The mathematician Benoit Mandelbrot is best known for his work on the geometry of what unusual shapes?
"music, painting, architecture, and stock market prices. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry: Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. —Mandelbrot, in his introduction to ""The Fractal Geometry of Nature"" Mandelbrot has been called a work of art, and a visionary and a maverick. His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion"
"to build larger strings of symbols, and a mechanism for translating the generated strings into geometric structures. In 1975, after centuries of slow development of the mathematics of patterns by Gottfried Leibniz, Georg Cantor, Helge von Koch, Wacław Sierpiński and others, Benoît Mandelbrot wrote a famous paper, ""How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension"", crystallising mathematical thought into the concept of the fractal. Living things like orchids, hummingbirds, and the peacock's tail have abstract designs with a beauty of form, pattern and colour that artists struggle to match. The beauty that people perceive in nature"
"12 days, when IBM decided to end pure research in his division. He joined the Department of Mathematics at Yale, and obtained his first tenured post in 1999, at the age of 75. At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences. Mandelbrot created the first-ever ""theory of roughness"", and he saw ""roughness"" in the shapes of mountains, coastlines and river basins; the structures of plants, blood vessels and lungs; the clustering of galaxies. His personal quest was to create some mathematical formula to measure the overall ""roughness"" of such objects in nature. He"
"Sam Vandervelde Samuel Kendrick Vandervelde (born 12 February 1971) is a mathematician who, along with Sandor Lehoczky and Richard Rusczyk, is most notable for creating the Mandelbrot Competition. and is listed first under ""Thanks"" in the mathematical textbook "" The Art of Problem Solving"". He contributes problems to the USA Math Olympiad. He was a member of the 1989 US International Mathematics Olympiad team. He was a grader at Math Olympiad Program, a seminar that prepares top high school students for the International Math Olympiad. Vandervelde founded the Stanford Math Circle. He currently is a coordinator of the San Francisco"
"theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred. Mandelbrot's awards include the Wolf Prize for Physics in 1993, the Lewis Fry Richardson Prize of the European Geophysical Society in 2000, the Japan Prize in 2003, and the Einstein Lectureship of the American Mathematical Society in 2006. The small asteroid 27500 Mandelbrot was named in his honor. In November 1990, he was made a Chevalier in France's Legion of Honour. In December 2005, Mandelbrot was appointed to the position of"
"Benoît and the Mandelbrots Benoît and the Mandelbrots, named after French American mathematician Benoît Mandelbrot, is a Computer Music band formed in 2009 in Karlsruhe, Germany. They are known for their live coded and Algorave performances, the Digital Arts practice of improvising with programming languages. The band consists of Juan A. Romero, Holger Ballweg, Patrick Borgeat and Matthias Schneiderbanger, who met while studying at the Institute for Musicology and Music Informatics at the University of Music Karlsruhe in Germany. They perform live using the SuperCollider programming language, writing code to improvise music in a range of electronic music genres, from"
"was inventing a new idea. He describes his feelings in a documentary with science writer Arthur C. Clarke: According to Clarke, ""the Mandelbrot set is indeed one of the most astonishing discoveries in the entire history of mathematics. Who could have dreamed that such an incredibly simple equation could have generated images of literally ""infinite"" complexity?"" Clarke also notes an ""odd coincidencethe name Mandelbrot, and the word ""mandala""—for a religious symbol—which I'm sure is a pure coincidence, but indeed the Mandelbrot set does seem to contain an enormous number of mandalas. Mandelbrot left IBM in 1987, after 35 years and"
"College. One of his published works is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure. Later, with Michel Kervaire, he showed that the 7-sphere has 15 differentiable structures (28 if one considers orientation).An ""n""-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently Milnor worked on the topology of isolated singular points of complex"
