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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 (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.

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.

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 ${\displaystyle 4\cdot {\frac {{\text{scale}}+1}{{\text{scale}}-1}}}$.

See also

• Mandelbulb
• Buddhabrot
• Lichtenberg figure

References

1. Lowe, Tom. "What Is A Mandelbox?". Archived from the original on 8 October 2016. Retrieved 15 November 2016.
2. Lowe, Thomas (2021). Exploring Scale Symmetry. Fractals and Dynamics in Mathematics, Science, and the Arts: Theory and Applications. Vol. 06. World Scientific. doi:10.1142/11219. ISBN 978-981-3278-55-4. S2CID 224939666.
3. ^ Leys, Jos (27 May 2010). "Mandelbox. Images des Mathématiques" (in French). French National Centre for Scientific Research. Retrieved 18 December 2019.
4. "Negative 1.5 Mandelbox – Mandelbox". sites.google.com.
5. "More negatives – Mandelbox". sites.google.com.
6. "Patterns of Visual Math – Mandelbox, tglad, Amazing Box". February 13, 2011. Archived from the original on February 13, 2011.
7. ^ Chen, Rudi. "The Mandelbox Set".

External links

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

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