Mandelbox


[Image:Mandelbox 20211127 1GP RGBA8.png|alt=A three-dimensional Mandelbox fractal of scale 2.|thumb|right|A "scale-2" Mandelbox]
[Image:Mandelbox 20211128 PWR3 RGBA8.png|alt=A three-dimensional Mandelbox fractal of scale 3.|thumb|right|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. 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 this: repeatedly transform a vector z, according to the following rules:
  1. First, for each component c of z, if c is greater than 1, subtract it from 2; or if c is less than -1, subtract it from −2.
  2. Then, depending on the magnitude of the vector, change its magnitude using some fixed values and a specified scale factor.

Generation

The iteration applies to vector z as follows:
function iterate:
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 / ^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 the mandelbox contains a solid core. Consequently, its fractal dimension is 3, or n when generalised to n dimensions.
For the mandelbox sides have length 4 and for they have length.