"sequence of plane algebraic curves, the ""Mandelbrot curves"", of the general type known as polynomial lemniscates. The Mandelbrot curves are defined by setting ""p"" = ""z"", ""p"" = ""p"" + ""z"", and then interpreting the set of points |""p""(""z"")| = 2 in the complex plane as a curve in the real Cartesian plane of degree 2 in ""x"" and ""y"". These algebraic curves appear in images of the Mandelbrot set computed using the ""escape time algorithm"" mentioned below. Upon looking at a picture of the Mandelbrot set, one immediately notices the large cardioid-shaped region in the center. This ""main cardioid"""
"formula_13—is variable instead, one obtains the corresponding Julia set for each point formula_1 in the parameter space of the simple function. Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The ""style"" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts. The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as"
"paper on self-similarity in the coastline of Britain. Various historical authorities credit him with also synthesizing centuries of complicated theoretical mathematics and engineering work and applying them in a new way to study complex geometries that defied description in usual linear terms. The earliest roots of what Mandelbrot synthesized as the fractal dimension have been traced clearly back to writings about undifferentiable, infinitely self-similar functions, which are important in the mathematical definition of fractals, around the time that calculus was discovered in the mid-1600s. There was a lull in the published work on such functions for a time after that,"
"John H. Hubbard, Sibony proved in the 1980s that the Mandelbrot set is locally connected. In 1985 he received the Vaillant Prize and in 2009 the Sophie Germain Prize from the French Academy of Sciences. For 2017 he received the Stefan Bergman Prize. In 1990 he was an Invited Speaker with talk ""Some recent results on weakly pseudoconvex domains"" at the ICM in Kyōto. He was a senior member of the Institut Universitaire de France from 2009 to 2014. Nessim Sibony Nessim Sibony (born October 1947) is a French mathematician, specializing in the theory of several complex variables and complex"
"rough surfaces via fractional Brownian motion was first proposed by Benoît Mandelbrot. Because the intended result of the process is to produce a landscape, rather than a mathematical function, processes are frequently applied to such landscapes that may affect the stationarity and even the overall fractal behavior of such a surface, in the interests of producing a more convincing landscape. According to R. R. Shearer, the generation of natural looking surfaces and landscapes was a major turning point in art history, where the distinction between geometric, computer generated images and natural, man made art became blurred. The first use of"
"curve of the second pencil orthogonally. Choosing other positions of the cardioid within the coordinate system results in different equations. The picture shows the 4 most common positions of a cardioid and their polar equations. In complex analysis, the image of any circle through the origin under the map formula_151 is a cardioid. One application of this result is that the boundary of the central bulb of the Mandelbrot set is a cardioid given by the equation The Mandelbrot set contains an infinite number of slightly distorted copies of itself and the central bulb of any of these smaller copies"
"to the discovery of Pickover stalks. However, these researchers did not filter out non-escaping trajectories required to produce the ghostly forms reminiscent of Hindu art. The inverse, ""Anti-Buddhabrot"" filter produces images similar to no filtering. Green first named this pattern Ganesh, since an Indian co-worker ""instantly recognized it as the god 'Ganesha' which is the one with the head of an elephant."" The name ""Buddhabrot"" was coined later by Lori Gardi. Mathematically, the Mandelbrot set consists of the set of points formula_1 in the complex plane for which the iteratively defined sequence does ""not"" tend to infinity as formula_3 goes"
"Pickover stalk Pickover stalks are certain kinds of details to be found empirically in the Mandelbrot set, in the study of fractal geometry. They are so named after the researcher Clifford Pickover, whose ""epsilon cross"" method was instrumental in their discovery. An ""epsilon cross"" is a cross-shaped orbit trap. According to Vepstas (1997) ""Pickover hit on the novel concept of looking to see how closely the orbits of interior points come to the x and y axes. In these pictures, the closer that the point approaches, the higher up the color scale, with red denoting the closest approach. The logarithm"
"the legs of an elephant and a mouse (so-called allometric scaling). Similarly, if a soft wax candle were enlarged to the size of a tall tree, it would immediately collapse under its own weight. A more subtle form of scale symmetry is demonstrated by fractals. As conceived by Benoît Mandelbrot, fractals are a mathematical concept in which the structure of a complex form looks similar at any degree of magnification, well seen in the Mandelbrot set. A coast is an example of a naturally occurring fractal, since it retains similar-appearing complexity at every level from the view of a satellite"
