# 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:
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.

## 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 $1<|{\text{scale}}|<2$ the mandelbox contains a solid core. Consequently, its fractal dimension is 3, or n when generalised to n dimensions.

For ${\text{scale}}<-1$ the mandelbox sides have length 4 and for $1<{\text{scale}}\leq 4{\sqrt {n}}+1$ they have length $4\cdot {\frac {{\text{scale}}+1}{{\text{scale}}-1}}$ .