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By Polina Vytnova
(This article was not written by Math Savvy Advisors. It was reposted from theconversation.com)
At the beginning of my third year at university studying mathematics, I spotted an announcement. A visiting professor from Canada would be giving a mini-course of ten lectures on a subject called complex dynamics.
It happened to be a difficult time for me. On paper, I was a very good student with an average of over 90%, but in reality I was feeling very uncertain. It was time for us to choose a branch of mathematics in which to specialise, but I hadn’t connected to any of the subjects so far; they all felt too technical and dry.
So I decided to take a chance on the mini-course. As soon as it started, I was captured by the startling beauty of the patterns that emerged from the mathematics. These were a relatively recent discovery, we learned; nothing like them had existed before the 1980s.
They were thanks to the maverick French-American mathematician Benoit Mandelbrot, who came up with them in an attempt to visualise this field – with help from some powerful computers at the IBM TJ Watson Research Center in upstate New York.
A fractal – the term he derived from the Latin word fractus, meaning “broken” or “fragmented” – is a geometric shape that can be divided into smaller parts which are each a scaled copy of the whole. They are a visual representation of the fact that even a process with the simplest mathematical model can demonstrate complex and intricate behaviour at all scales.
How the fractals are created
The system used by Mandelbrot was as follows: you choose a number (z), square it and then add another number (c). Then repeat over and over, keeping c the same while using the sum total from the previous calculation as z each time.
Starting, for example, with z=0 and c=1, the first calculation would be 0² + 1 = 1. By making z=1 for the next calculation, it’s 1² + 1 = 2, and so on.
To get a sense of what comes next, you can plot the value of c on a line and colour code it depending on how many iterations in the series it takes for the sum total to exceed 4 (the reason it’s 4 is because anything larger will quickly grow towards an infinitely large number in subsequent iterations). For example, you might use blue if the series never exceeds 4, red if it gets there after 1-5 iterations, black if it takes 6-9 iterations, and so on.
The Mandelbrot set is actually more complicated because you don’t plot c on a line but on a plane with x and y axes. This involves introducing several more mathematical concepts where c is a complex number and the y axis refers to imaginary values. If you want more on these, watch the video below. By plotting lots of different values of c on the plane, you derive the fractals.
