Fractal Power

Of earthquakes, snowflakes and computer images

Although Mandelbrot’s fractals began on the fringes of science, increasingly scientists have found the graphics cropping up all over their different subjects.

The strength of metals is one practical application: the surfaces of metals turn out to be fractal shaped and the Hausdorff dimension - the measure of roughness - provides a clue to the strength of the metal.

The prediction of earthquakes also involves fractals: graphs of the seismological activity across the Earth’s surface are not regular but demonstrate fractal qualities. Understanding the fractal geometry promises to provide clues to hidden patterns of behaviour that can be used to predict earthquakes.

One of the most mysterious processes in science is phase transition – when a liquid turns into a solid or vapourises into a gas. Again fractals and chaos seem to be at the heart of the shapes that a substance like ice forms as the water freezes. The reason behind the shape of a snowflake has taxed mathematicians for centuries, including the great astronomer Johannes Kepler in the early seventeenth century. Fractals have at long last provided the answer.

And fractals are now coming to a computer screen near you… One of the biggest challenges in computer systems has been encoding pictures in the smallest file that’s possible. If we tried to encode all the information in an image – its constituent pixels - we’d end up with a file that’s too huge to transmit. Formats like JPEG format reduce the size of the file, by discarding parts of the image that human eyes wouldn’t really notice.

In 1987, mathematician Michael Barnsley worked out how he could use fractals to encode an image. The encoding program works out how each part of the image can be described by a fractal formula. What’s recorded is not in any way the actual array of pixels, but a set of formulae that will recreate the original image.

The fractal compression technique works best – not surprisingly – on natural scenes such as faces and landscapes, rather than diagrams or text. The encoding process can be extremely lengthy and slow; but it’s very quick to decode.

At the moment, fractal compression is subject to several patents, so it hasn’t been rolled out into the public domain. But when you start seeing files labelled .fif (fractal image file) on your computer, you’ll know that fractal maths has really arrived in your life!

For more information

Create and explore your own Mandelbrot Set
http://www.h-schmidt.net/MandelApplet/mandelapplet.html

Fractals Unleashed website
hhttp://library.thinkquest.org/26242/full/