s, when home computers started to have sufficient power to simulate Brownian motion. ... A DLA cluster grown from a copper sulfate solution in an electrodeposition cell 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. ...

Classification of fractals

Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:

• Exact self-similarity — This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity.
• Quasi-self-similarity — This is a loose form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar but not exactly self-similar.
• Statistical self-similarity — This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of "fractal" trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar.

In mathematics, iterated functions are the objects of study in fractals and dynamical systems. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

Fractals in nature

A fractal that models the surface of a mountain (animation)

Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakes, crystals, mountain ranges, lightning, river networks, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels. Coastlines may be loosely considered fractal in nature. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... For other uses, see Snow (disambiguation). ... For other uses, see Crystal (disambiguation). ... For other uses, see Mountain (disambiguation). ... Not to be confused with lighting. ... For other uses, see River (disambiguation). ... Cauliflower within Brassica oleracea, in the family Brassicaceae. ... Broccoli is a plant of the Cabbage family, Brassicaceae (formerly Cruciferae). ... f you all The blood vessels are part of the circulatory system and function to transport blood throughout the body. ... It has been suggested that Pulmonary loop be merged into this article or section. ...

A fractal fern computed using an Iterated function system

Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples — a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. Image File history File links Download high resolution version (600x1000, 232 KB) Summary Made by own hands and own C++ program in 3·106 iterations by Kimbar 13:02, 30 December 2005 (UTC). ... Image File history File links Download high resolution version (600x1000, 232 KB) Summary Made by own hands and own C++ program in 3·106 iterations by Kimbar 13:02, 30 December 2005 (UTC). ... Menger sponge, created by using IFS. Iterated function systems or IFSs, are a kind of fractal which were conceived in their present form by John Hutchinson in 1981 and popularized by Michael Barnsleys book Fractals Everywhere. ... This article is about the concept of recursion. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... A fern with simple (lobed or pinnatifid) blades, the dissection of each blade not quite reaching to the rachis. ...

In 1999, certain self similar fractal shapes were shown to have a property of "frequency invariance" — the same electromagnetic properties no matter what the frequency — from Maxwell's equations (see fractal antenna).^[3] For thermodynamic relations, see Maxwell relations. ... A fractal antenna is an antenna that uses a [[self similar] design to maximize the length, or increase the perimeter (on inside sections or the outer structure), of material that can receive or transmit electromagnetic signals within a given total surface area. ...

Fractal pentagram drawn with a vector iteration program

Image File history File links Size of this preview: 600 × 600 pixelsFull resolution‎ (650 × 650 pixels, file size: 95 KB, MIME type: image/png) Designed by Vishva Kumara on a VB.net fractal application. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution‎ (650 × 650 pixels, file size: 95 KB, MIME type: image/png) Designed by Vishva Kumara on a VB.net fractal application. ... A pentagram A pentagram (sometimes known as a pentalpha or pentangle or, more formally, as a star pentagon) is the shape of a five-pointed star drawn with five straight strokes. ... Look up vector in Wiktionary, the free dictionary. ... The word iteration is sometimes used in everyday English with a meaning virtually identical to repetition. ...

Fractals in art

Fractal patterns have been found in the paintings of American artist Jackson Pollock. While Pollock's paintings appear to be composed of chaotic dripping and splattering, computer analysis has found fractal patterns in his work.^[4] Controversy swirls over the alleged sale of No. ...

Fractals are also prevalent in African art 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.^[5] Yoruba bronze head sculpture, Ife, Nigeria c. ...

A fractal is formed when pulling apart two glue-covered acrylic sheets. Natural fractal pattern - air displacing a vacuum formed by pulling two glue-covered acrylic sheets apart. ... The acryl group is one of the functional groups sorted in the chemical class of acryl where one of four hydrogen atoms in ethene is replaced with a different functional group. ...	High voltage breakdown within a 4″ block of acrylic creates a fractal Lichtenberg figure. Download high resolution version (800x726, 194 KB)Lichtenberg Figure High voltage dielectric breakdown within a block of plexiglas creates a beautiful fractal pattern called a Lichtenberg_figure. ... Lichtenberg figures are named after the German physicist Georg Christoph Lichtenberg, who originally discovered and studied them. ...	Fractal branching occurs in a fractured surface such as a microwave-irradiated DVD[6] Download high resolution version (1800x1813, 1802 KB) DVD irradiated for a shorter amount of time, showing how fractal branching occurs. ... DVD (also known as Digital Versatile Disc or Digital Video Disc) is a popular optical disc storage media format. ...	Romanesco broccoli showing very fine natural fractals Download high resolution version (1024x768, 125 KB)Romanesco broccoli fractals. ... Romanesco Broccoli, showing fractal forms Romanesco broccoli or fractal broccoli is a vegetable member of the Brassica oleracea L. species. ...
A DLA cluster grown from a copper sulfate solution in an electrodeposition cell Image File history File linksMetadata Download high-resolution version (996x672, 165 KB) [edit] Summary [edit] Licensing File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Fractal Brownian tree Diffusion-limited aggregation Metadata This file contains additional information, probably added... A DLA cluster grown from a copper sulfate solution in an electrodeposition cell 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. ... Copper (II) sulfate (CuSO4) is the most common copper salt, made by the action of sulfuric acid on the base copper oxide. ... Electroplating is the the coating of an electrically conductive item with a layer of metal using electrical current. ...	A magnification of the phoenix set Image File history File links No higher resolution available. ...	Pascal generated fractal Image File history File links Size of this preview: 800 × 600 pixelsFull resolution‎ (858 × 643 pixels, file size: 882 KB, MIME type: image/png) File historyClick on a date/time to view the file as it appeared at that time. ...	A fractal flame created with the program Apophysis (software) Image File history File links Size of this preview: 779 × 599 pixelsFull resolution‎ (2,560 × 1,970 pixels, file size: 4. ... Categories: Fractals | Math stubs ... For other uses, see Apophysis. ...

