Mandelbox

A 'scale 2' Mandelbox

A 'scale 3' 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. However, unlike the Mandelbrot set, the mandelbox is defined a map of continuous Julia sets, and thus can be defined in any number of dimensions.^[1] As a result, it is an example of a multifractal system. It is typically drawn in three dimensions for illustrative purposes.

Generation

The iteration applies to vector v as follows:

for (each axis)
    if (v[axis]>1) v[axis] = 2-v[axis];
    else if (v[axis]<-1) v[axis] = -2-v[axis];
 
if (v.magnitude() < 0.5) v *= 4;
else if (v.magnitude() < 1) v /= square(v.magnitude());
 
v = scale*v + c;

A notable property of the mandelbox, particularly for scale -1.5, is that it contains approximations of many well known fractals within it.^[2][3][4]

See also

• Mandelbulb

Notes

1. ^ [1]
2. ^ negative-mandelbox
3. ^ more-negatives
4. ^ mandelbox_3d_fractal

References

• Leys, Jos (2010), Mandelbox. Images des Mathématiques, CNRS, http://images.math.cnrs.fr/Mandelbox.html 

External links

• Images of some Mandelbox cubes