"Mandelbrot Competition Named in honor of Benoit Mandelbrot, the Mandelbrot Competition is a mathematics competition founded by Sam Vandervelde, Richard Rusczyk and Sandor Lehoczky that allows high school students to compete individually and in four-person teams. The Mandelbrot is a ""correspondence competition,"" meaning that the competition is sent to a school's coach and students compete at their own school on a predetermined date. Individual results and team answers are then sent back to the contest coordinators. The most notable aspects of the Mandelbrot competition are the difficulty of the problems (much like the American Mathematics Competition and harder American Invitational"
"particular projective geometry based on vector spaces over a finite field. In a well-known paper he proved the following theorem: In a Desarguesian plane of odd order, the ovals are exactly the irreducible conics. In 1959 he authored a survey ""Le geometrie di Galois"" on Galois geometry. According to J. W. P. Hirschfeld, it ""gave a comprehensive list of results and methods, and is to my mind the seminal paper in the subject."" Some critics felt that his work was no longer geometry, but today it is recognized as a separate sub-discipline: finite geometry or combinatorial geometry. According to Hirschfeld,"
"work of art was probably the famous one on the cover of ""Scientific American"", August 1985. This image showed a landscape formed from the potential function on the domain outside the (usual) Mandelbrot set. However, as the potential function grows fast near the boundary of the Mandelbrot set, it was necessary for the creator to let the landscape grow downwards, so that it looked as if the Mandelbrot set was a plateau atop a mountain with steep sides. The same technique was used a year after in some images in ""The Beauty of Fractals"" by Heinz-Otto Peitgen and Michael M."
"geometries developed and popularized by Benoit Mandelbrot. Fractals are usually associated with the self-similar case of a fractal curve, but other more complicated fractals are possible, e.g. considering not only curves, but also fractal surfaces or fractal volumes, as well as investigating fractal dimensions which have other values than 2, and which also vary with scale. Garnet Ord and Laurent Nottale both connected fractal space-time with quantum mechanics. Nottale coined the term ""scale relativity"" in 1992. He developed the theory and its applications with more than one hundred scientific papers, two technical books in English, and three popular books in"
"Gömböc A gömböc or gomboc () is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. Its existence was conjectured by Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by Hungarian scientists Gábor Domokos and Péter Várkonyi. The gömböc shape is not unique; it has countless varieties, most of which are very close to a sphere and all have very strict shape tolerance (about 0.1 mm per 100 mm). The most famous solution has a sharpened top, as shown in the photo. Its shape helped"
"others, mathematician Benoît Mandelbrot suggested as early as 1963 that the statistics prove this assumption incorrect. Mandelbrot observed that large movements in prices (i.e. crashes) are much more common than would be predicted from a log-normal distribution. Mandelbrot and others suggested that the nature of market moves is generally much better explained using non-linear analysis and concepts of chaos theory. This has been expressed in non-mathematical terms by George Soros in his discussions of what he calls reflexivity of markets and their non-linear movement. George Soros said in late October 1987, 'Mr. Robert Prechter's reversal proved to be the crack"
"of an object are relative to the observer and may be fractional. An object whose irregularity is constant over different scales (""self-similarity"") is a fractal (examples include the Menger sponge, the Sierpiński gasket, and the Koch curve or ""snowflake"", which is infinitely long yet encloses a finite space and has a fractal dimension of circa 1.2619). In 1982, Mandelbrot published ""The Fractal Geometry of Nature"", which became a classic of chaos theory. Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model. In December 1977, the New York Academy of Sciences organized"
