Geeky Math Equation Creates Beautiful 3-D World

By Alexis Madrigal

Editor’s note: We are rerunning this gallery of 3-D images inspired by the work of Benoit Mendelbrot, who is best known for popularizing fractal mechanics. Mandelbrot died Oct. 14, 2010, at the age of 85.

The quest by a group of math geeks to create a three-dimensional analogue for the mesmerizing Mandelbrot fractal has ended in success.

They call it the Mandelbulb. The 3-D renderings were generated by applying an iterative algorithm to a sphere. The same calculation is applied over and over to the sphere’s points in three dimensions. In spirit, that’s similar to how the original 2-D Mandelbrot set generates its infinite and self-repeating complexity.

If you were ever mesmerized by the Mandelbrot screen saver, the following images are worth a look. Each photo is a zoom on one of these Mandelbulbs.

Daniel White, the amateur fractal image maker who coordinated the Mandelbulb effort, admits this creation isn’t exactly the Mandelbrot in 3-D. It’s mesmerizing and beautiful, but as he notes, only some versions of their original formula generate the kind of detail and complexity they are looking for. Their original equation doesn’t work very well unless you take it beyond the 2nd power. The picture above, White says, doesn’t have the level of detail that should be there.

“That means the biggest secret is still under wraps, open to anyone who has the inclination, and appreciation for how cool this thing would look,” White wrote on his website.

If you take the original equation to the 8th power, it generates the most beautiful and detailed images. White and his collaborators aren’t sure why, but it’s hard to argue with images like the one above.

White and his collaborators spend quite a lot of time zooming through their creation looking for interesting structures. Here we see a slice of what they call the “Mandelbrot gateau,” a reference to the French cake.

Even though the entire structure is generated from just the single equation, what it produces after thousands of iterations can seem like the geological features of a planet we’ve yet to discover.

At times, the complexity of the structures inside the Mandelbulb astonish even White himself.

“Seriously, this universe has got to be quite messed up to be harboring math secrets capable of this kind of Baroquian beauty,” he wrote in the caption for this image of pillars and bridges.

Another area of interest is the “Cave of Lost Secrets.” They’ve even created an almost Tweetably short science-fiction story about the region.

“This ancient half-mile-high cave still exists (now underwater) from a planet several billions of light years away from Earth,” he wrote. “It was built by a (now extinct) intelligent race of beings who also discovered the 3-D Mandelbulb we are witnessing on this page. Inside the cave however, lies — amongst other technological and mathematical secrets — the last remaining scroll which contains the much deeper secret of the even more incredible real 3-D Mandelbrot formula.”

Some of the best evidence that a better Mandelbulb could be out there is that some sections of the current version just don’t have enough detail, even at the 8th power. “Whipped cream” parts like the one above are not what a true 3-D Mandelbrot would look like.

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