Mandelbrot and Julia Sets – Pictures of Infinity
The above video is of a Mandelbrot zoom. This is a infinitely large picture – which contains fractal patterns no matter how far you enlarge it. To put this video in perspective, it would be like starting with a piece of A4 paper and enlarging it to the size of the universe – and even at this magnification you would still see new patterns emerging.
To understand how to make the Mandelbrot set, we first need to understand Julia sets. Julia sets are formed by the iterative process:
Zn+1 = Zn2 + c
Here Z is a complex number (of the form a + bi) and c is a constant that we choose. So, for example if we choose Z1 = 1+i and c = 1 then:
Z2 = Z12 + 1
Z2 =(1+i)2 + 1
Z2 = 2i + 1
We then repeat this process:
Z3 = Z22 + 1
Z3 = (2i+1)2 + 1
Z3 = 4i-3
and so on – what we are looking for is whether this iterated Z value will diverge to infinity (i.e get larger and larger) or if it will remain bounded. If diverges to infinity we colour the initial point 1+i as red on a complex axis. If it remains bounded we will colour it in black. In this case our initial point 1 + i will diverge to infinity and so it will be coloured in red.
Next we do this for every single point in the complex plane – each time seeing what happens when we iterate it many times. Each time we colour it in as red if it diverges and black if it remains bounded. Once we have done that we will have a picture which represents what happens to every point in the complex plane. This then is our Julia set.
For example the Julia set for c = 1 looks like this:
This is because every single complex number when iterated by Zn+1 = Zn2 + 1 will diverge to infinity (get infinitely big).
Not very interesting so far, but different values of c provide some amazing patterns.
This above pattern is generated by c = 0.376 – 0.1566i.
and this pattern is for c = 0.376 – 0.1566i.
and this one is c = -0.78 + 0.1i.
This last one for c = 0.4 + 0.1i looks different to the others – this one has patterns but they are not connected together as in the other examples.
This brings us on to how to calculate the Mandelbrot set. We calculate every possible Julia set for all complex numbers c, and then for every Julia set which is connected then we colour the c value in black, and every value of c which the Julia set is disconnected we colour the c value in red. We then have a new plot in the complex plane of c values. This gives us the Mandelbrot set shown below:
Don’t worry if this seem a bit complicated – it is! You can play around making your own Julia sets by choosing a c value at this online generator. You might also like towatch the Numberphile video on the same topic:
If you enjoyed this post you might also like Dan Pearcy’s post on this topic which explains how Geogebra can be used to generate these sets. Also PlusMaths have a number of posts on this amazing subject