The Mandelbrot set is a famous example of a fractal. In Mathematics, the Mandelbrot set, named after Benoît

A closer view of the Mandelbrot set.

A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"^[1] a property called self-similarity. The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined In Mathematics, a self-similar object is exactly or approximately similar to a part of itself (i The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured. Benoît B Mandelbrot (born 20 November 1924 is a French mathematician, best known as the father of fractal geometry. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. "

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. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.
• It is self-similar (at least approximately or stochastically). In Mathematics, a self-similar object is exactly or approximately similar to a part of itself (i Stochastic (from the Greek "Στόχος" for "aim" or "guess" means Random.
• 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). In Mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative Real number associated In Mathematics, the Lebesgue covering dimension or topological dimension of a Topological space is defined to be the minimum value of n, such Space-filling curves or Peano curves are Curves first described by Giuseppe Peano (1858–1932 whose ranges contain the entire 2-dimensional Unit A Hilbert curve (also known as a Hilbert space-filling curve) is a continuous Fractal Space-filling curve first described by the German mathematician
• It has a simple and recursive definition. A recursive definition or inductive definition is one that defines something in terms of itself (that is recursively) albeit in a useful way ^[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. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a

History

Animated construction of a Sierpiński Triangle, only going nine generations of infinite—click for larger image. The Sierpiński triangle, also called the Sierpiński gasket or the Sierpiński Sieve, is a Fractal named after Wacław Sierpiński who described Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness

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. The Koch snowflake (or Koch star) is a mathematical Curve and one of the earliest Fractal curves to have been described " 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 1/3rd. Iteration means the act of repeating Mathematics Iteration in mathematics may refer to the process of iterating a function, or to the techniques used 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). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition

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 mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician In Mathematics, the Weierstrass function is a pathological example of a real -valued function on the Real line. Intuition is apparent ability to acquire knowledge without a clear inference or the use of reason In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, the Weierstrass function is a pathological example of a real -valued function on the Real line. 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. Niels Fabian Helge von Koch ( January 25, 1870 - March 11, 1924) was a Swedish Mathematician, who gave his name to the famous The Koch snowflake (or Koch star) is a mathematical Curve and one of the earliest Fractal curves to have been described In 1915, Waclaw Sierpinski constructed his triangle and, one year later, his carpet. Wacław Franciszek Sierpiński ( March 14 1882 — October 21 1969) (ˈvaʦwaf fraɲˈʨiʂɛk ɕɛrˈpʲiɲskʲi a Polish Mathematician The Sierpiński triangle, also called the Sierpiński gasket or the Sierpiński Sieve, is a Fractal named after Wacław Sierpiński who described The Sierpinski carpet is a plane Fractal first described by Wacław Sierpiński in 1916 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. Paul Pierre Lévy (15 September 1886 – 15 December 1971 was a French mathematician who was active especially in Probability theory, introducing martingales In Mathematics, the Lévy C curve is a Self-similar Fractal that was first described and whose differentiability properties were analysed by Ernesto

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 ( – January 6 1918) was a German Mathematician, born in Russia. In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith

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. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group Pierre Joseph Louis Fatou ( 28 February 1878, Lorient - 10 August 1929, Pornichet) was a French Mathematician Gaston Maurice Julia ( February 3, 1893 &ndash March 19, 1978) was a French mathematician who devised the formula for the 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 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. Benoît B Mandelbrot (born 20 November 1924 is a French mathematician, best known as the father of fractal geometry. How Long Is the Coast of Britain ? Statistical Self-Similarity and Fractional Dimension is a paper by Mathematician Benoît Mandelbrot Finally, in 1975 Mandelbrot coined the word "fractal" to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension. In Mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative Real number associated In Mathematics, the Lebesgue covering dimension or topological dimension of a Topological space is defined to be the minimum value of n, such 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".

