Fractal Power

New dimensions

One reason that fractal structure is so important for something like the lung is that it gets a large amount of surface area into a small volume like the chest. Indeed it is possible to construct geometric shapes with infinite length that fit into a finite amount of space. One of the earliest examples is called Koch’s snowflake.

By adding smaller and smaller triangles on to the edges of the snowflake one can prove mathematically that the length of the snowflake spirals off to infinity yet it is bounded in finite space. It was first described in 1904 by the Swedish mathematician Helge von Koch.

The infinite length of the perimeter of the snowflake led Mandelbrot to wonder how long the perimeter of a country like Great Britain might be. When scientists started measuring it on ever smaller and smaller scale, the length increased in a linear fashion. Indeed it seems that all the continents on the surface of the Earth appear to have infinite length – all that is except Africa. No one knows why the coastline of Africa is exceptional in having a finite length.

In 1918 a German mathematician, Felix Hausdorff introduced a way to measure the roughness of these shapes. A straight line has dimension one because it only moves in one direction. A square on the other hand has dimension two because it sweeps out two different directions. Hausdorff devised a new notion of dimension which allowed one to calculate that Koch’s snowflake had a dimension somewhere between a straight line and a square. In fact calculation gave the snowflake a dimension of log4/log3 or about 1.26.

The Mandelbrot Set

But it wasn’t clouds but the advent of the computer which once again revealed these fractals, the face of chaos, hiding everywhere. Mandelbrot’s enduring claim to fame is a graphic that embodies the idea of fractals.

Amazingly, this complex and beautiful diagram springs from a relatively simple equation. Mandelbrot effectively took a piece of graph paper, and wrote an equation that he could apply to each point on the two-dimensional sheet. The equation starts with the two co-ordinates of the point itself. From these, it calculates the co-ordinates of a “daughter point”. Mandelbrot then applied the equation to the daughter point, to calculate the position of a “grand-daughter point”; and so on.

For some points on the graph, the succession of “descendant points” would head out to infinity; for other points on the graph, the descendants all stayed within a particular region of the graph paper. It was like a feedback process in a loudspeaker: each new point is fed back into the equation. Some starting points cause a feedback which gets louder and louder. But other points stabilize.

Mandelbrot decided to colour a point in white if it whizzed off to infinity when you iterated the equation. A point was coloured black if it stayed within a bounded region of the graph. Mandelbrot first expected that he’d create a shape that would look rather like a simple black disc. Within the disc things behaved one way, outside the disc you got another behaviour. But the graphic produced by his computer was as shocking as it was beautiful.

The Mandelbrot Set, as the shape is now known, is a lovely kidney-shaped graphic that can be automatically coloured for impact. As you magnify any part of the Set, you see more and more intricate detail, forming beautiful abstract patterns. Because the Set is fractal, you can magnify any part of it as much as you like, and still see detail – and these details will never repeat, but always look fresh and different. The Set never gives away what magnification you are looking at.

It also demonstrated another example of chaos. Shift the point very slightly where you start the iteration, and you go from bounded behaviour to spiralling off to infinity. No could believe that such a simple law from which Mandelbrot began could produce anything so complex.

As we being beautiful, the Mandelbrot Set has many - unexpected - practical uses. Read on to find out more....