25 Fractals
In March 1980, the state-of-the-art mainframe computer at the IBM research centre at
Yorktown Heights, New York State, was issuing its instructions to an ancient Tektronix
printing device. It dutifully struck dots in curious places on a white page, and when it
had stopped its clatter the result looked like a handful of dust smudged across the
sheet. Benoît Mandelbrot rubbed his eyes in disbelief. He saw it was important, but what
was it? The image that slowly appeared before him was like the black and white print
emerging from a photographic developing bath. It was a first glimpse of that icon in the
world of fractals – the Mandelbrot set.
This was experimental mathematics par excellence, an approach to the subject
in which mathematicians had their laboratory benches just like the physicists and
chemists. They too could now do experiments. New vistas opened up – literally.
It was a liberation from the arid climes of ‘definition, theorem, proof’, though a
return to the rigours of rational argument would have to come albeit later.
The downside of this experimental approach was that the visual images
preceded a theoretical underpinning. Experimentalists were navigating without a
map. Although Mandelbrot coined the word ‘fractals’, what were they? Could
there be a precise definition for them in the usual way of mathematics? In the
beginning, Mandelbrot didn’t want to do this. He didn’t want to destroy the magic
of the experience by honing a sharp definition which might be inadequate and
limiting. He felt the notion of a fractal, ‘like a good wine – demanded a bit of
aging before being “bottled”.’
The Mandelbrot set
Mandelbrot and his colleagues were not being particularly abstruse
mathematicians. They were playing with the simplest of formulae. The whole
idea is based on iteration – the practice of applying a formula time and time
again. The formula which generated the Mandelbrot set was simply x^2 + c.
The first thing we do is choose a value of c. Let’s choose c = 0.5. Starting with
x = 0 we substitute into the formula x^2 + 0.5. This first calculation gives 0.5
again. We now use this as x, substituting it into x^2 + 0.5 to give a second
calculation: (0.5)^2 + 0.5 = 0.75. We keep going, and at the third stage this will