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	Mandelbrot set - Wikipedia, the free encyclopedia
		The Mandelbrot set was created by Benoît Mandelbrot as an index to the Julia sets: each point in the complex plane corresponds to a different Julia set.
		The Mandelbrot set is compact, and thus measurable; its area was estimated as 1.5065918 by Hill [1].
		Mandelbrot hobbyists quickly learn to recognize the "blobby" images caused by a program mistakenly placing points in the set, and will then up the iteration count (at the expense of slowing down the calculation of every point actually in the set).
	en.wikipedia.org /wiki/Mandelbrot_set (2122 words)

	Benoît Mandelbrot - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-22)
		Benoît B. Mandelbrot (born November 20, 1924) is a Polish-born French mathematician and leading proponent of fractal geometry.
		Born in Warsaw, Mandelbrot lived in France from the age of 12 to the end of his college studies.
		Mandelbrot's informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made The Fractal Geometry of Nature accessible to non-specialists.
	www.kernersville.us /project/wikipedia/index.php/Benoit_Mandelbrot (887 words)

	Benoît Mandelbrot - Wikipedia, the free encyclopedia
		Benoît Mandelbrot was the first to use a computer to plot the Mandelbrot set.
		Mandelbrot attended the Lycée Rolin in Paris until the start of World War II, when his family moved to Tulle.
		Mandelbrot brought these objects together for the first time and highlighted their common properties, such as self-similarity (sometimes partial or statistical), scale invariance and (usually) non-integer Hausdorff dimension.
	en.wikipedia.org /wiki/Beno%C3%AEt_Mandelbrot (909 words)

	The Mandelbrot Set
		The set is named for Benoit B. Mandelbrot, a research fellow at the IBM Thomas J. Watson Research Center in Yorktown Heights, N.Y. From his work with geometric forms Mandelbrot has developed the field he calls fractal geometry, the mathematical study of forms having a fractional dimension.
		The Mandelbrot set is the set of all complex numbers C for which the size Of z^2 + C is small even after an indefinitely large number of iterations.
		The magnification of the Mandelbrot set theoretically attainable with such precision is far greater than the magnification needed to resolve the nucleus of the atom.
	www.math.uwaterloo.ca /navigation/ideas/articles/mandelbrot/index.shtml (4301 words)

	Mandelbrot (Site not responding. Last check: 2007-10-22)
		Mandelbrot's family emigrated to France in 1936 and his uncle Szolem Mandelbrojt, who was Professor of Mathematics at the Collège de France and the successor of Hadamard in this post, took responsibility for his education.
		This brought a reaction from Mandelbrot against pure mathematics, although as Mandelbrot himself says, he now understands how Hardy's deep felt pacifism made him fear that applied mathematics, in the wrong hands, might be used for evil in time of war.
		In 1945 Mandelbrot's uncle had introduced him to Julia's important 1918 paper claiming that it was a masterpiece and a potential source of interesting problems, but Mandelbrot did not like it.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Mandelbrot.html (1593 words)

	Benoit Mandelbrot (Site not responding. Last check: 2007-10-22)
		Mandelbrot received his diploma from L'École Polytechnique, Paris, in 1947, his Master of Science in Aeronautics from the California Institute of Technology in 1948, and his Ph.D. in Mathematical Sciences from the University of Paris in 1952.
		In the mid-1970s Mandelbrot coined the word "fractal" (from the Latin word "fractus", meaning fractured, broken) to label objects, shapes or behaviors that have similar properties (self-similarity) at all levels of magnification or across all times, and which dimension, being greater than one but smaller than two, cannot be expressed as an integer.
		Mandelbrot's own work is a case of multidisciplinarity: his doctoral thesis (a mathematical analysis of the distribution of words in the English language, U. de Paris, 1952) combined linguistics with the tools of statistical thermodynamics.
	www.fractovia.org /people/mandelbrot.html (851 words)

	Mandelbrot (Site not responding. Last check: 2007-10-22)
		Mandelbrot is a program that allows you to view and zoom in and out of the mandelbrot set.
		The mandelbrot set is a mathematic representation of a subset of the complex plane.
		The mandelbrot set itself is not a fractal, but it's boundary is a fractal.
	www.csc.calpoly.edu /~bfriesen/software/mandelbrot.shtml (332 words)

