View source for Mandelbox

{{Short description|Fractal with a boxlike shape}}
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{{Primary sources|date=December 2023}}
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[[Image:Mandelbox 20211127 1GP RGBA8.png|alt=A three-dimensional Mandelbox fractal of scale 2.|thumb|right|A "scale-2" Mandelbox]]
[[Image:Mandelbox 20211128 PWR3 RGBA8.png|alt=A three-dimensional Mandelbox fractal of scale 3.|thumb|right|A "scale-3" Mandelbox]]
[[File:Mandelbox 20211129 -1,5 RGBA8.png|alt=A three-dimensional Mandelbox fractal of scale -1.5.|thumb|right|A "scale -1.5" 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 set]]s, but, unlike the Mandelbrot set, can be defined in any number of dimensions.<ref>{{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 }}</ref> It is typically drawn in three dimensions for illustrative purposes.<ref name="Lowe">{{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
 }}</ref><ref name="leys">{{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}}</ref>

== 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.<ref name="leys" />

== Properties ==
A notable property of the mandelbox, particularly for scale −1.5, is that it contains approximations of many well known fractals within it.<ref>{{Cite web |url=https://sites.google.com/site/mandelbox/negative-1-5-mandelbox |title=Negative 1.5 Mandelbox – Mandelbox |website=sites.google.com}}</ref><ref>{{Cite web |url=https://sites.google.com/site/mandelbox/more-negatives |title=More negatives – Mandelbox |website=sites.google.com}}</ref><ref>{{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}}</ref>

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

For <math>\text{scale} < -1</math> the mandelbox sides have length 4 and for <math>1 < \text{scale} \leq 4 \sqrt{n} + 1</math> they have length <math>4 \cdot \frac{\text{scale} + 1}{\text{scale} - 1}</math>.<ref name = "properties"/>

==See also ==
* [[Mandelbulb]]
*[[Buddhabrot]]
*[[Lichtenberg figure]]

==References==
{{Reflist}}

==External links==
{{Commons category|Mandelboxes}}
* [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]

{{Fractal software}}
{{Fractals}}
{{Mathematics and art}}

[[Category:Fractals]]


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