A fractal is "a rough or fragmented geometric shape The shape of an object located in some space is the part of that space occupied by the object, as determined by its external boundary – abstracting from other properties such as colour, content, and material composition, as well as from the object's other spatial properties (position and orientation in space; size) 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 In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Roots of mathematically rigorous treatment of fractals can be traced back to functions studied by Karl Weierstrass Karl Theodor Wilhelm Weierstrass (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the "father of modern analysis", Georg Cantor Georg Ferdinand Ludwig Philipp Cantor was a mathematician, best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the and Felix Hausdorff in studying functions that were continuous In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". An intuitive though imprecise idea of continuity is but not differentiable In calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain. As a result, the graph of a differentiable function must be relatively smooth, and cannot contain any breaks, bends, or cusps, or any; however, the term fractal was coined by Benoît Mandelbrot Benoît B. Mandelbrot is a French and American mathematician, best known as the father of fractal geometry. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Center; and Battelle Fellow at the Pacific Northwest National Laboratory. Mandelbrot was born in Poland. His in 1975 and was derived from the Latin Latin or sometimes Roman is an Italic language originally spoken in Latium and Ancient Rome. Although often considered a dead language, in view of the fact that it has no native speakers, a small number of scholars can fluently speak it and it continues to be taught in schools and universities and has been, and currently is, used in the process of fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation An equation is a mathematical statement that asserts the equality of two expressions. Equations consist of the expressions that have to be equal on opposite sides of an equal sign, as in that undergoes iteration Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration", and the results of one iteration are used as the starting point for the next iteration, a form of feedback Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same (i.e. same defined) event / phenomenon (or the continuation / development of the original phenomenon) in the present or future. When an event is part of a chain of cause-and-effect that forms a circuit or based on recursion Recursion, in mathematics and computer science, is a method of defining functions in which the function being defined is applied within its own definition; specifically it is defining an infinite statement using finite components. The term is also used more generally to describe a process of repeating objects in a self-similar way. For instance,.[2]
A fractal often has the following features:[3]
- It has a fine structure at arbitrarily small scales.
- It is too irregular to be easily described in traditional Euclidean geometric Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, whose Elements is the earliest known systematic discussion of geometry. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Although many of Euclid's results had been language.
- It is self-similar In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals (at least approximately or stochastically Stochastic means random. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E. Nelson, any kind of time development (be it deterministic or essentially probabilistic) which is).
- It has a Hausdorff dimension In mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional vector space equals n. This means, for example the Hausdorff dimension of a point is zero, the which is greater than its topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space X is defined to be the minimum value of n, such that every open cover of X has an open refinement in which no point is included in more than n+1 elements. If no such minimal n exists, the space is said to be of infinite covering dimension (although this requirement is not met by space-filling curves In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square . Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are commonly called Peano curves such as the Hilbert curve A Hilbert 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 curves discovered by Giuseppe Peano in 1890).[4]
- It has a simple and recursive definition A recursive definition or inductive definition is one that defines something in terms of itself , albeit in a useful way. For it to work, the definition in any given case must be well-founded, avoiding an infinite regress.
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that are approximated by fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns. However, not all self-similar objects are fractals—for example, the real line In mathematics, the real line is the line whose points correspond to the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space. It is the Euclidean space of dimension one, and can be thought of as a vector space , a metric space, a topological space, or simply as a linear continuum (a straight Euclidean In Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely long. Lines are a fundamental concept in some line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms.
Images of fractals can be created using fractal-generating software Many different features are included in fractal-generating software packages. Most feature some form of algorithm selection, an interactive image zoom, and the ability to save files in JPEG, TIFF or PNG format, as well as the ability to save parameter files, allowing the user to easily return to previously created images for later modification or. Images produced by such software are normally referred to as being fractals even if they do not have the above characteristics, such as when it is possible to zoom into a region of the fractal that does not exhibit any fractal properties. Also, these may include calculation or display artifacts In natural science and signal processing, an artifact is any error in the perception or representation of any visual or aural information introduced by the involved equipment which are not characteristics of true fractals.
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History
To create a Koch snowflake The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a continuous curve without tangents, constructible from elementary geometry" (original French title: "Sur une courbe continue sans tangente, obtenue par, one begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral "bump." One then performs the same replacement on every line segment of the resulting shape, ad infinitum. With every iteration Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration", and the results of one iteration are used as the starting point for the next iteration, the perimeter of this shape increases by one third of the previous length. The Koch snowflake Snowflakes are conglomerations of frozen ice crystals which fall through the Earth's atmosphere. They begin as two snow crystals which develop when microscopic supercooled cloud droplets freeze. Snowflakes come in a variety of sizes and shapes. Complex shapes emerge as the flake moves through differing temperature and humidity regimes. Individual 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 Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions behind fractals began to take shape in the 17th century when mathematician and philosopher Gottfried Leibniz Gottfried Wilhelm Leibniz (German pronunciation: [ˈɡɔtfʁiːt ˈvɪlhɛlm fɔn ˈlaɪpnɪts] (July 1, 1646 - June 21, 1716) was a German mathematician and philosopher. He wrote primarily in Latin and French considered recursive Recursion, in mathematics and computer science, is a method of defining functions in which the function being defined is applied within its own definition; specifically it is defining an infinite statement using finite components. The term is also used more generally to describe a process of repeating objects in a self-similar way. For instance, self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).
