Fractal geometry
A revolutionary system helps measure and understand the wondrous complexity of nature
A close-up view of a head of Romanesco cauliflower, a natural example of fractal geometry

For thousands of years, the smooth and regular shapes of classical geometry — lines and planes, triangles and cones, circles and spheres — have helped humans measure and understand the world around them. But when it comes to describing the complexity of nature, these smooth constructions often turn out to be the wrong kind of abstraction. Nature is rough and irregular, and until recently this irregularity was impossible to measure. That changed, thanks to the work of Benoît Mandelbrot, a maverick mathematician and research scientist. He noted that clouds are not perfect spheres, mountains are not symmetric cones, and lightning doesn’t travel in a straight line.

Mandelbrot, who spent 35 years at IBM, pioneered the discovery of the principles that would later form an entirely new field — fractal geometry. He coined the term “fractal” to refer to a new class of mathematical shapes whose uneven contours mimicked the irregularities found in nature.

Since its discovery, fractal geometry has informed breakthroughs in everything from biology and telecommunications to climate science and filmmaking. Of central importance to its discovery was the advent of the computer. Because Mandelbrot had access to IBM’s high-powered computers, he was among the first to use computer graphics to create and display complex fractal geometric images and reveal hidden patterns.

Mandlebrot believed that fractals were a more natural way to describe the world around us
Thinking visually
Mandlebrot finds patterns in noise

In 1961, Mandelbrot was working for IBM as a research scientist at the Thomas J. Watson Research Center in Yorktown Heights, New York. The company was involved in transmitting computer data over phone lines, but a kind of white noise kept disturbing the information flow, and Mandelbrot was tasked with providing a new perspective on the problem.

Since he was a boy, Mandelbrot had thought in visual terms, so he instinctively viewed the white noise in terms of the shapes it generated. 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 a day, an hour or a second, the pattern of disturbance was surprisingly similar. There was a larger structure at work.

Mandelbrot knew he was onto something with potentially far-reaching implications. The seemingly chaotic white noise hid a degree of order, and this insight could be applied to many aspects of not only the natural world but also human constructs such as the seemingly random fluctuations of financial markets.

A new word for a new type of geometry
Measuring a jagged landscape

To describe such structures, Mandelbrot coined the term fractal, a derivative from the Latin word fractus, meaning broken or uneven. Fractals are a form of geometric repetition, in which successively smaller copies of a pattern are nested inside each other, so that the same intricate shapes appear repeatedly.

Mandelbrot’s work on fractals has roots in a question he encountered as a young researcher: How long is the coast of Britain? The answer, he was surprised to discover, depends on how closely one looks. On a map, the contours of Britain may appear relatively smooth, but zooming in will reveal jagged edges. When these edges are properly accounted for, the coast becomes far longer. Zooming in further reveals even more coastline, all the way down to atomic scale. As a result, the length of the coastline is, in a sense, infinite. Mandelbrot believed that fractals were in many ways a more natural way to describe the world around us than the artificially smooth objects of traditional geometry.

At IBM, Mandelbrot found a group of like-minded colleagues and was afforded a great deal of freedom to pursue his investigations. The combination of computing power, an inquisitive research team and intellectual independence provided the necessary tools to develop a new branch of geometry.

 

Fractals everywhere
Clouds, trees, human heartbeats

It wasn’t until the 1982 book, The Fractal Geometry of Nature, in which Mandelbrot highlighted the many occurrences of fractal objects in nature, that his insights would receive widespread attention. For example, each split in a tree – from trunk to limb to branch to twig – is remarkably similar, yet with subtle differences that provide increasing detail and insight into the inner workings of the entire tree. His fractals explorations touched off a revolution in numerous areas of science, industry and art. For instance, the outlines of clouds and coastlines, once considered unmeasurable, could now be approached in a rigorously quantitative fashion. The geometry of fractals has also helped explain how galaxies cluster, how mammalian brains fold as they grow, and how landscapes fragment in an earthquake zone.

Fractal geometry is also used to model the human lung, blood vessels, neurological systems, and many other physiological processes. The human heart was always thought to beat in a regular, linear fashion, but studies have shown that its true rhythm fluctuates in a distinctively fractal pattern. Blood is also distributed throughout the body in a fractal manner, and researchers have created models of blood flows for early detection of cancerous cells. Graphic designers and filmmakers took advantage of Mandelbrot’s creation to model lifelike “fractal worlds,” prominent in films such as Star Trek II: The Wrath of Khan and Return of the Jedi.

Fractals also play a role in climate science. Researchers have shown that the distribution of large branches to smaller branches in a single tree exactly replicates the distribution of large trees to smaller trees in an entire forest. This information is used to measure how much carbon dioxide a single forest is capable of processing, with the goal of applying the findings to every forest on Earth, quantifying how much carbon dioxide the entire world can safely absorb.

We have merely scratched the surface of what fractal geometry can teach us about the world
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