Years ago, many fractal enthusiasts viewed these intricate and majestic patterns and thought "Wouldn't this be incredible in 3-D?". Dozens of methods have been developed to view fractals in an arbitrary number of dimensions. Quaternions are 3D shadows of 4D Julia Sets, which, if sliced in a plane, reveal the corresponding 2 dimensional Julia Set.
Other attractive developments include 'Quasi-Fuchian" fractals and the recent popularity of the 'Mandelbulb" 3D Mandelbrot - which was exclusively discovered and implemented collaboratively by the inquisitive genius folks at Fractalforums.com. Go there to see what is literally the cutting edge of fractal discovery on a daily basis. Really!
Listed Below: The many ways fractals have been rendered in 3 Dimensions, culminating in the Mandelbulb & Mandelbox!
Iteration-based 3-D Texture Mapping
Mandelbrot and Julia fractals with escape time translated into height values. Essentially a textured 2-D fractal.
"Flame" Fractals, 3D Iterative Function Systems
A beautiful variety of lace-like and organic texture fractals, generally IFS and Strange Attractors mapped in 3D..
Quaternion Julia Sets
Quaternion Julia sets are constructed just like complex Julia sets. Each point in four-space can be represented by a quaternion. That quaternion is then run through the function zn+1 = zn2 + c many times. If the result goes to infinity, the point is not in the set. If the result does not go to infinity, that point is in the set. Since quaternions have four parts, we can graph them in four dimensions..
BuddhaBrot 3D Mandelbrot Method
The Buddhabrot is a special rendering of the Mandelbrot set which, when traditionally oriented, resembles to some extent certain depictions of the Buddha. The image can be thought of as a map of effective viscosity acting against a particle traveling through the positions of a point not in the mandelbrot set, as it is repeatedly iterated until it escapes the set boundary, with bright areas representing high 'viscosity'.
Organic 3D Fractals
The edible fractal food "Romanesco" broccoli displays a stunning spiral/fibonacci surface texture. Some 3D fractals are a close match..
Quasi-Fuchsian Fractals approximate a similar topology.
The Mandelbulb & Variants
Daniel White and Paul Nylander constructed the 3D Mandelbulb, a 3-dimensional analog of the Mandelbrot set, using an hypercomplex algebra based on spherical coordinates..
The image below shows a rotation of the 8th order variation of this fractal. Daniel White was very excited when he first saw type of rendering because it has significantly more intricate 3D details than the quadratic version. He wrote a beautiful article about it, calling it the "Mandelbulb" (I believe it was Rudy Rucker who coined the name). The Mandelbulb rapidly grew in popularity, and it was presented in many popular articles including Slashdot, Frankfurter Allgemeine newspaper, New Scientist magazine, Wikipedia, Science & Vie magazine, Discover Magazine, and Discovery News and hundreds of blogs.
More details emerge in these zooms into the Mandelbulb found on Fractalforums.com.
Cross-Section of Mandelbulb reveals Mandelbrot-like shapes. Image by Jos Leys Sections of this page were compiled from various sources, including:
Paul Nylander http://bugman123.com/Hypercomplex/index.html - Ross Hilbert http://www.fractalsciencekit.com/
http://www.dhushara.com/DarkHeart/DarkHeart.htm - http://www.brainjam.ca/fractals.html