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# Topic: Mandelbrot set

###### In the News (Wed 26 Jun 19)

 Mandelbrot set The whole Mandelbrot set lies within a circle of radius 2.5 centered at the origin of the complex plane. One region of the Mandelbrot set containing spiral shapes is known as Seahorse Valley because it resembles a seahorse's tail. The Mandelbrot set was created by Mandelbrot as an index to the Julia sets. www.daviddarling.info /encyclopedia/M/Mandelbrot_set.html   (535 words)

 Mandelbrot set - Wikivisual That is, the Mandelbrot set is the subset of the complex plane consisting of those parameters $c$ for which the Julia set of $f_c$ is connected. The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 by a result of Mitsuhiro Shishikura.Mitsuhiro Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set. en.wikivisual.com /index.php/Mandelbrot_set   (3135 words)

 PlanetMath: Mandelbrot set The higher the resolution, the easier it is to see that the entire Mandelbrot set is simply connected, with no holes. Most of the Mandelbrot set lies within the unit disk of the complex plane. This is version 8 of Mandelbrot set, born on 2007-04-21, modified 2007-06-14. planetmath.org /encyclopedia/MandelbrotSet.html   (284 words)

 Mandelbrot biography 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/Biographies/Mandelbrot.html   (1693 words)

 Mandelbrot Set Although the iterative function that produces the Mandelbrot set is quite simple, the complexity of the set itself is mind-boggling. In the case of the Mandelbrot set, the resulting sequence is determined by the initial value of z. Mandelbrot has conjectured that the boundary of the set, which he modestly refers to as the M-set, is a curve whose fractal dimension is home.comcast.net /~davebowser/fractals/mandelbrot.html   (943 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 set - Wikipedia, the free encyclopedia The boundary of the Mandelbrot set is exactly the bifurcation locus of the quadratic family; that is, the set of parameters As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set. ^ Mitsuhiro Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. en.wikipedia.org /wiki/Mandelbrot_set   (3080 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. The Mandelbrot set is a dynamic calculation based on the iteration (calculation based on constant feedback) of complex numbers with zero as the starting point. 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)

 More about the Mandelbrot Set The condition for a point's membership of the Mandelbrot Set is that the value remains finite, and a monochrome version of the representation of the set is formed by making the corresponding pixel fl if the point belongs to the set, or white otherwise. A Julia Set depends on an iteration exactly like that for the Mandelbrot Set except that the initial value of z is the complex number representing the point whose membership is to be tested, and c is a parameter of the set. The fractal nature of the Mandelbrot Set can be verified by computer generation of the figure with successively greater magnification, and this has been taken to extremes where extra precision of numerical representation is necessary and the enlargement is comparable to that which would reveal individual molecules of a solid object. www.cybsoc.org /about-soc/about-fractal.htm   (1210 words)

 Math Forum - Fractals - Mandelbrot Set   (Site not responding. Last check: ) For some types of functions, the set of numbers that yield chaotic or unpredictable behavior in the plane is called the Julia set after the French mathematician Gaston Julia, who first formulated many of the properties of these sets in the 1920s. The fl points in graphic representations of these sets are the non-chaotic points, representing values that under iteration eventually tend to cycle between three different points in the plane so that their dynamical behavior is predictable. First viewed in 1980 by Benoit Mandelbrot and others, the Mandelbrot set completely characterizes the Julia sets of quadratic functions, and has been called one of the most intricate and beautiful objects in mathematics. mathforum.org /~sarah/mandelbrot.all.html   (329 words)

 Area of the Mandelbrot Set The Mandelbrot set (M) has been called the most complex object in mathematics, and continues to be the subject of active research. Attempts to determine the area of M have generally involved "pixel counting," wherein M is approximated by a set of points or pixels, and the number of points determined to be inside the set is directly related to the area estimate. Thus, it is concluded that the area of the Mandelbrot set is approximately 1.506484, with a 95% confidence interval from 1.506480 to 1.506488. www.fractalus.com /kerry/articles/area/mandelbrot-area.html   (1811 words)

 The Mandelbrot Set The Mandelbrot set is a collection of numbers, a mathematical set, named after its discoverer; Polish born mathematician Benoit B. Mandelbrot. Mandelbrot in fact, is widely considered to be the father of fractal geometry. A basic Google search for the Mandelbrot set will return about 203,000 hits, so those interested in the geometry and mathematics behind the pictures will easily find answers to all of their questions on the web. home.comcast.net /~hal9002/mandelbrot_set.htm   (685 words)