"Supersingular elliptic curve In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic ""p"" > 0 with unusually large endomorphism rings. Elliptic curves over such fields which are not supersingular are called ""ordinary"" and these two classes of elliptic curves behave fundamentally differently in many aspects. discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that in positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and developed their basic theory. The term ""supersingular"" has nothing to do with singular points"
"which are at present in use.” The legacy of Tissot’s method is still vivid today, as suggested by the authors of ""Map Projections for Europe"", who argue that since Tissot’s famous analysis regarding distortion, the only major scientific development in the metric interpretation of deformation has been Eduard Imhof's ""Verzerrungsgitter"", or ""deformation grid"". Nicolas Auguste Tissot Nicolas Auguste Tissot (; 1824–1897) was a 19th-century French cartographer, who in 1859 and 1881 published an analysis of the distortion that occurs on map projections. He devised Tissot's indicatrix, or ""distortion circle"", which when plotted on a map will appear as an ellipse"
"to the algorithm for computing the Mandelbrot set. The cover featured an image located at -0.909 + -0.275 and was created by Peitgen, et al. The Mandelbrot set became prominent in the mid-1980s as a computer graphics demo, when personal computers became powerful enough to plot and display the set in high resolution. The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics and abstract mathematics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. An exhaustive list of all who have contributed to the understanding of"
"this set since then is long but would include Mikhail Lyubich, Curt McMullen, John Milnor, Mitsuhiro Shishikura and Jean-Christophe Yoccoz. The Mandelbrot set is the set of values of ""c"" in the complex plane for which the orbit of 0 under iteration of the quadratic map remains bounded. That is, a complex number ""c"" is part of the Mandelbrot set if, when starting with ""z"" = 0 and applying the iteration repeatedly, the absolute value of ""z"" remains bounded however large ""n"" gets. This can also be represented as For example, letting ""c"" = 1 gives the sequence 0, 1,"
"Michel Kervaire Michel André Kervaire (26 April 1927 – 19 November 2007) was a French mathematician who made significant contributions to topology and algebra. He introduced the Kervaire semi-characteristic. He was the first to show the existence of topological ""n""-manifolds with no differentiable structure (using the Kervaire invariant), and (with John Milnor) computed the number of exotic spheres in dimensions greater than four. He is also well known for fundamental contributions to high-dimensional knot theory. The solution of the Kervaire invariant problem was announced by Michael Hopkins in Edinburgh on 21 April 2009. He was the son of André Kervaire"
"effect of formula_77 on points near the origin consists of a counterclockwise rotation about the origin of formula_78, or very nearly formula_79, followed by scaling (dilation) by a factor of formula_80. Douady rabbit The Douady rabbit is any of various particular filled Julia sets associated with the parameter near the center period 3 buds of Mandelbrot set for complex quadratic map. The rabbit is a parabolic Julia set for internal angle 1/3 (parameter ""c"" is a root point between period 1 and period 3 components of Mandelbrot set). Douady's rabbit or the Douady rabbit is named for the French mathematician"
"John H. Hubbard John Hamal Hubbard (born October 6 or 7, 1945; the actual date is unknown) is an American mathematician and professor at Cornell University and the Université de Provence. He is well known for the mathematical contributions he made with Adrien Douady in the field of complex dynamics, including a study of the Mandelbrot set. One of their most important results is that the Mandelbrot set is connected. Hubbard graduated with a Doctorat d'État from Université de Paris-Sud in 1973 under the direction of Adrien Douady; his thesis was entitled ""Sur Les Sections Analytiques de La Courbe Universelle"
"External ray An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set. Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray. External rays are used in complex analysis, particularly in complex dynamics and geometric function theory. External rays were introduced in Douady and Hubbard's study of the Mandelbrot set External rays of (connected) Julia sets on dynamical plane are often called dynamic rays. External rays of the Mandelbrot set (and similar one-dimensional connectedness loci) on parameter plane are called parameter rays."
"pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour. Geometry has also had a large effect on other areas of mathematics. For instance, the introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. The subject of geometry was further enriched by the study of the intrinsic structure"