Mandelbox
Page Module:Message box/ambox.css has no content.Page Template:Multiple issues/styles.css has no content.
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages)
|
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:
- 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 in pseudocode:[<span title="Script error: No such module "decodeEncode".">clarification needed]
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.[3]
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 the mandelbox contains a solid core. Consequently, its fractal dimension is 3, or n when generalised to n dimensions.[7]
For the mandelbox sides have length 4 and for they have length .[7]
See also
References
Page Template:Reflist/styles.css has no content.
- ^ Page Module:Citation/CS1/styles.css has no content.Lowe, Tom. "What Is A Mandelbox?". Archived from the original on 8 October 2016. Retrieved 15 November 2016.
- ^ Page Module:Citation/CS1/styles.css has no content.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.
- ^ a b Page Module:Citation/CS1/styles.css has no content.Leys, Jos (27 May 2010). "Mandelbox. Images des Mathématiques" (in français). French National Centre for Scientific Research. Archived from the original on 19 December 2019. Retrieved 18 December 2019.
- ^ Page Module:Citation/CS1/styles.css has no content."Negative 1.5 Mandelbox – Mandelbox". sites.google.com.
- ^ Page Module:Citation/CS1/styles.css has no content."More negatives – Mandelbox". sites.google.com.
- ^ Page Module:Citation/CS1/styles.css has no content."Patterns of Visual Math – Mandelbox, tglad, Amazing Box". February 13, 2011. Archived from the original on February 13, 2011.
- ^ a b Page Module:Citation/CS1/styles.css has no content.Chen, Rudi. "The Mandelbox Set".
External links
Page Module:Side box/styles.css has no content.Page Template:Sister project/styles.css has no content.
- Gallery and description
- Images of some Mandelbox cubes Script error: No such module "webarchive".
- Video : zoom in the Mandelbox cube
Template:Fractal software Lua error in package.lua at line 80: module 'Module:Navbox/configuration' not found. Lua error in package.lua at line 80: module 'Module:Navbox/configuration' not found.