 Mandelbrot Set The set of parameter c for which the trajectory does not diverge is called as Mandelbrot set. The set is colored with fl, and the points which do not belong to it are colored depending on the time to diverge. Shishikura mathematically proved that the dimension of the boundary of the Mandelbrot set is 2, and it is connective. brain.cc.kogakuin.ac.jp /~kanamaru/Chaos/e/Mandelbrot   (277 words)

 Sustainability through the Dynamics of Strategic Dilemmas: in the light of the coherence and visual form of the ... The Mandelbrot set is the most complex mathematical object known to man. This set is composed by iteratively taking each point and multiplying it by itself and measuring the rate at which it escapes toward infinity. Sets and connectedness -- J-set and M-set: The J-sets and M-set emerge in a 4 dimensional space that is populated by iterations. Mandelbrot set (M-set): As the M-set is the set of all parameters c that give rise to connected J-sets, it is also called the connectedness locus for complex quadratic polynomials. www.laetusinpraesens.org /docs00s/cardrep.php   (11030 words)

 2.3 Mandelbrot Sets Given a family of complex iterative maps, the set of all parameter values that produce wholly connected Julia sets is determined by the behavior of a single seed value: the origin. 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. Julia sets are slices parallel to the z-axis while the Mandelbrot set is a slice along the c-axis through the origin. hypertextbook.com /chaos/23.shtml   (1172 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   (232 words)

 Iterations and the Mandelbrot Set from Interactive Mathematics Miscellany and Puzzles However, Douady and Hubbard made a major contribution to the understanding of the Mandelbrot set and its role in the description of dynamics of iterative processes. In 1980, Mandelbrot has discovered the set M of parameter values c for which Julia sets are connected. The main purpose of the Mandelbrot set is to index Julia sets corresponding to various values of the parameter c. www.cut-the-knot.org /blue/Mandel.shtml   (1051 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)

 Fabio Cesari: Fractal Explorer 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. In either case, we set a maximum number of iterations, after which we assume it is part of the set (and we paint it fl). www.geocities.com /CapeCanaveral/2854/mandelbrot.html   (698 words)

 FRACTINT The Mandelbrot Set This set is the classic: the only one implemented in many plotting programs, and the source of most of the printed fractal images published in recent years. Like most of the other types in Fractint, it is simply a graph: the x (horizontal) and y (vertical) coordinate axes represent ranges of two independent quantities, with various colors used to symbolize levels of a third quantity which depends on the first two. Notice that the boundary of the M-set becomes more and more convoluted (the technical terms are "wiggly," "squiggly," and "utterly bizarre") as the Z-magnitudes for points that were still within the set after 150 iterations turn out to exceed 2 after 200, 500, or 1200. spanky.triumf.ca /www/fractint/mandelbrot_type.html   (667 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. Note important, as it is, the classification of Julia set in terms of disconnected sets, this still doesn't allow one to visualize the shape of the set of points, in the parameter space, for which the Julia set is connected. Figure 3: The buds of the Mandelbrot set corresponding to Julia sets that bound the basins of attraction (trapping sets) of periodic orbits. chaos.phy.ohiou.edu /~thomas/fractal/mandel.html   (1324 words)

 Mandelbrot Set and Julia Sets - Permutation City Both the Mandelbrot Set and Julia Sets are pictorial representations of a simple recurrence formula. Here both types of image can be seen, Mandelbrot Set to the left and Julia Set to the right, those with Java capable browsers will have to wait for rendering to take place. The non-divergent (red, fl) region of the Julia Set may come as several separate components surrounded by divergence (green) or may be connected together. www.permutationcity.co.uk /programming/mandelbrot.html   (382 words)

 The Mandelbrot Set   (Site not responding. Last check: ) A second, equivalent, way to define the Mandelbrot set is given in terms of filled Julia sets. The "reset" button draws the Mandelbrot set using the default coarseness level (7) and the default viewing window ([-2,.5] x [-1.25, 1.25]). For points belonging to the Mandelbrot set, this algorithm takes longer than it does for points not in the set. faculty.gvsu.edu /fishbacp/dynamics/MandelbrotDemo.htm   (415 words)

 The Mandelbrot Set This is a little applet (which should appear as a pop-up) I wrote that lets you explore the Mandelbrot set. The Mandelbrot set is a type of infinitely complex mathematical object known as a fractal. There are many strange and beautiful sights to see when you explore the Mandelbrot set, ranging from the sublime to the psychedelic. www.mindspring.com /~chroma/mandelbrot.html   (364 words)

 The Mandelbrot set   (Site not responding. Last check: ) 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)

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