Although you may never have heard of it, you’ve most likely seen it before. It is, in my opinion, one of the most beautiful sets in mathematics you will encounter. The set I’m referring to is called the Mandelbrot set. The iterative function (I will later elaborate on the meaning of this) which defines this set is not that exceptional itself, it is the visualisation of the Mandelbrot set that makes you question whether there is actual mathematics behind this. In this article, I introduce you to the mathematical meaning of the Mandelbrot set and its remarkable appearance.
The History of the Mandelbrot Set
Before we take a look at the mathematics behind this Mandelbrot set, I will first give you a short introduction to the history of it. The Mandelbrot set is named after Benoit Mandelbrot (20 November 1924 – 14 October 2010), a French-American mathematician at the IBM Thomas J. Watson Research Centre. On 1 March 1980, Mandelbrot was the first to see a visualisation of the set (see the picture above). However, this discovery has its roots way earlier. The Mandelbrot set has its origin in complex dynamics, a field which studies the iteration of functions on complex number spaces. The French mathematicians Pierre Fatou and Gaston Julia were the first to investigate this field of study. The latter conceived the formula for the Julia set, which lays the foundation for the Mandelbrot set. In the next paragraph we will discuss the connection between the Julia set and the Mandelbrot set.
What is the Mandelbrot Set?
Now that we have discussed the history of the Mandelbrot set, you might wonder what the meaning is behind this set. It is all about iteration, which is the process of repeatedly applying the same function. Since the Mandelbrot set has a strong connection with the Julia set, I will first explain what the second set is. The Julia set consists of values such that an arbitrarily small disruption can cause extreme changes in the sequence of iterated (process of repeatedly applying a function) function values. Formally, for a complex number c, the filled-in Julia set of c is the set of all z for which the iteration does not diverge to infinity. The Julia set is the boundary of the filled-in Julia set. For example, the Julia set of is the set of all z for which the iteration does not diverge to infinity.
Now that we’ve discussed the definition of the Julia set we can move on to the Mandelbrot set. The Mandelbrot set, formally stated, is the set of all c for which the iteration , starting from , does not diverge to infinity. Regarding the Julia set, the Mandelbrot set is the set of those c for which the Julia set is connected.
Like I stated in the beginning, the mathematics behind the Mandelbrot set is not that extraordinary itself. Hence, in the next paragraph I introduce you to the visualisation of the set.
Now that I have explained what the mathematical meaning of the Mandelbrot set is, you might wonder what this set actually looks like. The visual representation of the Mandelbrot set (see Mandel_zoom_00_mandelbrot_set.jpg (2560×1920) (wikimedia.org) for an example) is, in my opinion, one of the most beautiful looking things ever to be discovered in mathematics. So how are these images of the Mandelbrot set established? Like I explained in the previous paragraph, this set is all about iteration and divergence. First we need to define “diverging to infinity”. That can be done by defining as the radius for which holds that points lying outside this radius are considered as infinitely far away from the origin. Now it is time to explain where the many colours in the images of the Mandelbrot set come from. The Mandelbrot set is the collection of complex values for c, usually displayed on a two-dimensional plane with on the x-axis the real part and on the y-axis the imaginary part of the complex numbers. The values of c are coloured according to the number of steps required to reach . For example, if r-max equals 3, and , then the first steps of the iteration are given by:
Thus we see that after three iterations we reached , which is greater than our thus by applying our definition of infinity we can conclude that we’ve reached infinity after three iterations. Now the main idea is to give all those points which reach infinity after the same amount of iterations the same colour. Notice how all of these colours actually do not represent the Mandelbrot set itself, since this set contains the points that do not diverge.
In conclusion, the Mandelbrot set is a very pleasing collection of complex numbers to look at. Although the mathematical meaning behind it might not be the most exciting part of this article, you can’t deny that the Mandelbrot set is an intriguing piece of mathematics to lay your eyes on. Hopefully having completed reading this article, you understand a bit more of those famous images of the Mandelbrot set!