Posted: Wednesday 17 November 2010
Mandelbrot will live on
The Polish mathematician Benoit Mandelbrot died on the 14th October of this year, aged 85. He is best known for his discovery of mathematical formulae which describe fractal shapes, and for the iconic picture of one of them, known as the Mandelbrot Set (pictured). Fractals have the remarkable property of having the same degree of detail in their outline, no matter how often they are magnified. If you zoom in on one edge of the computer–generated image of a fractal set you will keep seeing smaller versions which grow larger, and be able to keep zooming in forever without the shape becoming any simpler.
There are several reasons why fractal geometry is important, and not just a mathematical curiosity. Despite the complexity of the shapes, the formulae which produce them are disarmingly simple. They are iterative equations which are in more general cases central to our understanding of the boundaries between order and randomness. Chaos Theory is a new science which attempts to explain many natural phenomena which have resisted all of our attempts to model them using computer simulation, and fractals are a good analogy for what may be going on here.
Consider the process you go through in learning about something in science, for example how chemical reactions work. To begin with you need to understand that the world is composed of chemical elements, which have different properties. Going deeper, you learn that these elements can combine to form compounds, which again have different properties. Under certain conditions these compounds can react with each other. Going deeper you learn about the nature of bonding, and activation energy, enthalpy and entropy and consider ‘equilibrium’. Moving deeper still you find that reaction rates can be controlled by surface chemistry and crystal defects, by diffusion and by the pressure and temperature conditions. You can keep on looking and find that sadly, equilibrium is only a crude approximation and that a kind of apparent static equilibrium may be operating, or maybe all reactions are really non-equilibrium processes. No matter how deep you look, or hard you think... there is always a further level of complexity which can be explored.
Much scientific work in the last two centuries has been an attempt to simplify, categorise and put in neat pigeon-holes everything that we see around us (‘All science is either physics or stamp-collecting’ – Ernest Rutherford). But new models of natural processes are beginning to accept that there is a degree of randomness which controls the final outcome. Paradoxically, small changes in starting conditions can lead to massive changes in the end result, and while this may lead one to assume that natural processes are totally random, in fact they may often keep tracking towards a set number of possible outcomes or ‘attractors’. The same Mandelbrot Set is always produced from the relevant equations, and so it is really a picture of a small area of order or ‘anti-chaos’ within the wider ‘chaos’. The challenge is to find the right equations to produce similar areas of anti-chaos representing what we actually see happening in chemical reactions, or in local weather patterns, or even in fluctuations in the stock market....
The power of that idea makes it likely that Mandelbrot’s name will be remembered for a very long time.