	Fractal Geometry
		His uncle, Szolem Mandelbrot, was a member of an elite group of French mathematicians in Paris known as the "Bourbaki." Benoit Mandelbrot was born in Warsaw in 1924 to a Lithuanian Jewish family.
		But Mandelbrot conceived and developed a new fractal geometry of nature based on the fourth dimension and Complex numbers which is capable of describing mathematically the most amorphous and chaotic forms of the real world.
		Mandelbrot discovered that the fourth dimension of fractal forms includes an infinite set of fractional dimensions which lie between the zero and first dimension, the first and second dimension and the second and third dimension.
	www.fractalwisdom.com /FractalWisdom/fractal.html (2550 words)

	Last Year in Mandelbrot
		Mandelbrot’s shocking conclusion, published in 1963, was that the time series was in no way Gaussian: in fact, he argued that the departures from normality could be accounted for by using distribution functions with infinite variance, which are termed L-stable.
		Mandelbrot’s heirs are primarily physicists entering the field of finance, who recognize the fundamental importance of fat tails and are able to elaborate and extend Mandelbrot’s suggestive results.
		Mandelbrot writes with economy and felicitously, and intersperses the more mathematical sections with frank historical anecdotes, such as the events which led up to his work on cotton pricing, and the embarrassment caused by interpreting United States Department of Agriculture data for weekly averages as "Sunday closing prices".
	guava.physics.uiuc.edu /~nigel/articles/mandelbrot.html (1111 words)

	Explore: Mandelbrot Set
		The Mandelbrot set was introduced in 1980, showing how complex phenomena could be generated from simple rules iterated repeatedly.
		Benoit Mandelbrot, often referred to as the father of fractals, almost single-handedly created a new geometry of nature.
		He introduced the concept of fractal dimension by suggesting that the dimension of a coastline, for example, must fall somewhere between the dimensions of a smooth curve (with dimension one) and a smooth surface (with dimension two).
	library.wolfram.com /explorations/explorer/Mandelbrot.html (220 words)

	The Mandelbrot Set
		The color of a pixel outside the Mandelbrot set indicates the number n of iterations of (1) that it took until the distance of z(n) from the origin exceeded the square root of 5.
		So we are not really drawing the Mandelbrot set itself, only an approximation to it that is the better the larger maxit is. The default value for maxit is 100 which is OK for drawing the whole set, but is too small for drawing small parts of it.
		Points outside the Mandelbrot set are assigned a color that is a combination, usually a convex combination, of the RGB values of the colors C and F. Here's another intriguing picture (bay) with lots of interesting parts to explore.
	www.math.utah.edu /~alfeld/math/mandelbrot/mandelbrot.html (5294 words)

	2.3 Mandelbrot Sets
		This trick was discovered by the Polish-American mathematician Benoit Mandelbrot and in his honor the set of all parameter values whose Julia sets are wholly connected is called a Mandelbrot set.
		The main body of the Mandelbrot set is a cardioid with a series of successively smaller circles attached to it in a chain running along the x-axis in the negative direction.
		The Mandelbrot and Julia sets are therefore two-dimensional cross sections through a four-dimensional parent set; the mother of all iterated quadratic mappings so to speak.
	www.hypertextbook.com /chaos/23.shtml (1172 words)

	[No title]
		Mandelbrot coined the word "fractal" to describe his new object and those like it.
		Mandelbrot adopted a much more abstract "definition of dimension than that used in Euclidean geometry, stating that the dimension of a fractal must be used as an exponent when measuring its size.
		Mandelbrot has suggested that mountains, clouds, aggregates, galaxy clusters, and other natural phenomena are similarly fractal in nature, and fractal geometry's application in the sciences has become a rapidly expanding field.
	www.cc.utah.edu /~jtm26960/chaos/frac.htm (1830 words)

