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In mathematics, themandelbox is afractal with a boxlike shape found by Tom Lowe in 2010. It is defined in a similar way to the famousMandelbrot 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 continuousJulia sets, but, unlike the Mandelbrot set, can be defined in any number of dimensions.[1] It is typically drawn in three dimensions for illustrative purposes.[2][3]
The simple definition of the mandelbox is this: repeatedly transform a vectorz, according to the following rules:
The iteration applies to vectorz as follows:[clarification needed]
function iterate(z):for each componentinz:if component > 1: component := 2 - componentelse if component < -1: component := -2 - componentif magnitude ofz < 0.5:z :=z * 4else if magnitude ofz < 1:z :=z / (magnitude ofz)^2z :=scale *z +c
Here,c is the constant being tested, andscale is a real number.[3]
A notable property of the mandelbox, particularly for scale −1.5, is that it contains approximations of many well known fractals within it.[4][5][6]
For the mandelbox contains a solid core. Consequently, itsfractal dimension is 3, orn when generalised ton dimensions.[7]
For the mandelbox sides have length 4 and for they have length.[7]
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