By Soler97 (Own work) [Public domain], via Wikimedia Commons

By Soler97 (Own work) [Public domain], via Wikimedia Commons

By Soler97 (Own work) [Public domain], via Wikimedia Commons

By Soler97 (Own work) [Public domain], via Wikimedia Commons

By Soler97 (Own work) [Public domain], via Wikimedia Commons

By Soler97 (Own work) [Public domain], via Wikimedia Commons

The **Mandelbulb** is a three-dimensional analogue of the Mandelbrot set, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009. (wikipedia)

A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions. However, this set does not exhibit detail at all scales like the 2D Mandelbrot set does.

White and Nylander’s formula for the “*n*th power” of the 3D vector is

where , , and .

They use the iteration where is defined as above and is a vector addition. For *n* > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of “lobes” controlled by the parameter *n*. Many of their graphic renderings use *n* = 8. The equations can be simplified into rational polynomials for every odd n, so it has been speculated by some that odd-numbered powers are more elegant. For example *n* = 3 is simplified to:

Mandelbulber is an experimental application that helps to make rendering 3D Mandelbrot fractals much more accessible. A few of the supported 3D fractals:*Mandelbulb, Mandelbox, BulbBox, JuliaBulb, Menger Sponge, Quaternion, Trigonometric, Hypercomplex, and Iterated Function Systems (IFS).* All of these can be combined into infinite variations with the ability to hybridize different formulas together.

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