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. In Complex dynamics, the Julia set J(f\ of a Holomorphic function f\ informally consists of those points whose long-time behavior under In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith The Sierpiński triangle, also called the Sierpiński gasket or the Sierpiński Sieve, is a Fractal named after Wacław Sierpiński who described The Sierpinski carpet is a plane Fractal first described by Wacław Sierpiński in 1916 In Mathematics, the Menger sponge is a Fractal curve It is the universal curve, in that it has Topological dimension one and any other 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 Space-filling curves or Peano curves are Curves first described by Giuseppe Peano (1858–1932 whose ranges contain the entire 2-dimensional Unit The Koch snowflake (or Koch star) is a mathematical Curve and one of the earliest Fractal curves to have been described Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. In Mathematics, Lyapunov fractals (also known as Markus-Lyapunov fractals) are bifurcational Fractals derived from an extension of the In Mathematics, a Kleinian group, named after Felix Klein, is a finitely generated Discrete group &Gamma of orientation preserving conformal Fractals can be deterministic (all the above) or stochastic (that is, non-deterministic). Determinism is the philosophical Proposition that every event including human cognition and behaviour decision and action is causally determined Stochastic (from the Greek "Στόχος" for "aim" or "guess" means Random. For example, the trajectories of the Brownian motion in the plane have a Hausdorff dimension of 2. This article is about the physical phenomenon for the stochastic process see Wiener process.

Chaotic dynamical systems are sometimes associated with fractals. In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that Objects in the phase space of a dynamical system can be fractals (see attractor). In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position An attractor is a set to which a Dynamical system evolves after a long enough time Objects in the parameter space for a family of systems may be fractal as well. In Generative art people talk about parameter space as the set of possibleparameters for a generative system An interesting example is the Mandelbrot set. In Mathematics, the Mandelbrot set, named after Benoît 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. For a different notion of boundary related to Manifolds see that article Mitsuhiro Shishikura ( Japanese: 宍倉 光広 Shishikura Mitsuhiro) is a Japanese Mathematician working in the field of Complex dynamics A closely related fractal is the Julia set. In Complex dynamics, the Julia set J(f\ of a Holomorphic function f\ informally consists of those points whose long-time behavior under

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 that coincides with

Generating fractals

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

Three common techniques for generating fractals are:

• Escape-time fractals — (also known as 'orbits' fractals) These are defined by a recurrence relation at each point in a space (such as the complex plane). "Difference equation" redirects here It should not be confused with a Differential equation. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal, the Nova fractal and the Lyapunov fractal. In Mathematics, the Mandelbrot set, named after Benoît In Complex dynamics, the Julia set J(f\ of a Holomorphic function f\ informally consists of those points whose long-time behavior under The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E Nova fractal refers to a family of fractals related to the newton fractal. In Mathematics, Lyapunov fractals (also known as Markus-Lyapunov fractals) are bifurcational Fractals derived from an extension of the Curiously, 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) is passed through this field repeatedly.
• Iterated function systems — These have a fixed geometric replacement rule. In Mathematics, iterated function systems or IFS s are a method of constructing Fractals the resulting constructions are always Self-similar. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals. In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith The Sierpinski carpet is a plane Fractal first described by Wacław Sierpiński in 1916 The Sierpiński triangle, also called the Sierpiński gasket or the Sierpiński Sieve, is a Fractal named after Wacław Sierpiński who described Space-filling curves or Peano curves are Curves first described by Giuseppe Peano (1858–1932 whose ranges contain the entire 2-dimensional Unit The Koch snowflake (or Koch star) is a mathematical Curve and one of the earliest Fractal curves to have been described 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 In Mathematics, the T-square is a two-dimensional Fractal. As all two-dimensional fractals it has a boundary of infinite length bounding a finite area In Mathematics, the Menger sponge is a Fractal curve It is the universal curve, in that it has Topological dimension one and any other
• Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapes and the Brownian tree. This article is about the physical phenomenon for the stochastic process see Wiener process. 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 " A fractal landscape is a surface generated using a Stochastic algorithm designed to produce Fractal behaviour which mimics the appearance of natural terrain 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 The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters. 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

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. In Mathematics, iterated functions are the objects of deep study in Computer science, Fractals and Dynamical systems An iterated function is
• 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. "Difference equation" redirects here It should not be confused with a Differential equation.
• 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. In Fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a Fractal appears to fill space as ) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar.

