Scitech  Practical fractals

Looking back on Mandelbrot’s contributions to math and beyond

History is a thousand teachers, and we would certainly do well to pay attention to lessons imparted to us by its prominent figures. One such figure is Benoit Mandelbrot who, on October 14, 2010, died from pancreatic cancer at the age of 85. Mandelbrot will be missed by many – but his impact will remain forever inscribed in history books.

Throughout his lifetime, Mandelbrot made many contributions in fields ranging from fluid dynamics to information technology to economics and financial markets. His most significant achievement, however, was in mathematics where he defined the Mandelbrot set and coined the term “fractal.” Now, weeks after Mandelbrot has left us, his intriguing bequest warrants appreciation.

A fractal is a geometric shape that is defined by the invariance of scale: if you look at a figure and then look at it again under a microscope and you see the same shape, you’ve got yourself a fractal. Fractals are regarded as infinitely complex and irregular to the point that they cannot be described in Euclidean geometry (the x, y, and z-axes that we all dread as university students) – but they can be used to model commonplace things like clouds, coastlines, and even traffic.

Consider Ste. Catherine on a crowded day. Look every other car, then every three cars, then every four, five, six and so on. If the traffic looks the same – namely, a line of cars, no matter how much you zoom out– then a fractal is present.

Fractals are more than simply amusing: they have a rich breadth of applications that have gone a long way in numerous disciplines. Take Microsoft’s Encarta Encyclopedia. In 1992, the disc version contained several thousand articles and photographs, as well as hundreds of animations and maps. Yet Microsoft managed to cram all of this into less than 600 megabytes of data. That’s the equivalent of a little over one hundred songs or one relatively low-quality movie. How did they achieve this? The data was compressed using the principles of fractals.

Or how about medicine? To track esophageal pH, measurements can be made once every six seconds for 24 hours. But fluctuations in esophageal pH are consistent with fractal patterns. Instead of taking the pH every six seconds, measurement could be made, say, every 36 seconds. The shape of the data would look the same, yet would be achieved with only one sixth of the measurements.

As a final example, let’s look at the universe. It too is a fractal! Consider a planet with a moon revolving around it. Zoom out and you have a star system (ours is called the Solar System). Go out further and you have a cluster of such systems. Even further, galaxies. Further again, clusters of galaxies. And finally, gigantic superclusters. Keep zooming out, and the picture remains roughly the same: clusters of celestial bodies, held together by gravity.

There is no doubt that numerous advancements in science and mathematics, as well as several other fields, can be attributed to the application of fractals. For this, we have Mandelbrot to thank. And although he will be dearly missed, he will be remembered for his inquisitive mind and his ingenious discoveries.