# Pretty maths

Bear with this post as it goes through some equations at the beginning, but it is worth it. We’ll be doing some of the calculations to get this picture:

This is the set of numbers “c” such that  is bounded. These z are complex numbers, which we’ll ignore for now. It is much easier to understand if we look at some examples:

Let’s say c = -1.

We start with $z_0 = 0$

$z_1 = z_0^2 + c = 0^2 - 1 = -1$

$z_2 = z_1^2 + c = (-1)^2 - 1 = 0$

$z_3 = z_2^2 + c = (0)^2 - 1 = -1$

This is repeating, and the numbers are bounded.

Let’s now try c = 0.5.

We start with $z_0 = 0$

$z_1 = z_0^2+c = 0^2 + 0.5 = 0.5$

$z_2 = z_1^2 +c = (0.5)^2 + 0.5 = 0.75$

$z_3 = z_2^2+c = (0.75)^2 + 0.5 = 1.0625$

$z_4 = z_3^2 + c = (1.0625)^2 + 0.5 = 1.62890625$

$z_5 = z_4^2 + c = (1.62890625)^2 + 0.5 = 3.15333557129$

$z_6 = ...$

We can see that these numbers are getting bigger and bigger, and it is not bounded.

One more: c=-1.9

$z_0 = 0$

$z_1 = z_0^2 + c = 0^2 - 1.9 = -1.9$

$z_2 = z_1^2 + c = (-1.9)^2 - 1.9 = 1.71$

$z_3 = z_2^2 + c = (1.71)^2 - 1.9 = 1.0241$

$z_4 = z_3^2 + c = (1.0241)^2 - 1.9 = -0.85121919$

$z_5 = z_4^2 + c = (-0.85121919)^2 - 1.9 = -1.17542589058$

$z_6 = ...$

It bounces around a lot, never getting very big or very small, so it is bounded. It is kinda fun to sit with a calculator and try this.

Mathematicians call this kind of system “chaos”, as it is very sensitive to the starting conditions. Sometimes this is called the butterfly effect. Note that chaotic is not the same as random: in chaotic systems if you know everything about the initial conditions you know what will happen, whereas in random systems even if you knew everything about the initial conditions you wouldn’t know what was going to happen.

Benoit Mandelbrot was one of the first mathematicians to have access to a computer. Hopefully you can also see now why Benoit Mandelbrot needed a computer to work these out. He repeated this for lots of values of c. The pretty picture we started with is really a plot of the set of c (called the Mandelbrot set), where the colours indicate what happens to the sequence (eg how quickly it converges, if it does).

You can zoom into the colourised picture to see how complex this is here. Lots of people (me included) think it is pretty cool. It is really worth taking a look to appreciate the complexity.

## Other than being pretty, why does this matter?

Stepping back: This picture is made from the formula . This is so simple, and yet gives rise to infinite complexity. In the words of Jonathan Coulton,

Infinite complexity can be defined by simple rules

Benoit Mandelbrot went on to apply this to the behaviour of economic markets, among other things. Later people have applied this to fluid dynamics (video), medicine, engineering, and many other areas. Apparently there is even a Society for Chaos Theory in Psychology & Life Sciences..!