Mandelbox

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.

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.[2]

A notable property of the mandelbox, particularly for scale −1.5, is that it contains approximations of many well known fractals within it.[3][4][5]

For ${\displaystyle 1<|{\text{scale}}|<2}$ the mandelbox contains a solid core. Consequently, its fractal dimension is 3, or n when generalised to n dimensions.[6]

For ${\displaystyle {\text{scale}}<-1}$ the mandelbox sides have length 4 and for ${\displaystyle 1<{\text{scale}}\leq 4{\sqrt {n}}+1}$ they have length ${\displaystyle 4\cdot {\frac {{\text{scale}}+1}{{\text{scale}}-1}}}$.[6]

• Mandelbulb

Wikimedia Commons has media related to Mandelboxes.

• Gallery and description
• Images of some Mandelbox cubes
• Video : zoom in the Mandelbox cube