3D Fractals

What Are 3D Fractals?

In 2009, after a two-year effort, a group of innovators from Fractal Forums found a way to project the Mandelbrot set and similar equations into three dimensional space.

A typical 8th power Mandelbulb, rendered with Mandelbulber.

A typical Mandelbulb, generated with Mandelbulber

This mathematical transformation that manifested the two-dimensional Mandelbrot set as the three-dimensional Mandelbulb, Daniel White and Paul Nylander (and the rest of the group at Fractal Forums) opened the door into a world populated by a previously unknown kind of object: the 3D fractal.

The Mandelbulb and the Mandelbox (discovered in 2010 by Tom Lowe) are ‘pure’ manifestations of the Mandelbrot equation and exhibit the same kind of bottomless, self-similar detail. The Mandelbulb and Mandelbox are sometimes described as “cousins.”

Beyond these two shapes exist a wild variety of endlessly detailed 3D fractals. They are formed using different projections of the Mandelbrot set, projections of other equations, folding and symmetry-making transformations, and hybridizations that mix the Mandelbrot set with other equations. As fractal-generating/imaging software evolves, the range of 3D fractal objects is growing.


A typical Mandelbox, generated with Mandelbulb 3D (MB3D)

A typical Mandelbox, generated with Mandelbulb 3D (MB3D)

3D fractals are a range of chaotic equation-based objects—most often derived from- or related to- the Mandelbrot set. These are also called “Mandelmorphs.” The term “Mandelmorphic art” is used to describe art made with with these kinds of forms. Most Mandelmorphic art is image and video based—but the Mandelbulb was even 3D printed shortly after its discovery.

Mandelmorphosis is 3D fractal generation, or formation. The term combines the prefix “Mandel-” referring to the work of Benoit Mandelbrot with the suffix “-morph” meaning “form.”

Visit the Mandelmorphosis page on Facebook, administrated by 3D fractal artist John Vega, for an astounding range Mandelmorphic art.

An Emerging Field

The hunt for the Mandelbulb took two years and concluded in November 2009, when the Fractal Forums group announced the discovery of the Mandelbulb, and the news broke to the world. Since then, the community of 3D fractal enthusiasts has not stopped innovating—developing new maths, new imaging methods, and exploring new kinds of 3D fractal objects beyond the Mandelbulb.

A hybrid 3D fractal created with Mandelbulb 3D (MB3D). 'Base Response #8' © Matthew Haggett, 2014

A hybrid 3D fractal created with Mandelbulb 3D (MB3D). ‘Base Response #8’ © Matthew Haggett, 2014

The New Scientist article from Nov 18, 2009 that broke the news of the Mandelbulb’s discovery, provides a concise description of the mathematical transformation that manifests the Mandelbrot set as the Mandelbulb: “(Daniel White) took the geometrical properties of the ‘complex plane,’ where multiplication becomes rotation and addition becomes movement of the plane in a particular direction, and applied them to a three-dimensional space.” [link]

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