Mandelbox Explained

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] [3]}

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 (which corresponds to a dimension), 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(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.^{[4] [5] [6]}

For

1 < | scale | < 2 the mandelbox contains a solid core. Consequently, its fractal dimension is 3, or n when generalised to n dimensions.^[7]

For

scale < -1 the mandelbox sides have length 4 and for 1 < scale \leq 4 \sqrt{n} + 1 they have length 4 ⋅ (scale +1)/(scale -1) .^[7]

See also

• Mandelbulb
• Buddhabrot
• Lichtenberg figure

External links

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

Notes and References

1. Web site: Lowe . Tom . What Is A Mandelbox? . 15 November 2016 . https://web.archive.org/web/20161008202249/https://sites.google.com/site/mandelbox/what-is-a-mandelbox . 8 October 2016 . dead .
2. Book: Lowe , Thomas . 2021 . Exploring Scale Symmetry . Fractals and Dynamics in Mathematics, Science, and the Arts: Theory and Applications . 06 . World Scientific . 10.1142/11219 . 978-981-3278-55-4. 224939666 .
3. Web site: Jos . Leys . Mandelbox. Images des Mathématiques . fr . . 27 May 2010 . 18 December 2019.
4. Web site: Negative 1.5 Mandelbox – Mandelbox . sites.google.com.
5. Web site: More negatives – Mandelbox . sites.google.com.
6. Web site: Patterns of Visual Math – Mandelbox, tglad, Amazing Box . https://web.archive.org/web/20110213231307/http://www.miqel.com/fractals_math_patterns/mandelbox_3d_fractal.html . dead . February 13, 2011 . February 13, 2011.
7. Web site: Chen . Rudi . The Mandelbox Set .