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.[2]

A 'scale 2' Mandelbox
A 'scale 3' Mandelbox

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.[2]

Properties

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 the mandelbox contains a solid core. Consequently its fractal dimension is 3, or n when generalised to n dimensions.[6]

For the mandelbox sides have length 4 and for they have length [6]

See also

References
  1. Lowe, Tom. "What Is A Mandelbox?". Archived from the original on 8 October 2016. Retrieved 15 November 2016.
  2. Leys, Jos (27 May 2010). "Mandelbox. Images des Mathématiques". French National Centre for Scientific Research. Retrieved 18 December 2019.
  3. "Negative 1.5 Mandelbox - Mandelbox". sites.google.com.
  4. "More negatives - Mandelbox". sites.google.com.
  5. "Patterns of Visual Math - Mandelbox, tglad, Amazing Box". web.archive.org. February 13, 2011.
  6. Chen, Rudi. "The Mandelbox Set".
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