Mandelbox

A three-dimensional Mandelbox fractal of scale 2.
A 'scale 2' Mandelbox
A three-dimensional Mandelbox fractal of scale 3.
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.[1]. It is typically drawn in three dimensions for illustrative purposes.

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.[2][3][4]

For 1<|scale|<2 the mandelbox contains a solid core. Consequently its fractal dimension is 3, or n when generalised to n dimensions.[5]

For scale < -1 the mandelbox sides have length 4 and for 1 < scale <= 4√n+1 they have length 4(scale+1)/(scale-1)[5]

See also

Notes

  1. ↑ Lowe, Tom. "What Is A Mandelbox?". Archived from the original on 8 October 2016. Retrieved 15 November 2016.
  2. ↑ negative-mandelbox
  3. ↑ more-negatives
  4. ↑ mandelbox_3d_fractal
  5. 1 2 Chen, Rudi. "The Mandelbox Set".

References

  • Leys, Jos (2010), Mandelbox. Images des Mathématiques, CNRS
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