I have previously authored a post on mathematical beauty, which as we know here at RTU is one of the purest forms a human can revel in. Today I intend to present an elegant piece of mathematics which through it’s simplicity can give insight into seemingly disordered systems, along with producing imagery that looks more like a magic eye that mathematics. The skills required are reasonably elementary until the final stages, where we take a little more care to define the concept of a complex number. A minimal comprehension in this area is required, and will be contained within the post.

The beginnings

In mathematics (or any other walk of life I assume) an iterative process is one which is repeated. If I do my process 4 times, I can say that I have done 4 iterations for example – the point is I have repeated the same process four times. We can express iterative processes as very simple expressions. Say for example my process was to take a number, then add 2 to the number with each iteration. In order to express this mathematically I would write:

x₁ = x₀ + 2

What I am saying here is that the next term in the series, which here is x subscript 1 is simply the previous term (x subscript 0) plus 2. Now I could write the next term as:

x₂ = x₁ + 2

but really you didn’t need this information – you have all the information you need from the first expression and can iterate away as many times as you like – there is no limit, your time is your own. There is of course no explicit advantage to expressing these things things as formulas, we can communicate the same information with words, there is just an efficiency benefit since as the expressions become more complex words become unwieldy, while mathematical expressions stay more compact. It also means if someone is reading this blog for example who speaks no word of English, but is a mathematician they will be able to tell you that I have used one hell of a lot of words to talk about the iterative process of adding 2 to a number, in their mother tongue.

We need to briefly discuss sets, before we proceed to the Mandelbrot. A set is a collection of objects, or in this case numbers that has some rule which defines if they should be inside or outside of the set. Generally a set is enclosed in curly brackets; as a basic example I might write {1,3,7,23,51}. There is no real logic to this, but if I give you any number one can quickly tell from inspection if an item is inside or outside the set – 1 is inside, 2 is outside. A slightly more fanciful set might look like A = {a ε ℝ : a > 4} . This looks wonderful, but reading left to right all this says is the set A is defined as the set equal to all real numbers greater than 4. Clearly we had to use more powerful notation here, since this set has an open upper bound and we could not have listed out all the elements.

Mandelbrot sets curiously involve a function no more complicated than the types taught at GCSE grade – but don’t beat yourself up if you are rusty! We consider the functions of the form

f(x) = x² + c

Where c is a constant. Now this function can be altered lightly to make a series of iterations like we had before –

x₁ = x₀² + c

x₂ = x₁² + c

and so on, where I take the previous number, square it and add my constant. If I asked you to write me out 100 iterations – hopefully you would ask me for two things – the value of c and the value of x₀. The value of x₀ is of great importance, so it will have a special name – the seed, for it is the seed which will drive the growth. The value of the constant is also of tremendous importance for Mandelbrot sets, although depending on the relationship with the seed in some examples it is quite trivial.

 A few Mandelbrots

I want to present a few very basic sets for simple values of c and x₀ before going on to bigger things. Say for example I take my constant to be 0, and my seed to be 1. Then I have a very boring set indeed! The first term is 1² + 0, so 1. Then the next term is the previous term squared plus 0 – uh oh, looks like I’m stuck in a loop. So by taking these values I have the set {1}. If I had of let c equal zero as before, but let my seed also take the value of zero I would of had the most boring set of all – the set {0}. Moving on to something slightly more interesting – see if it follows to you that if we let c equal 0 and we have our seed as -1, we get the set {-1,1}.

If I take less trivial values, say for example letting the seed and the constant take the value of 1 then hopefully it is clear to see this thing will blow up pretty quickly – the first term will be a nice sensible 2, then a manageable 5, followed by 26 and we are off – the next term is 677 and the one after that into the hundreds of thousands. This is because, if my seed and constant allow the squared term to dominate the growth is explosive giving a set where the terms quickly get larger. Unlike the other sets we discussed, this one is infinite in size. Interestingly when we put in different values the system tends to different behavior. If for example we hold our seed at 0, then a value of c equal to -1.6 gives us chaotic behavior however a value of c equal to -1.75 gives us a repeating pattern with a period of 3 – this is by no means intuitive. It is the changing nature of the sets relative to a seed of zero that interests us today.

