Fans of British fantasy author Piers Anthony's Mode series may recall that the second book, Fractal Mode, speaks of a strange world that represents a perfect three-dimensional model of the famous (to mathematicians, anyway) Mandelbrot Set -- a stunning geometrical shape that results when you take a particular equation and apply it to a number, and then to the result, and then to each subsequent result after, ad infinitum.

I've written about chaos and fractal patterns before, most notably in the work of painter Jackson Pollock. “Chaos” — a word that typically denotes utter randomness — has a different meaning in the context of math and science. It applies to systems that only appear to be random on the surface; underneath is a hidden order. The stock market is a chaotic system, for example: a slight blip can be amplified many times over until the system “goes critical” and the market crashes. It’s known as the “butterfly effect”: a butterfly flaps its wings in Brazil, and the air disturbance amplifies over time and distance, eventually causing a tornado in Texas.

Fractal patterns are the mathematical offspring of chaos theory, the remnant of chaotic motion — wreckage strewn in the wake of a hurricane, for example. It's kind of like fossilized footprints left behind by now-extinct dinosaurs: those patterns are left behind by the movement of a chaotic system

2D Mandelbrot Set. Created by Wolfgang Beyer with the program Ultra Fractal 3. Via Wikimedia Commons.

What makes fractals so unusual -- and so visually appealing, and hence hugely popular -- is that such an image might appear to be haphazard on the surface, but look closer and you realize that there is, in fact, a single geometric pattern repeated thousands of times over at different size scales, just like those nested Russian dolls. That telltale pattern is known as “self-similarity.”

The Mandelbrot set is named after the late mathematician Benoit Mandelbrot, often called the "father of fractals," although he certainly wasn't the only person who developed them. Even the set that bears his name actually dates back to work by Pierre Fatou and Gaston Julia early in the 20th century.

Mandelbrot saw the first visualizations of the set back in 1980, while at IBM's T.J. Watson Research Center. He absolutely deserves credit for popularizing the Mandelbrot set, making it one of the most readily recognizable fractal shapes.

It's a wonderful example, too, of how complex elaborate patterns can emerge from a few simple rules (in this case, the iterative application of a single equation). No matter how many times you "zoom" into the generated image, you will still see those same exquisite details repeating at ever-smaller size scales.

But what happens when you go from two dimensions to three? You get a "Mandelbulb." I was reminded of this over the weekend when I stumbled across this stunning computer animation of a Mandelbulb modeling the movement of 250,000,000 particles:

It's actually no easy feat to translate the 2D Mandelbrot set into that third dimension. Mathematician and sci-fi author Rudy Rucker -- one of the founders of cyberpunk -- speculated about the possibility two decades ago, even writing a short story ("As Above, So Below") based on the concept in 1987. (You can read it here.) But computer processing power just wasn't up to the task back then; it required billions of calculations, for starters.

A Mandelbulb, created by Ondrej Karlik. Source: Wikimedia Commons.

It wasn't for lack of trying. People tried spinning the 2D fractal image, for example, or toyed with mathematical tricks in higher dimensions. But none produced a "true" fractal, according to Daniel White, an amateur fractal enthusiast in the U.K. who became fascinated by the challenge.

Rather than relying on complicated mathematics, he approached the problem geometrically. There are lots of ways of doing this, but two possibilities involve extending the initial flat 2D plane into a box or a sphere and then performing the same iterative process in the newly 3D space.

White came up with a basic equation in 2007 that almost did the trick. It wasn't until 2009 when White and his fellow enthusiast Paul Nylander collaborated successfully to produce the first Mandelbulb. Nylander realized he could raise White's equation to a higher power to produce a Mandelbulb.

Still, their resulting image wasn't an entirely true 3D fractal either. "There are still 'whipped cream' sections, where there isn't detail," White told New Scientist in 2009. "If the real thing does exist -- and I'm not saying 100% that it does -- one would expect even more variety than we are currently seeing."

True fractal or not, the Mandelbulb is a truly beautiful geometric shape.