Fractals: Gods Artwork, Part I

I thought I’d go a little ‘off piste’ and rabbit on about fractals for a while. In the end we’ll tie them in with fluid turbulence but that’s a ways away yet. I first saw a computer generated fractal in 1989 for sale in a shopping mall in Uxbridge. It was as stunning as it was intricate, as intense as it was mesmerising. Obviously not hand painted, I wondered how on earth it was created. A trip to the library later I was surprised to find how simply fractals were defined…

The most common fractal is the Mandelbrot set.  I’m not big on maths so the wiki  definition:

“Is the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial z_n+1 = z_n² + c remains bounded“

is a bit off putting. More simply put I like to think of it as:

1. Choose a number
2. Put it through an equation (square it and add it to the chosen number)
3. Get the result and put it through the same equation
4. Go back to 3

If the result is that the iterated number zooms off to infinity then the chosen number is not in the Mandelbrot set. The set is made up of lots of different values of chosen (starting) numbers.

The equation is:

z_n+1 = z_n² + c

Z is considered a complex number (z=x+iy). C is also complex but remains constant in the iterative loop. Graphically this equates to picking a point on the screen (x,y), putting it through the equation again and again and colouring the point black if it doesn’t diverge off the screen, i.e. if it’s in the set. Go onto the next pixel x,y location and repeat the process. The pixels that don’t diverge off look like:

with a scale of  -2 to 1 on the real axis, -1 to 1 on the imaginary axis.

You can zoom in real close to the boundary between those numbers that diverge and those that do not:

In fact you can zoom in for ever, you’ll keep seeing the same sort of structures, this is a key feature of fractals known as ‘scale similarity’. Mandelbrot himself pointed this out with a cauliflower, same can be said for cloud boundaries and fluid turbulence generally. The latter summed up beautifully by Richardson:

Big whorls have little whorls
That feed on their velocity,
And little whorls have lesser whorls
And so on to viscosity.

— Lewis F. Richardson

But I’ll cover fractals and turbulence in more detail later…

July 6th Ross-on-Wye