## Infinity in a grain of sand

The wonder I felt when I first came across the mathematics of complexity has never really left me.

It brought me face to face with the reality that even the seemingly simplest thing can be vastly more complex than my mind can comprehend. It goes against intuitive understanding and just leaves me feeling amazed. I’ll try and show you what I mean.

I think a good example is the very well known fractal, the Mandelbrot set.

The equation that generates the Mandelbrot set is almost the simplest equation possible yet leads to an infinity complex pattern. That means a pattern that when you zoom in to some detail just keeps getting more detailed every time you zoom in to take a closer look….forever!!!

Check out the video of zooming in. You might want to turn the sound off though :-)

Here is the Mandelbrot equation:

## *x*_{n+1} = *x*_{n}^{2} + *c *

The Mandelbrot set is generated by iterating this equation for different values of C which is a complex number. C is in the Mandelbrot set if X remains bounded under iteration starting from 0 i.e. X does not go to infinity. The pattern is formed by plotting the results in the complex plane formed by the values of C.For those without a mathematics background, X & C are a special type of number called a complex number, which has two parts; a real part and an imaginary part.

imaginary numbers are the numbers you get if you multiple any real number by the square root of minus 1. Remember that the square of any real number is always positive even minus numbers, so there is no real number whose square can be -1. That doesn’t stop your mathematician though. They simply defined the symbol i to be the square root of minus and multiple it by a real number. So a complex number has the form x+iy. The imaginary number line is orthogonal to the real number line as they never cross except at zero (cause i*0 is still 0). This then forms what is called the complex plane.

The colours you see in the Mandelbrot set are indications of how quickly non Mandelbrot values shot off to infinity.

Spend a few minus looking at the youtube video of zooming into the Mandelbrot set. Look at the complexity that goes on forever. Now look at that simple little equation.

Where on earth does that pattern and beauty come from. Where is that contained in

*x*

_{n+1}=

*x*

_{n}

^{2}+

*c?*

This equations is like a grain of sand in mathematically terms, it doesn’t get much simpler. Yet it is infinity complex as well.

I think this also shows that we really need to appreciate what mathematics and physics has to tell us about complex system. We would love to rely on intuition but it is very hard without an appreciation of complexity theory to see how to embrace complexity rather than trying to stamp it out.

The reality is that the world is full of complex systems. The economy is a complex system. Worse the economy is a complex adaptive system ( a complex system that evolves). We can try and remove the complexity as in the command economy, or we can work with it as in the free market economy.

As technology connects businesses and people every more closely the complexity of any given part of the system is rising rapidly. From the web to manufacturing, business will need to understand how to deal with ecosystems rather than supply chains (i.e. from top down control to emergent behaviour).

For more reading on complexity, economics and business check out the comprehensive The Origin of Wealth and you may find this recent new scientist article interesting as well.

## Reader Comments