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. "Snowfall" redirects here For other uses see Snow (disambiguation or Snowfall (disambiguation. In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating A mountain is a Landform that extends above the surrounding Terrain in a limited area with a peak Lightning is an atmospheric discharge of Electricity, which typically occurs during Thunderstorms and sometimes during volcanic eruptions or "Riverine" redirects here For the use of that term in Maritime geography, see there Cauliflower is one of several vegetables in the species Brassica oleracea, in the family Brassicaceae. Broccoli is a plant of the Cabbage family Brassicaceae (formerly Cruciferae The blood vessels are part of the Circulatory system and function to transport Blood throughout the body Pulmonary circulation is the portion of the Cardiovascular system which carries Oxygen -depleted Blood away from the heart to the Lungs, and Coastlines may be loosely considered fractal in nature. How Long Is the Coast of Britain ? Statistical Self-Similarity and Fractional Dimension is a paper by Mathematician Benoît Mandelbrot

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. In Mathematics, iterated function systems or IFS s are a method of constructing Fractals the resulting constructions are always Self-similar. Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation 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. A frond is a large Leaf with many divisions to it and the term is typically used for the leaves of palms Ferns or Cycads A frond is

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). In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric A fractal antenna is an antenna that uses a Fractal, self-similar design to maximize the length or increase the perimeter (on inside ^[3]

Fractal pentagram drawn with a vector iteration program

In creative works

Fractal patterns have been found in the paintings of American artist Jackson Pollock. Early history Sumer The first known uses of the pentagram are found in Mesopotamian writings dating to about 3000 BC Iteration means the act of repeating Mathematics Iteration in mathematics may refer to the process of iterating a function, or to the techniques used Paul Jackson Pollock (January 28 1912 &ndash August 11 1956 was an influential American painter and a major force in the abstract expressionist movement While Pollock's paintings appear to be composed of chaotic dripping and splattering, computer analysis has found fractal patterns in his work. ^[4]

Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns. Decalcomania, from the French décalcomanie, is a decorative technique by which engravings and prints may be transferred to pottery or other materials Max Ernst ( 2 April 1891 &ndash 1 April 1976) was a German painter, Sculptor, Graphic artist, and ^[5] It involves pressing paint between two surfaces and pulling them apart.

Fractals are also prevalent in African art and architecture. African art constitutes one of the most diverse legacies on earth 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. ^[6]

A fractal is formed when pulling apart two glue-covered acrylic sheets. In Organic chemistry, the acryl group is the Functional group with structure H 2 C =CH-C(= O)- it is the Acyl group	High voltage breakdown within a 4″ block of acrylic creates a fractal Lichtenberg figure. Lichtenberg figures (Lichtenberg-Figuren or "Lichtenberg Dust Figures" are branching Electric discharges that sometimes appear on the surface or the interior of	Fractal branching occurs in a fractured surface such as a microwave-irradiated DVD[7]	Romanesco broccoli showing very fine natural fractals
A DLA cluster grown from a copper(II) sulfate solution in an electrodeposition cell	A "woodburn" fractal	A magnification of the phoenix set	Pascal generated fractal
A fractal flame created with the program Apophysis

Applications

As described above, random fractals can be used to describe many highly irregular real-world objects. DVD (also known as " Digital Versatile Disc " or " Digital Video Disc " - see Etymology)is Romanesco broccoli is an edible flower of the species Brassica oleracea and a variant form of Cauliflower. 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 is the Chemical compound with the formula Cu[[Sulfur S]] O 4 Fractal flames are a member of the Iterated function system class of Fractals created by Scott Draves in 1992 Other applications of fractals include:^[8]

• Classification of histopathology slides in medicine
• Fractal landscape or Coastline complexity
• Enzyme/enzymology (Michaelis-Menten kinetics)
• Generation of new music
• Generation of various art forms
• Signal and image compression
• Seismology
• Fractal in Soil Mechanics
• 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
• Small angle scattering theory of fractally rough systems
• Neo-hippies' t-shirts and other fashion
• Generation of patterns for camouflage, such as MARPAT
• Digital sundial
• Technical analysis of price series (see Elliott wave principle)

See also

References

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. Benoît B Mandelbrot (born 20 November 1924 is a French mathematician, best known as the father of fractal geometry. 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. Clifford A Pickover is an American author editor and columnist in the fields of Science, Mathematics, and Science fiction, and is employed at the 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. Jesse Holman Jones (also known as Jesse H Jones) ( April 5, 1874 &ndash June 1, 1956) was a Houston Texas politician 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. Sir Arthur Charles Clarke, CBE (16 December 1917–19 March 2008 was a British Science fiction Author, Inventor, and In Mathematics, the Mandelbrot set, named after Benoît
• Gouyet, Jean-François. Physics and Fractal Structures (Foreword by B. Mandelbrot); Masson, 1996. ISBN 2-225-85130-1, and New York: Springer-Verlag, 1996. ISBN 0-387-94153-1. Out-of-print. Available in PDF version at [1].

External links

• Fractals at the Open Directory Project