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.
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 + c
Here, c is the constant being tested, and scale is a real number.
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√+1 they have length 4(scale+1)/(scale-1)
- Leys, Jos (2010), Mandelbox. Images des Mathématiques, CNRS