|
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] |
List Of Fractals By Hausdorff Dimension
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff dimension, Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension. Deterministic fractals Random and natural fractals See also * Fractal dimension * Hausdorff dimension * Scale invariance Notes and references Further reading * * * * External links The fractals on MathworldOther fractals on Paul Bourke's websiteFractals on mathcurve.com* [https://web.archive.org/web/20060923100014/http://library.thinkquest.org/26242/full/index.html Fractals unleashed] IFStile - software that computes the dimension of the boundary of self-affine tiles {{DEFAULTSORT:Fractals By Hausdorff Dimension Fractals, Hausdorff Dimension Fractal curves, Hausdorff Dimension Mathematics-related lists ... [...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] |
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] |
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] |
Annihilation (film)
''Annihilation'' is a 2018 science fiction psychological horror film written and directed by Alex Garland, based on the 2014 novel of the same name by Jeff VanderMeer. It stars Natalie Portman, Jennifer Jason Leigh, Gina Rodriguez, Tessa Thompson, Tuva Novotny, and Oscar Isaac. The story follows a group of explorers who enter "The Shimmer", a mysterious quarantined zone of mutating plants and animals caused by an alien presence. ''Annihilation'' was released theatrically in the United States by Paramount Pictures on February 23, 2018, and in China on April 13, 2018. It was released digitally by Netflix in a number of other countries on March 12, 2018. According to ''Empire'' magazine, the film addresses "depression, grief, and the human propensity for self-destruction". Plot Lena, a biology professor and army veteran, loses her husband, Kane, after he is deployed on a special forces mission. She believes Kane accepted a suicide mission because she cheated on him. Kane suddenly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