Applications

As described above, random fractals can be used to describe many highly irregular real-world objects. Other applications ^[7] of fractals include:

• Classification of histopathology slides in medicine
• Enzyme/enzymology (Michaelis-Menten kinetics)
• Generation of new music
• Generation of various art forms
• Signal and image compression
• Seismology
• Computer and video game design, especially computer graphics for organic environments and as part of procedural generation
• Fractography and fracture mechanics
• Fractal antennas — Small size antennas using fractal shapes
• Neo-hippies t-shirts and other fashion
• Generation of patterns for camouflage, such as MARPAT
• Digital sundial
• Generation of Price Series

For Wikipedias categorization projects, see Wikipedia:Categorization. ... Histopathology is a field of pathology which specialises in the histologic study of diseased tissue. ... For the chemical substances known as medicines, see medication. ... Michaelis-Menten kinetics describes the kinetics of many enzymes. ... This article is about the philosophical concept of Art. ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ... Fractal compression is a lossy compression method used to compress images using fractals. ... Seismology (from the Greek seismos = earthquake and logos = word) is the scientific study of earthquakes and the propagation of elastic waves through the Earth. ... Game design is the process of designing the content and rules of a game. ... This article is about the scientific discipline of computer graphics. ... For other uses, see Life (disambiguation). ... Procedural generation is a widely used term to indicate the possibility to create content on the fly, as opposed to creating it before distribution. ... This does not cite its references or sources. ... A fractal antenna is an antenna that uses a [[self similar] design to maximize the length, or increase the perimeter (on inside sections or the outer structure), of material that can receive or transmit electromagnetic signals within a given total surface area. ... T-Shirt A T-shirt (or tee shirt) is a shirt with short or long sleeves, a round neck, put on over the head, without pockets. ... For other uses, see Fashion (disambiguation). ... General Hagee (CMC) in MARPAT combat utilities Marines wearing woodland MARPAT during Exercise Talisman Saber 2007 at Shoalwater Bay, Australia. ... Digital sundial is a clock that indicates the current time with numerals formed by the sunlight. ...

See also

• Bifurcation theory
• Butterfly effect
• Chaos theory
• Complexity
• Constructal theory
• Contraction mapping theorem
• Diamond-square algorithm
• Droste effect
• Feigenbaum function
• Fractal art
• Fractal compression
• Fractal flame
• Fractal landscape
• Fracton
• Graftal
• List of fractals by Hausdorff dimension
• Publications in fractal geometry
• Newton fractal
• Recursion
• Recursionism
• Reentrant
• Sacred geometry
• Self-reference
• Strange loop
• Turbulence

In mathematics, specifically in the study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden qualitative or topological change in the systems long-term dynamical behaviour. ... Point attractors in 2D phase space. ... For other uses, see Chaos Theory (disambiguation). ... Complexity in general usage is the opposite of simplicity. ... Constructal design of a cooling system The constructal theory of global optimization under local constraints explains in a simple manner the shapes that arise in nature. ... The Banach fixed point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points. ... The Diamond-Square Algorithm is a method for generating highly realistic heightmaps for computer graphics. ... The Droste effect is a Dutch term for a specific kind of recursive picture[1], one that in heraldry is termed mise en abyme. ... Feigenbaum functions are iterating nonlinear functions discovered by the mathematician Mitchell Feigenbaum. ... Computer-generated fractal image. ... Fractal compression is a lossy compression method used to compress images using fractals. ... Categories: Fractals | Math stubs ... A fractal landscape is essentially a two-dimensional form of the fractal coastline, which can be considered a stochastic generalization of the Koch curve. ... There are very few or no other articles that link to this one. ... A graftal or L-system is a formal grammar used in computer graphics to recursively define branching tree and plant shapes in a compact format. ... A fractal is a geometric object whose Hausdorff dimension (δ) strictly exceeds its topological dimension. ... This is a list of important publications in mathematics, organized by field. ... The Newton fractal is a boundary set in the complex plane which is characterized by Newtons method applied to a fixed polynomial p(Z)∈ℂ[Z]. It divides the complex plane into regions Gk, each of which is associated with a root ζk of the polynomial, . In this way the... This article is about the concept of recursion. ... Recursionism means a variety of things to different people. ... A computer program or routine is described as reentrant if it can be safely called recursively or from multiple processes. ... The Parthenons facade showing an interpretation of golden rectangles in its proportions. ... A self-reference occurs when an object refers to itself. ... M.C. Escher - Drawing Hands, 1948. ... In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic, stochastic property changes. ...