	Mandelbrot and Julia sets
		Their contribution (Ref [3]), although regarded as a masterpiece, was largely ignored by the mathematical community until a revival in the late 1970s spawned by the discovery of fractals by Benoit Mandelbrot.
		This is the set that consists of the enterior cardioid-like shape with a circle attached on its left.
		Once the Mandelbrot set is drawn, you can select a value of c by clicking anywhere (any time) inside the left portion of the display.
	www.cut-the-knot.org /blue/julia.shtml (550 words)

	The Mandelbrot Set (Site not responding. Last check: 2007-10-22)
		As I mentioned, Mandelbrot sets and Julia sets are related in a very special way.
		Mandelbrot was looking for some sort of clue as to which c numbers made disconnected sets, and which made connected sets.
		Mandelbrot detail is never the same twice, and there are some very exotic and bizarre shapes in the set.
	www.icd.com /tsd/fractals/beginner3.htm (382 words)

	The Mandelbrot set (Site not responding. Last check: 2007-10-22)
		Note that the Mandelbrot set in general is _not_ strictly self-similar; the tiny copies of the Mandelbrot set are all slightly different, mainly because of the thin threads connecting them to the main body of the Mandelbrot set.
		The boundary of the Mandelbrot set and the Julia set of a generic c in M have Hausdorff dimension 2 and have topological dimension 1.
		This follows from a theorem of Douady and Hubbard that there is a conformal isomorphism from the complement of the Mandelbrot set to the complement of the unit disk.
	www.faqs.org /faqs/fractal-faq/section-6.html (954 words)

	Classic Mandelbrot/Julia Set
		The Mandelbrot Set is probably one of the most well known fractals, and probably one of the most widely implemented fractal in fractal plotting programs.
		By thinking of the Mandelbrot Set as an index for Julia Sets, the question comes up about what happens if a point on the edge of the Mandelbrot Set is chosen as the value of c.
		This is one of the few points on the edge of the Mandelbrot Set with rational coordinates which is why I name it.
	www.jamesh.id.au /fractals/mandel/Mandel.html (580 words)

	Known Geometry of the Julia and Mandelbrot Sets (Site not responding. Last check: 2007-10-22)
		Finally, the Mandelbrot set can be considered a true fractal, in the sense that it's boundary (topologically of dimension one) has Hausdorff dimension greater than one.
		In fact, Mitsuhiro Shishikura [9] proved that the Hausdorff dimension of the boundary of the Mandelbrot set is exactly 2.
		This remarkable result indicates that the boundary of the Mandelbrot is indeed very complicated, as it appears to be.
	www.math.vt.edu /people/hoggard/FracGeom/node16.html (301 words)

	Mandelbrot Set
		In the Mandelbrot set, nature (or is it mathematics) provides us with a powerful visual counterpart of the musical idea of 'theme and variation': the shapes are repeated everywhere, yet each repetition is somewhat different.
		This was indeed the key result that clued Mandelbrot, in 1979, to visualize a set in the complex parameter space c which is called the Mandelbrot set.
		Zero is the critical point of Mandelbrot equation given by 2.
	chaos.phy.ohiou.edu /~thomas/fractal/mandel.html (1324 words)

	The Mandelbrot Set (Site not responding. Last check: 2007-10-22)
		The outline of the Mandelbrot set is produced by a trick called the escape time algorithm.
		The logistic function is a quadratic, the sine function is trigonometric, the Gaussian is exponential and the Mandelbrot set is none of these.
		The whole Mandelbrot set is contained in the complex plane such that a circle of radius 2.5 centered at zero would completely enclose it.
	www.mcasco.com /mset.html (1692 words)

	Citations: Self-similar Error Clusters in Communication Systems and the Concept of Conditional Stationarity - ... (Site not responding. Last check: 2007-10-22)
		Mandelbrot, Self-similar error clusters in communication systems and the concept of conditional stationarity, IEEE Transactions on Communication Technology COM-13 (1965) 71--90.
		The class of model proposed by Berger and Mandelbrot belongs to that which is known in the literature as a renewal process model [8] 9] It is worth noting that....
		In the second case A e (t) is well defined (though it itself has infinite expectation) Hence we study the ERP and consider the connection between it and the covariant stationary processes (CoVSP s) with unsummable auto covariance functions used in [16] to describe long term correlation and....
	citeseer.ist.psu.edu /context/88583/0 (2839 words)

