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.
The simple definition of the mandelbox is, for a vector z, for each component in z (which corresponds to a dimension), if the absolute value of the component is greater than 1, subtract it from either 2 or −2, depending on the z.
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.
For the mandelbox sides have length 4 and for they have length .
- Lowe, Tom. "What Is A Mandelbox?". Archived from the original on 8 October 2016. Retrieved 15 November 2016.
- Lowe, Thomas (2021). Exploring Scale Symmetry. World Scientific. ISBN 978-981-3278-55-4.
- Leys, Jos (27 May 2010). "Mandelbox. Images des Mathématiques" (in French). French National Centre for Scientific Research. Retrieved 18 December 2019.
- "Negative 1.5 Mandelbox – Mandelbox". sites.google.com.
- "More negatives – Mandelbox". sites.google.com.
- "Patterns of Visual Math – Mandelbox, tglad, Amazing Box". February 13, 2011. Archived from the original on February 13, 2011.
- Chen, Rudi. "The Mandelbox Set".
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