From 0 to Infinity in 26 Centuries (18 page)

It is this sensitivity to initial conditions that causes unpredictable, chaotic behaviour. Even when we completely understand the motion of the particles involved, we can never gather enough
detail in our measurements of their speed or mass or temperature to be able to predict an accurate long-term outcome.

The same applies to Turing’s observations in morphogenesis – although the formation of stripes on a zebra is governed by very simple mathematical and chemical rules, immeasurable,
tiny differences in each zebra means that they have huge variation in their stripes.

The computer has been key in the study of chaos mathematics because it allows you to take a model of a situation and run it forward, recalculating the many variables at each
stage. Without a computer it can be incredibly time consuming to perform each step, called an
iteration
. Computers are good for iterative processes, which helps in the next branch of
mathematics we shall look at:
fractals
.

Under the microscope

Although the term ‘fractal’ wasn’t coined until 1975, mathematicians have been fascinated by them for hundreds of years. A fractal is a geometric figure that
has what mathematicians call
self-similarity
: no matter at what scale you look at the image, you still see the same features.

A coastline is a good example of a fractal. Imagine you had a satellite image of a coastline. Without any clues like buildings, trees or boats, you would find it hard to tell whether you were
looking at 1,000 miles of coast, or 10 miles of an island. This is because the features – the river inlets, headlands, bays, etc. – look very similar at different
scales. The same is true of many features of the natural landscape. Astronauts on the moon found it very hard to gauge the size of boulders – without the haze of atmosphere, they could not
tell whether they were looking at a car-sized boulder 100m away or a much larger boulder 500m off.

Famous fractals

Swedish mathematician Helge von Koch (1870–1924) devised one of the first mathematical fractals, known as the
Koch snowflake
, which follows a very simple set of
rules: you start with an equilateral triangle, and then with each iteration you add another equilateral triangle to the middle third of each line in the diagram. This builds up a snowflake
structure as shown:

Besides looking pretty, this snowflake has some very interesting properties. It has the
self-similarity property
– if you zoom in on an edge you would not be able
to tell how many iterations had been performed. If you continued the iterations ad infinitum, the snowflake would have an infinite perimeter even though the snowflake has a finite area.

Polish mathematician Wacław Sierpi
ń
ski (1882–1969) invented another straightforward geometrical fractal, again based on a triangular theme. For
Sierpi
ń
ski’s
triangle
, you start with a solid equilateral triangle. You then cut out an equilateral triangle from the middle, which gives you three smaller triangles. At each iteration you cut a triangle
out of the middle of any remaining triangles to produce the fractal.

Again, we see self-similarity, as each part of the triangle looks like the whole.

It turns out that there are two other unexpected ways to draw Sierpi
ń
ski’s triangle. The first harks back to Blaise Pascal and his triangle. If you shade in all the odd numbers in the
triangle, you get Sierpi
ń
ski’s triangle.

The second way is an example of a
Chaos Game
. You draw the three corners of a triangle and pick a random point inside it. You then, again at
random, pick one of the corners of the triangle and move halfway towards it, making a dot at your new position. If you continue this iterative process, you build up Sierpi
ń
ski’s triangle
again! It can take a while by hand, but a computer can do this sort of thing very quickly.

The ultimate fractal

There are many other fractals, but the daddy of them all was created by French-American mathematician Benoit Mandelbrot (1924–2010). As a researcher for the computer
company IBM, Mandelbrot was initially interested in self-similarity because he felt it was a common feature of many things, including the stock markets and the way the stars are spread out across
the universe. Mandelbrot’s access to the computers at IBM enabled him to perform many iterations quickly, which allowed his studies to thrive. In 1975 he coined the term
‘fractal’.

In the early 1980s he used a computer to investigate for the first time the fractal known as the
Mandelbrot set
. Like so many of the fractals we have looked at, it has a remarkably simple
basis:

This iterative equation says that you get the next number in the sequence, z
_n+1
, by taking the current number z
_n
, squaring it and adding another number, c,
to it.

You choose a number, c, and test it using the iteration (starting with z
₀
= 0). Some values of c cause z to get larger and larger, but for others it gets closer and closer to a stable
value or else keeps moving around without ever getting larger than 2.

z and c are both complex numbers, which means they have both a real part and an imaginary part (see
here
). So each value of c has, in effect, two values, which we can plot on a graph as a
point. If you plot all the points where c doesn’t ever lead to a value larger than 2, you get an image like this:

If you shade the unstable points on the graphs to indicate how many iterations it took them to become larger than two (the brighter the shade, the more iterations), the famous
image emerges.

We can see many self-similar features on the Mandelbrot set, but what is perhaps most amazing is the complexity that arises even though the governing equation is so simple. You
can literally zoom into the fractal for ever, a mystery that has led to this fractal assuming the mantle the Thumbprint of God.

As we have seen, the increasing use of computers has also enabled mathematicians to find numeric solutions to equations that cannot be solved using algebra. It is now commonplace for scientists,
engineers, designers and inventors to use computer methods to help them in their work. The experiments at the Large Hadron Collider in Europe produce 15 million gigabytes of data each year.

This experiment and many others that require a large amount of number-crunching are now able to exploit the redundant computing power of home computers via the Internet. This
distributed
computing
allows the data to be dealt with far more quickly and allows members of the public to be involved in the frontiers of scientific research.

Modern Mathematics