	Exploration #3
		If c lies in the Mandelbrot set, then the corresponding filled Julia set is connected, meaning it is just one piece.
		If c lies outside the Mandelbrot set, then the filled Julia set shatters into infinitely many pieces (what is known technically as a "Cantor set" or, more popularly, "fractal dust").
		Note that the filled Julia sets from inside the Mandelbrot set assume many different shapes, but they are all connected (= one piece).
	math.bu.edu /DYSYS/explorer/tour3.html (258 words)

	Fabio Cesari: Fractal Explorer
		The Mandelbrot set is the domain of convergence of the series built up by the complex sequence defined by the recursion law:
		Their colour depends on how many iterations have been required to determine that they are outside the Mandelbrot set, and it can be interpreted as their "distance" from the Mandelbrot set.
		In the first case, we say that C belongs to the Mandelbrot set (it is one of the fl points in the image); otherwise, we say that it goes to infinity and we assign a colour to C depending on the speed the point "escapes" from the origin.
	www.geocities.com /CapeCanaveral/2854/mandelbrot.html (698 words)

	Fractal eXtreme: Fractal Theory (Site not responding. Last check: 2007-10-22)
		The fl, barnacle covered pear is the Mandelbrot set proper - all the bands of colour outside of it are simply curious artifacts that help to expose the detail of the Mandelbrot set itself.
		What this means is that the boundary between the fl area that is the Mandelbrot set and the surrounding area that isn't the Mandelbrot set is not a simple line or a curve (one dimensional), but it also isn't a filled in circle or square (two dimensional).
		The Mandelbrot set, the internal fl area, itself is not excluded from this rule.
	www.cygnus-software.com /theory/theory.htm (1926 words)

	Buddhabrot fractal method
		To produce the image only requires some very simple modifications to the traditional mandelbrot rendering technique: Instead of selecting initial points on the real-complex plane one for each pixel, initial points are selected randomly from the image region.
		For those familiar with the mandelbrot technique, the size of the image region is 0.125 units, and the coordinates of the image center is (-1.15, 0.0).
		I've not bothered to link a related underlying mandelbrot image to this one since it is almost exactly identical to the first mandelbrot image.
	www.superliminal.com /fractals/bbrot/bbrot.htm (1154 words)

	Philosophy and Computers --- Links - Mandelbrot Set (Site not responding. Last check: 2007-10-22)
		Introduction to the Mandelbrot set - Introduces the basic concepts behind the Mandelbrot set, with iteration examples and images from the Mandelbrot set.
		Mandelbrot Pictures - An organized gallery of named images of different zooms of the Mandelbrot Set.
		Mandelbrot Set Java Applet - A java applet with complete instructions of use to discover regions of the Mandelbrot Set.
	www.sinc.sunysb.edu /Class/phi365/mandelbrot_set.html (348 words)

	Mandelbrot Set Chaos (Site not responding. Last check: 2007-10-22)
		There are also various ways to express the Mandelbrot set in terms of a single time-delayed scalar variable.
		If chaotic orbits are limited to the boundary of the Mandelbrot set, as appears to be the case, then they occur with a probability less than or equal to the probability that a point falls on the boundary of the set.
		Thus the Mandelbrot set, with all its complexity, apparently admits a negligibly small number of truly chaotic orbits.
	sprott.physics.wisc.edu /chaos/manchaos.htm (1174 words)

	Chaffey's Fractals - The Mandelbrot
		The mandelbrot was first discovered about 1980, so it is fairly new to the mathematical world.
		Mandelbrot himself (depicted to the here on the right).
		He is a mathematician, born in 1924 in Warsaw.
	www.chaffey.org /fractals/mandelbrot (79 words)

	The Mandelbrot Set
		The Mandelbrot Set is a ring of online journals which are unique, intricate, and beautiful.
		The boundary of the Mandelbrot set is a very complicated fractal with a Hausdorff dimension of 2.
		The area of the Mandelbrot set is unknown, but it's fairly small.
	www.jade-leaves.com /mandelbrot_set/index.shtml (422 words)

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