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Mandelbox
In mathematics, the mandelbox is a fractal with a boxlike shape found by Tom Lowe in 2010. It is defined in a similar way to the famous Mandelbrot set as the values of a parameter such that the origin does not escape to infinity under iteration of certain geometrical transformations. The mandelbox is defined as a map of continuous Julia sets, but, unlike the Mandelbrot set, can be defined in any number of dimensions. It is typically drawn in three dimensions for illustrative purposes. Simple definition The simple definition of the mandelbox is, for a vector ''z'', for each component in ''z'' (which corresponds to a dimension), if the absolute value of the component is greater than 1, subtract it from either 2 or −2, depending on the ''z''. Generation The iteration applies to vector ''z'' as follows: function iterate(''z''): for each component in ''z'': if component > 1: component := 2 - component else if component < -1: c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mandelbox 20211127 1GP RGBA8
In mathematics, the mandelbox is a fractal with a boxlike shape found by Tom Lowe in 2010. It is defined in a similar way to the famous Mandelbrot set as the values of a parameter such that the origin does not escape to infinity under iteration of certain geometrical transformations. The mandelbox is defined as a map of continuous Julia sets, but, unlike the Mandelbrot set, can be defined in any number of dimensions. It is typically drawn in three dimensions for illustrative purposes. Simple definition The simple definition of the mandelbox is, for a vector ''z'', for each component in ''z'' (which corresponds to a dimension), if the absolute value of the component is greater than 1, subtract it from either 2 or −2, depending on the ''z''. Generation The iteration applies to vector ''z'' as follows: function iterate(''z''): for each component in ''z'': if component > 1: component := 2 - component else if component < -1: c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mandelbox 20211128 PWR3 RGBA8
In mathematics, the mandelbox is a fractal with a boxlike shape found by Tom Lowe in 2010. It is defined in a similar way to the famous Mandelbrot set as the values of a parameter such that the origin does not escape to infinity under iteration of certain geometrical transformations. The mandelbox is defined as a map of continuous Julia sets, but, unlike the Mandelbrot set, can be defined in any number of dimensions. It is typically drawn in three dimensions for illustrative purposes. Simple definition The simple definition of the mandelbox is, for a vector ''z'', for each component in ''z'' (which corresponds to a dimension), if the absolute value of the component is greater than 1, subtract it from either 2 or −2, depending on the ''z''. Generation The iteration applies to vector ''z'' as follows: function iterate(''z''): for each component in ''z'': if component > 1: component := 2 - component else if component < -1: c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mandelbox 20211129 -1,5 RGBA8
In mathematics, the mandelbox is a fractal with a boxlike shape found by Tom Lowe in 2010. It is defined in a similar way to the famous Mandelbrot set as the values of a parameter such that the origin does not escape to infinity under iteration of certain geometrical transformations. The mandelbox is defined as a map of continuous Julia sets, but, unlike the Mandelbrot set, can be defined in any number of dimensions. It is typically drawn in three dimensions for illustrative purposes. Simple definition The simple definition of the mandelbox is, for a vector ''z'', for each component in ''z'' (which corresponds to a dimension), if the absolute value of the component is greater than 1, subtract it from either 2 or −2, depending on the ''z''. Generation The iteration applies to vector ''z'' as follows: function iterate(''z''): for each component in ''z'': if component > 1: component := 2 - component else if component < -1: c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal
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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Mandelbulb
The Mandelbulb is a three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis and in 2009 further developed by Daniel White and Paul Nylander using spherical coordinates. A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions and bicomplex numbers. White and Nylander's formula for the "''n''th power" of the vector \mathbf v = \langle x, y, z\rangle in is : \mathbf v^n := r^n \langle\sin(n\theta) \cos(n\phi), \sin(n\theta) \sin(n\phi), \cos(n\theta)\rangle, where : r = \sqrt, : \phi = \arctan\frac = \arg(x + yi), : \theta = \arctan\frac = \arccos\frac. The Mandelbulb is then defined as the set of those \mathbf c in for which the orbit of \langle 0, 0, 0\rangle under the iteration \mathbf v \mapsto \mathbf v^n + \mathbf c is bounded. For ''n'' > 3, the result is a 3-dimensional ... [...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] |
French National Centre For Scientific Research
The French National Centre for Scientific Research (french: link=no, Centre national de la recherche scientifique, CNRS) is the French state research organisation and is the largest fundamental science agency in Europe. In 2016, it employed 31,637 staff, including 11,137 tenured researchers, 13,415 engineers and technical staff, and 7,085 contractual workers. It is headquartered in Paris and has administrative offices in Brussels, Beijing, Tokyo, Singapore, Washington, D.C., Bonn, Moscow, Tunis, Johannesburg, Santiago de Chile, Israel, and New Delhi. From 2009 to 2016, the CNRS was ranked No. 1 worldwide by the SCImago Institutions Rankings (SIR), an international ranking of research-focused institutions, including universities, national research centers, and companies such as Facebook or Google. The CNRS ranked No. 2 between 2017 and 2021, then No. 3 in 2022 in the same SIR, after the Chinese Academy of Sciences and before universities such as Harvard University, MIT, or Stanford ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer. The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed ''fractional dimensions''. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used ( see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buddhabrot
The Buddhabrot is the probability distribution over the trajectories of points that escape the Mandelbrot fractal. Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddha, seated in a meditation pose with a forehead mark ('' tikka''), a traditional topknot (''ushnisha'') and ringlet hair. Discovery The ''Buddhabrot'' rendering technique was discovered by Melinda Green, who later described it in a 1993 Usenet post to sci.fractals. Previous researchers had come very close to finding the precise Buddhabrot technique. In 1988, Linas Vepstas relayed similar images to Cliff Pickover for inclusion in Pickover's then-forthcoming book ''Computers, Pattern, Chaos, and Beauty''. This led directly to the discovery of Pickover stalks. Noel Griffin also implemented this idea in the 1993 "Mandelcloud" option in the Fractint renderer. However, these researchers did not filter out non-escaping trajectories required to produce the ghostly forms reminiscent of Hindu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lichtenberg Figure
A Lichtenberg figure (German ''Lichtenberg-Figuren''), or Lichtenberg dust figure, is a branching electric discharge that sometimes appears on the surface or in the interior of insulating materials. Lichtenberg figures are often associated with the progressive deterioration of high voltage components and equipment. The study of planar Lichtenberg figures along insulating surfaces and 3D electrical trees within insulating materials often provides engineers with valuable insights for improving the long-term reliability of high-voltage equipment. Lichtenberg figures are now known to occur on or within solids, liquids, and gases during electrical breakdown. Lichtenberg figures are natural phenomena which exhibit fractal properties. History Lichtenberg figures are named after the German physicist Georg Christoph Lichtenberg, who originally discovered and studied them. When they were first discovered, it was thought that their characteristic shapes might help to reveal the nature ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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