References

1. ^ Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W. H. Freeman and Company.. ISBN 0-7167-1186-9. 
2. ^ Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd., xxv. ISBN 0-470-84862-6. 
3. ^ Hohlfeld,R., and Cohen, N.,"SELF-SIMILARITY AND THE GEOMETRIC REQUIREMENTS FOR FREQUENCY INDEPENDENCE IN ANTENNAE ", Fractals, Vol. 7, No. 1 (1999) 79-84
4. ^ Richard Taylor, Adam P. Micolich and David Jonas. Fractal Expressionism : Can Science Be Used To Further Our Understanding Of Art?
5. ^ Ron Eglash. African Fractals: Modern Computing and Indigenous Design. New Brunswick: Rutgers University Press 1999.
6. ^ Peng, Gongwen; Decheng Tian (21 July 1990). "The fractal nature of a fracture surface". Journal of Physics A (14): 3257-3261. doi:10.1088/0305-4470/23/14/022. Retrieved on 2007-06-02. 
7. ^ Applications. Retrieved on 2007-10-21.

Journal of Physics A is a peer-reviewed scientific journal published by the Institute of Physics (IOP) in the United Kingdom. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 153rd day of the year (154th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 294th day of the year (295th in leap years) in the Gregorian calendar. ...

Further reading

• Barnsley, Michael F., and Hawley Rising. Fractals Everywhere. Boston: Academic Press Professional, 1993. ISBN 0-12-079061-0
• Falconer, Kenneth. Techniques in Fractal Geometry. John Willey and Sons, 1997. ISBN 0-471-92287-0
• Jürgens, Hartmut, Heins-Otto Peitgen, and Dietmar Saupe. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992. ISBN 0-387-97903-4
• Benoît B. Mandelbrot The Fractal Geometry of Nature. New York: W. H. Freeman and Co., 1982. ISBN 0-7167-1186-9
• Peitgen, Heinz-Otto, and Dietmar Saupe, eds. The Science of Fractal Images. New York: Springer-Verlag, 1988. ISBN 0-387-96608-0
• Clifford A. Pickover, ed. Chaos and Fractals: A Computer Graphical Journey - A 10 Year Compilation of Advanced Research. Elsevier, 1998. ISBN 0-444-50002-2
• Jesse Jones, Fractals for the Macintosh, Waite Group Press, Corte Madera, CA, 1993. ISBN 1-878739-46-8.
• Hans Lauwerier, Fractals: Endlessly Repeated Geometrical Figures, Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991. ISBN 0-691-08551-X, cloth. ISBN 0-691-02445-6 paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.
• Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850839-5 and ISBN 978-0-19-850839-7. 
• Bernt Wahl, Peter Van Roy, Michael Larsen, and Eric Kampman Exploring Fractals on the Macintosh, Addison Wesley, 1995. ISBN 0-201-62630-6
• Nigel Lesmoir-Gordon. "The Colours of Infinity: The Beauty, The Power and the Sense of Fractals." ISBN 1-904555-05-5 (The book comes with a related DVD of the Arthur C. Clarke documentary introduction to the fractal concept and the Mandelbrot set.

Benoît Mandelbrot in 2006 Benoît B. Mandelbrot (born November 20, 1924) is a French mathematician, best known as the father of fractal geometry. Benoît Mandelbrot was born in Poland, but his family moved to France when he was a child; he is a French citizen and was... Clifford A. Pickover is an author, editor, and columnist in the fields of science, mathematics, and science fiction. ... Jesse Holman Jones Jesse Holman Jones (also known as Jesse H. Jones) (April 5, 1874 – June 1, 1956) was a Houston, Texas politician and entrepreneur. ... Wikiquote has a collection of quotations related to: Arthur C. Clarke Sir Arthur Charles Clarke, CBE (born 16 December 1917) is a British science-fiction author and inventor, most famous for his novel 2001: A Space Odyssey, and for collaborating with director Stanley Kubrick on the film of the same... Initial image of a Mandelbrot set zoom sequence with continuously coloured environment The Mandelbrot set is a set of points in the complex plane that forms a fractal. ...

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Fractal. ... Image File history File links Sound-icon. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... is the 167th day of the year (168th in leap years) in the Gregorian calendar. ... Image File history File links Sound-icon. ... Image File history File links Commons-logo. ... Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... The Open Directory Project (ODP), also known as dmoz (from , its original domain name), is a multilingual open content directory of World Wide Web links owned by Netscape that is constructed and maintained by a community of volunteer editors. ...