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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. As a result, it is an example of a multifractal system. It is typically drawn in three dimensions for illustrative purposes.
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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: component := -2 - component if magnitude of z < 0.5: z := z * 4 else if magnitude of z < 1: z := z / (magnitude of z)^2 z := scale * z + cHere, c is the constant being tested, and scale is a real number.
Properties
A notable property of the mandelbox, particularly for scale -1.5, is that it contains approximations of many well known fractals within it.
For 1<|scale|<2 the mandelbox contains a solid core. Consequently its fractal dimension is 3, or n when generalised to n dimensions.
For scale < -1 the mandelbox sides have length 4 and for 1 < scale <= 4√n+1 they have length 4(scale+1)/(scale-1)