Fractal geometry has broad appeal because of the amazing colorful shapes that can be created, but it’s easy to forget that there is actual information — and some pretty rigorous math — underlying the pretty pictures. Algebra and geometry are inextricably linked: in calculus, for example, mathematical functions can be represented as geometric curves — what David Berlinski, author of A Tour of the Calculus, whimsically dubbed the “face” of a particular function.
Fractals are the result of chaotic activity, much like the wreckage strewn in the wake of a tornado. Hypothetically, at least, you could glean information about the tornado’s path by examining the patterns in the wreckage. So with fractals: examine them at every level of detail and you can extract useful information about the underlying processes that created them.
That’s why folks were so excited earlier this week when an article appeared in New Scientist announcing the creation by an amateur fractal image maker named Daniel White of the first truly three-dimensional image of the most famous fractal of them all: the Mandelbrot set. The Mandelbrot set is a two-dimensional equation devised by mathematician Benoit Mandelbrot, who coined the term “fractals” in 1975 to describe the geometric shapes he created.
It’s an “iterative” process. As the New Scientist article explains so clearly, fractals are created by applying an equation to a number, getting an answer, and then applying the same equation to that result. Do this over and over, and eventually you will get a fractal, with the telltale mark of self-similarity: the same shapes recur at different levels of magnification
Other 3D fractals do exist, most notably the so-called “Menger sponge,” but to date, only White has produced the most accurate 3D version (so far) of the Mandelbrot set, which is tricky to bring to a higher dimension. White accomplished this not with complex mathematics, but by approaching the problem geometrically, specifically by exploiting something called the “complex plane.” Per New Scientist, this is
“… a mathematical landscape where ordinary numbers run from ‘east’ to ‘west,’ while ‘imaginary’ numbers, based on the square root of -1, run from ‘south’ to ‘north.’ Multiplying numbers on the complex plane is the same as rotating it, and addition is like shifting the plane in a particular direction.”
There’s an animation that demonstrates this brain-twisting concept here. White adapted the 2D complex plane to 3D and came up with a series of “near-miss” images, such as “spinning the 2D Mandelbrot fractal like wood on a lathe.” The problem is that you end up with an image that is too smooth, and lacks the degree of self-similar details — actual information — at different levels of magnification.
But eventually he found what he was looking for: The Mandelbulb. And a pretty little image it is, too! It’s not yet perfect, White admits, since there are still parts of the image that lack sufficient self-similar detail for a true fractal. He’s set the bar pretty high, however. And delighted fractal fans by producing the beauty of the Mandelbrot set in a new and higher dimension,
Image credit: Daniel White