The fancy header image

The above image is of a few iterations of a Newton fractal.

Assuming that isn’t quite enough information for you, a Newton fractal is a 2D plot of the complex plane – that is to say real numbers along the horizontal axis and imaginary along the vertical – where each point is coloured by the value of a complex number there.

http://en.wikipedia.org/wiki/Newton_fractal

Newton of course didn’t have access to enough computing power to generate fractals (we assume…), so the name comes from the fact the fractals are generated through a series of Newton-Raphson iterations.

http://en.wikipedia.org/wiki/Newton%27s_method

In this process we are interested in solving the equation f(z) = 0, where z is a complex number and f(z) an arbitrary complex-valued function. Through successive iterations

z \rightarrow z - \frac{f(z)}{f'(z)}

we will gradually converge on the value of z solving our equation – the root. There will in general be many roots depending on the function we choose (a polynomial of degree n will have up to n distinct roots – the Fundamental Theorem of Algebra).

However, the root we eventually land upon depends on the initial value we choose for z, call it z_0. There is then a ‘basin of attraction’ around z_0, a region which draws z into a local root as we perform iterations. For simple functions such as f(z) = z^2 - 1 we expect simple boundaries between basins. The two roots here are z = \pm 1, and the Newton-Raphson iterations proceed like

z \rightarrow z - \frac{z^2 - 1}{2z} .

After a number of iterations we can plot the complex plane as a function of our initial iteration guess z_0 – specifically we can plot the phase of the final iteration step as a function of starting position:

FractalSimp

The two roots are marked by circles, and as expected if we start with a number whose real value is larger then zero we end up at the positive root (pink basin of attraction), and below zero we get to the negative one (white basin of attraction). Great.

If we take the next logical step and move to the simplest polynomial with 3 roots, we can try solving f(z) = z^3 - 1 and expect to see the complex plane neatly partitioned into 3 areas:

Fractal3

The three roots are again surrounded by their large respective basins of attraction (white/black/orange), but the boundaries between them are much more interesting this time, and repeat on ever-finer scales if we were to zoom in. How…fractal.

As an example of how this structure develops, the animated GIF below shows what happens as we increase the number of iterations – the fractal ‘copies’ itself along boundary lines and shrinks as it goes. If we were to repeat this process ‘to infinity’ we would end up with an infinitely detailed, infinitely complex object. Who knew cube roots were so much fun?

Fractal3

To generate the fractal you see in the header, I cheat a little and don’t use a proper Newton-Raphson iteration step but just any old nonsense that made a pretty picture. If you’d like to replicate it, try iterating

z \rightarrow z - e^{0.34i\pi}\frac{z^3 - 1}{3z^2}

and zoom in to the region around (1, 1/\sqrt{2}). Happy procrastinating!

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