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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 this: repeatedly transform a vector ''z'', according to the following rules: #First, for each component ''c'' of ''z'' (which corresponds to a dimension), if ''c'' is greater than 1, subtract it from 2; or if ''c'' is less than -1, subtract it from −2. #Then, depending on the magnitude of the vector, change its magnitude using some fixed values and a specified ''scale'' factor. Generation The iteration applies to vector ''z'' as fol ... [...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 further developed in 2009 by Daniel White and Paul Nylander using Spherical coordinate system, spherical coordinates. A canonical form, 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 number, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
