Mandelbox

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Image:Mandelbox 20211127 1GP RGBA8.png

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File:Mandelbox 20211129 -1,5 RGBA8.png

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.{{cite web |last1=Lowe |first1=Tom |title=What Is A Mandelbox? |url=https://sites.google.com/site/mandelbox/what-is-a-mandelbox |accessdate=15 November 2016 |archiveurl=https://web.archive.org/web/20161008202249/https://sites.google.com/site/mandelbox/what-is-a-mandelbox |archivedate=8 October 2016 |url-status=dead }} It is typically drawn in three dimensions for illustrative purposes.{{cite book

|last=Lowe

|first=Thomas

|date=2021

|title=Exploring Scale Symmetry

|series=Fractals and Dynamics in Mathematics, Science, and the Arts: Theory and Applications

|volume=06

|url=https://www.worldscientific.com/worldscibooks/10.1142/11219

|publisher=World Scientific

|doi=10.1142/11219

|isbn=978-981-3278-55-4

|s2cid=224939666

}}{{cite web |first=Jos |last=Leys |title= Mandelbox. Images des Mathématiques |lang=fr |publisher= French National Centre for Scientific Research |date=27 May 2010 |url= http://images.math.cnrs.fr/Mandelbox.html |accessdate=18 December 2019}}

Simple definition

The simple definition of the mandelbox is this: repeatedly transform a vector z, according to the following rules:

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.
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:{{clarify|reason=What does the function return? Or is z sent to the iterate function by reference, like in C++?|date=January 2024}}

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.{{Cite web |url=https://sites.google.com/site/mandelbox/negative-1-5-mandelbox |title=Negative 1.5 Mandelbox – Mandelbox |website=sites.google.com}}{{Cite web |url=https://sites.google.com/site/mandelbox/more-negatives |title=More negatives – Mandelbox |website=sites.google.com}}{{Cite web |url=http://www.miqel.com/fractals_math_patterns/mandelbox_3d_fractal.html |archive-url=https://web.archive.org/web/20110213231307/http://www.miqel.com/fractals_math_patterns/mandelbox_3d_fractal.html |url-status=dead |archive-date=February 13, 2011 |title=Patterns of Visual Math – Mandelbox, tglad, Amazing Box |date=February 13, 2011}}

For $1 < |\text{scale}| < 2$ the mandelbox contains a solid core. Consequently, its fractal dimension is 3, or n when generalised to n dimensions.{{cite web |last1=Chen |first1=Rudi |title=The Mandelbox Set |url=http://digitalfreepen.com/mandelbox370}}

For $\text{scale} < -1$ the mandelbox sides have length 4 and for $1 < \text{scale} \leq 4 \sqrt{n} + 1$ they have length $4 \cdot \frac{\text{scale} + 1}{\text{scale} - 1}$ .

References

External links

[https://sites.google.com/site/mandelbox/ Gallery and description]
[http://images.math.cnrs.fr/Mandelbox.html Images of some Mandelbox cubes]
[https://www.youtube.com/watch?v=7Pf6jZWguCc Video : zoom in the Mandelbox cube]

Category:Fractals