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Mandelbrot Set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York. Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications; mathematically, one would say that the boundary of the Mandelbrot set is a ''fractal curve''. The "style" of this recursive detail depends on the region of the set boundary being examined. Mandelbrot set images may be created by sampling the complex numbers and testing, for each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Geometry
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Pickover
Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research Center in Yorktown, New York where he was Editor-in-Chief of the '' IBM Journal of Research and Development''. He has been granted more than 700 U.S. patents, is an elected Fellow for the Committee for Skeptical Inquiry, and is author of more than 50 books, translated into more than a dozen languages. Pickover.com Biography Pickover was elected as a Fellow for the Committee for Skeptical Inquiry for his “significant contributions to the general public’s understanding of science, reason, and critical inquiry through their scholarship, writing, and work in the media.” Other Fellows have included Carl Sagan and Isaac Asimov. He has been awarded almost 700 United States patents, and his '' The Math Book'' was winner of the 2011 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbit Trap
In mathematics, an orbit trap is a method of colouring fractal images based upon how close an iterative function, used to create the fractal, approaches a geometric shape, called a "trap". Typical traps are points, lines, circles, flower shapes and even raster images. Orbit traps are typically used to colour two dimensional fractals representing the complex plane. Examples Point based A point-based orbit trap colours a point based upon how close a function's orbit comes to a single point, typically the origin. Line based A line-based orbit trap colours a point based upon how close a function's orbit comes to one or more lines, typically vertical or horizontal (x=a or y=a lines). Pickover stalks are an example of a line based orbit trap which use two lines. Algorithm Orbit traps are typically used with the class of two-dimensional fractals based on an iterative function. A program that creates such a fractal colours each pixel, which represent discrete points in the complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior Points
In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the closure of the complement of . In this sense interior and closure are dual notions. The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty). Definitions Interior point If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in . (This is illustrated in the introductory section to this article.) This definition generalizes to any subset of a metric space with metric : is an interior point of if there exists r > 0, such that i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biomorph
Biomorph may refer to: * A shape resembling that of a living organism (such as bacteria), though not necessarily of biotic origin * One of the virtual creatures in a computer simulation described by Richard Dawkins in his book ''The Blind Watchmaker'' * In biomorphism, shapes that derive their form from nature as with contemporary architecture art * One of the organic creatures in the art of surrealist painters such as Salvador Dalí or Yves Tanguy * One of the mysterious alien creatures in the book ''Империя Превыше Всего (Empire Above All)'' by Nick Perumov Nick Perumov (russian: link=no, Ник Перумов) is the pen name of Nikolay Daniilovich Perumov (russian: link=no, Николай Даниилович Перумов; born 21 November 1963), a Russian fantasy and science fiction writer. Bi ... * Various fractals, particularly Pickover biomorphs, which are computer generated graphics from mathematical chaos modelisation * ''Biomorph'' (video game) [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julia Set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") 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 that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaotic". The Julia set of a function is commonly denoted \operatorname(f), and the Fatou set is denoted \operatorname(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century. Formal definition Let f(z) be a non-constant holomorphic function from the Riemann sphere on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Dawkins
Richard Dawkins (born 26 March 1941) is a British evolutionary biologist and author. He is an emeritus fellow of New College, Oxford and was Professor for Public Understanding of Science in the University of Oxford from 1995 to 2008. An atheist, he is well known for his criticism of creationism and intelligent design. Dawkins first came to prominence with his 1976 book ''The Selfish Gene'', which popularised the gene-centred view of evolution and introduced the term '' meme''. With his book ''The Extended Phenotype'' (1982), he introduced into evolutionary biology the influential concept that the phenotypic effects of a gene are not necessarily limited to an organism's body, but can stretch far into the environment, for example, when a beaver builds a dam. His 2004 The Ancestor's Tale set out to make understanding evolution simple for the general public, by tracing common ancestors back from humans to the origins of life. Over time, numerous religious people challenged th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
