Fractal geometry

Geometry. Its principles are taught to young students across the world. The Pythagorean theorem. Surface area and volume. Pi. This classical, or Euclidean, geometry is perfectly suited for the world that humans have created. But if one considers the structures that are present in nature, that which are beyond the realm of smooth human construction, many of these rules disappear. Clouds are not perfect spheres, mountains are not symmetric cones, and lightning does not travel in a straight line. Nature is rough, and until very recently this roughness was impossible to measure. The discovery of fractal geometry has made it possible to mathematically explore the kinds of rough irregularities that exist in nature.
In 1961, Benoit Mandelbrot was working as a research scientist at the Thomas J. Watson Research Center in Yorktown Heights, NY. A bright young academic who had yet to find his professional niche, Mandelbrot was exactly the kind of intellectual maverick IBM had become known for recruiting. The task was simple enough: IBM was involved in transmitting computer data over phone lines, but a kind of white noise kept disturbing the flow of information – breaking the signal – and IBM looked to Mandelbrot to provide a new perspective on the problem.
Since he was a boy, Mandelbrot had always thought visually, so instead of using the established analytical techniques, he instinctually looked at the white noise in terms of the shapes it generated – an early form of IBM’s now-renowned data visualization practices. A graph of the turbulence quickly revealed a peculiar characteristic. Regardless of the scale of the graph, whether it represented data over the course of one day or one hour or one second, the pattern of disturbance was surprisingly similar. There was a larger structure at work.
The problem was familiar to Mandelbrot, and he recalled the advice his mathematician uncle, Szolem Mandelbrojt, had given him years ago in France – attempt to make something of the obscure theories of iteration established by French mathematicians Pierre Fatou and Gaston Julia. Their work intrigued mathematicians around the world and revolved around the simplest of equations: z = z & sup2; + c. With a variable of z and parameter of c, this equation maps values on the complex plane – where the x-axis measures the real part of complex number and the y-axis measures the imaginary part ( i) of a complex number.
At the time of the advice, Mandelbrot couldn’t find any breakthrough, but the intellectual freedom he found at IBM allowed him to fully engage this new project. In 1980, building on the technology and talent of IBM, Mandelbrot used high-powered computers to iterate the equation, or use the equation’s first output as its next input. With these computers, Mandelbrot crunched and manipulated the numbers a thousand times over, a million times over, and graphed the outputs.
The result was an awkwardly shaped bug-like formation, and it was perplexing to say the least. But as Mandelbrot looked closer, he saw the detailed edges of this formation held smaller, repeating versions of the larger bug-like formation. What’s more, every smaller version held more complex detail than the previous version. These structures were not exactly alike, but the general shape was strikingly similar, it was only the details that differed. The specificity of these details, it turned out, was limited only to the power of the machine computing the equation, and similar shapes could continue on forever – revealing more and more detail, on an infinite scale.

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Fractal geometry