It was not until 1872 that a function appeared whose graph In mathematics, the graph of a function f is the collection of all ordered pairs (x, f). In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching. If the would today be considered fractal, when Karl Weierstrass Karl Theodor Wilhelm Weierstrass (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the "father of modern analysis" gave an example of a function with the non-intuitive Intuition is the apparent ability to acquire knowledge without inference or the use of reason. “The word ‘intuition’ comes from the Latin word 'intueri', which is often roughly translated as meaning ‘to look inside’ or ‘to contemplate’." Intuition provides us with beliefs that we cannot necessarily justify. For this reason, it property of being everywhere continuous In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". An intuitive though imprecise idea of continuity is but nowhere differentiable. In 1904, Helge von Koch, dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch curve The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a continuous curve without tangents, constructible from elementary geometry" (original French title: "Sur une courbe continue sans tangente, obtenue par. (The image at right is three Koch curves put together to form what is commonly called the Koch snowflake The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a continuous curve without tangents, constructible from elementary geometry" (original French title: "Sur une courbe continue sans tangente, obtenue par.) Waclaw Sierpinski constructed his triangle in 1915 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 Paul Pierre Lévy was a French mathematician who was active especially in probability theory, introducing martingales and Lévy flights. Lévy processes, Lévy measures, Lévy's constant, the Lévy distribution, the Lévy skew alpha-stable distribution, the Lévy area and the fractal Lévy C curve are also named after him, 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. Georg Cantor Georg Ferdinand Ludwig Philipp Cantor was a mathematician, best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the also gave examples of subsets In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B of the real line with unusual properties—these Cantor sets In mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883 , is a set of points lying on a single line segment that has a number of remarkable and deep properties. Through consideration of it, Cantor and others helped lay the foundations of modern general topology. Although Cantor himself defined the set in a general, are also now recognized as fractals.
Iterated functions in the complex plane In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along were investigated in the late 19th and early 20th centuries by Henri Poincaré Jules Henri Poincaré (French pronunciation: [ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe]) was a French mathematician, theoretical physicist, and a philosopher of science. Poincaré is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime, Felix Klein Felix Christian Klein 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. His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day, Pierre Fatou Pierre Joseph Louis Fatou was a French mathematician working in the field of complex analytic dynamics. He entered the École Normale Supérieure in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an astronomy post in the Paris Observatory. Fatou continued his mathematical explorations and studied iterative and and Gaston Julia. Without the aid of modern computer graphics, however, they lacked the means to visualize the beauty of many of the objects that they had discovered.
In the 1960s, Benoît Mandelbrot Benoît B. Mandelbrot is a French and American mathematician, best known as the father of fractal geometry. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Center; and Battelle Fellow at the Pacific Northwest National Laboratory. Mandelbrot was born in Poland. His started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension is a paper by mathematician Benoît Mandelbrot, first published in Science in 1967. In this paper Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this, 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 In mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional vector space equals n. This means, for example the Hausdorff dimension of a point is zero, the is greater than its topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space X is defined to be the minimum value of n, such that every open cover of X has an open refinement in which no point is included in more than n+1 elements. If no such minimal n exists, the space is said to be of infinite covering 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".
Examples
A Julia set In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such, a fractal related to the Mandelbrot setA class of examples is given by the Cantor sets In mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883 , is a set of points lying on a single line segment that has a number of remarkable and deep properties. Through consideration of it, Cantor and others helped lay the foundations of modern general topology. Although Cantor himself defined the set in a general,, 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.
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.
Generating fractals
| Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set. |
Four common techniques for generating fractals are:
- Escape-time fractals – (also known as "orbits" fractals) These are defined by a formula or 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, the Nova fractal and the Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.
- Iterated function systems – These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Highway 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.
- Strange attractors – Generated by iteration of a map or the solution of a system of initial-value differential equations that exhibit chaos.
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Wed, 07 Jul 2010 23:10:12 GMT+00:00
in Ghana Nanotechnology and Development News [More] Finding and Teaching Fractals in Ghana -- The New York Times (7/7/2010) This article, by Ron Eglash, a professor of science and technology studies at ...
Daniel Shultz
Wed, 14 Jul 2010 23:17:38 GM
I just got an email from Will Murray, a juggler and professor of mathematics at Cal State Long Beach. He figured out some crazy cool way to make . fractals. out of siteswaps. To get his explanation and a bunch of examples click here. ...
Q. i need 2 examples of fractals using real world situations... please help
Asked by mmf7492 - Sat Jun 27 18:45:01 2009 - - 1 Answers - 0 Comments
A. HI: Here some websites : it use in computer video game , modeling weather patterns , Cool t- shirt designs, Modeling coastlines, seimetry making a digital sundial. And here are some books on the subject ( all books are available at any book store or public library) The Fractal Geometry of Nature by Benoit B. Mandelbrot ISBN-13: 9780716711865 Introducing Fractal Geometry by Nigel Lesmoir-Gordon, Ralph Edney, Will Rood ISBN-13: 9781840467130 Fractal Geometry 2e by Falconer, K. J. Falconer ISBN-13: 9780470848623 Fractal Geometry And Number Theory by Michel L. Lapidus, Machiel Van Frankenhuysen, Machiel Van Frankenhuysen ISBN-13: 9780486237015 Fractals and Chaos in Geology and Geophysics by Turcotte,… [cont.]
Answered by iroc70 - Sat Jun 27 21:27:45 2009
