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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-2

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.

Mandelbrot Set

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

IB Revision

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

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