Getting complex

I need you to make a conceptual leap and consider complex numbers. We won’t learn about them in full, but let me tell you a few facts about them. A complex number has a real and imaginary element to it – the real part is the bit you are comfortable with, the ordinary numbers. The imaginary element is just another number added to the real element and denoted i. It is worth knowing that i² is equal to -1, so these numbers are very hand as we can square root negative numbers.

You might think this sounds totally made up – and if you do feel that way you are correct, however what is spooky about complex numbers is that the addition of the imaginary element allows us to model and solve real world problems. So can we really say they are imaginary when they solve things which are real? Or can we say the things are real when they are solved by numbers which are imaginary? One for another day.

Rather than representing complex numbers in a line, like we would ordinary numbers we are required to introduce an additional axis – we have two number lines, at 90 degrees to each-other with one representing the complex numbers and one representing the real numbers. Because the complex numbers occupy all of this space we say the complex numbers are represented in the complex plane – this isn’t anything technical, it’s just a result of the fact that we can plot all the real numbers on a line but we need an area, or a plane for the complex numbers. Here are a few complex numbers plotted in the complex plane.

So here we have the number represented by a as 1 + 4i; 1 being the real part and 4i being the imaginary part we just introduced. Lovely.

The Mandelbrot we love

Now we have battled through all that maths we can finally produce something artistic! Now earlier we discussed a seed of 0, and we highlighted the fact that when the value of c takes different values we have different options;

• We have a null set, when c =0;
• We have a finite set, when c takes certain values possibly following a repeating pattern; or
• We have a set which tends to infinity.

The Mandelbrot set is interested in the entire complex plane – so to give you a few examples if I let c = i and my seed as zero, I have;

x₀ = 0x₁ = 0² + i = ix₂ = i² + i = -1 + i

This set does not escape to infinity it stays locked up with fairly small values. There are many sets for which this happens – and many for which it does not. We want to be able to “see” this.

A visual representation of the Mandelbrot set is interested in considering the values within the complex plane where I can have this phenomena – if I let my seed equal zero, what values of c will give me a finite set. This is the result.

The black areas indicate the regions where this is possible, with the areas outside being values which tend to infinity.So any area within the black represents a value of c, with a seed of zero which produces a finite set. It should be clear this is a highly intricate pattern from this very zoomed out image, but what happens when we zoom in closer is truly breathtaking. Have a look at this zoomed in image.

 

What is most interesting is that the more we zoom in on these boundaries, they still look just as crinkly. The detail is almost beyond comprehension! The main heart of the Mandelbrot is decorated finely with bulbs and antenna all different in shape – the nature of these crinkles is one of the big problems still open in mathematics today.

There are however many interesting things we do know. For example, if we choose any number within a bulb of the Mandelbrot we will know the period the pattern tends towards. This is shown here – so for example if my c value is in the area labeled 3, then my iterations tend to the same value with each 3 goes.

The conclusions we can draw from these sets is an underlying order among chaos. We are using geometry to understand the dynamics of the function, and using the dynamical information to understand the shape of the geometry – this is a big leap forwards in human thought. Whilst this interplay is not fully understood, it has fundamentally shaped the way we look at the world. Benoit Mandelbrot once said;

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

This roughness, or chaos in the world is something Mandelbrot celebrated – and in fact on a small scale these chaotic things are similar to the overall larger things. Small clouds for example is very similar to the overall thing; allowing some logic and unity to be applied. This is the very heart of fractals and has found successful application in the fields of cosmology, medicine, engineering, genetics, art and music. The discoveries into this area are relatively new, due to the fact that they required computers for the powerful visualizations – something Mandelbrot had access to while working at IBM. There is a particularly interesting application he was able to use; through fractals he could accurately predict profits and losses made by traders over time. In 2005 he warned that traders have a tendency to act as if the market is predictable and immune to large swings; something we may well argue was vindicated a few years later. This is truly a powerful area of mathematics, which through comprehension can only deepen the human need to bring structure to chaos.

Finally, in case you need it:

 

A big thank you to all our followers, new and old. This is post 100 here at RTU.