The Mandelbrot Set

When anyone speaks about fractals, one fractal that comes into the mind of every mathematician is the Mandelbrot set. This set was discovered by and named after Benoit Mandelbrot himself who coined the term ‘fractals’ for these unusual and unique images.

This set consists of numbers that are complex. Assuming we all already know about these numbers and it’s properties, I’m not going to go in detail about it. So, the study of the Mandelbrot set is all happening in the world of complex numbers. It is plotted on the complex plane also referred to as the Argand plane.

Now, how do you find the the Mandelbrot set from complex numbers?

Let’s take a complex number c. We’ll now associate this complex number to a function,

where z is another complex number which is an input fed into the equation.

The first number that is used in the equation to get the first result is called the ‘seed’ of the iteration.

here, z_0 is the seed of the iteration. This result is then fed back into the equation to get a new result. Then that is again fed back into the equation to get another new result. This process of iteration goes on and on.

and so on……

Here, z_0, z_1, z_2, z_3, z_4, z_5, etc, are called the orbit of z_0 under iteration of

First, we’ll take the seed to be 0 for the iteration of the above simple quadratic equation. So, we are going to check the behaviour of 0 under iteration for a particular value of c.

For example, if we take c = 1.

and so on……

So what is happening to the size of these numbers? By ‘size’ I mean, the distance of these points from the origin in the complex plane. It’s increasing, tending to infinity.

Now, what if we take c = 0?

and so on……

Here, the size is fixed to one value, 0.

What if c = -1? Let’s try.

It’s 0 again. But, we already know what happens when we feed 0 to the equation, right?

and so on……

We can observe that the size is now bouncing between two values, 0 and -1. This is what we call a ‘2-cycle’ or ‘a cycle with a period of 2’.

Through these examples we can say that for any value of c, there are two options for the fate of

One, the size or the distance from the origin in the complex plane just grows arbitrarily larger and larger and tends to infinity, (in the case c = 1) or,

Two, the value stays within certain boundaries, or more specifically, the size or the distance from the origin in the complex plane is ≤ 2. (in the case c = 0, -1)

This means that, suppose we take any value of c and iterate it. Then, if it’s size is within the radius of 2 units with origin as centre, and does not tend to infinity, that value of c will be within the Mandelbrot set (the black region in the diagram given below). Or else, it will be outside it.

Thus, we can define the Mandelbrot set as the set that consists of all the complex numbers ‘c’ where the function

is iterated with zero as it’s initial value such that the results obtained do not escape to infinity.

In the above diagram, as we had observed in the cases of c = 0,1 and -1, we can see that 0 and -1 which did not tend to infinity lies well within the Mandelbrot set, while, 1 lies outside the Mandelbrot set.

It is impossible for a human to manually work out all the iterations, thus, the algorithm is fed into a computer. A sort of grid is created which contains all points in a complex plane or the complex numbers c. It then takes each point or each complex number c and iterates it with

several times. The computer checks whether the result is escaping to infinity or it is fixed or bouncing between values. If the result is bounded, let’s give it one particular colour, say, black and if it escapes to infinity, lest give it say red. Different colours are assigned to the results that slowly tend to infinity and other colours assigned to the results that rapidly tend to infinity. This is how we obtain a shade of colours at the boundary of the Mandelbrot set as shown in the following diagram.

Everything in the Mandelbrot set is mathematical. There is nothing in it only for the purpose of beauty or art. Every colour, every pattern has a mathematical reason to it.

So now, let’s observe the diagram given above a little bit more carefully. We can see that it has a few obvious characteristics. It has a heart shaped main structure called the ‘main cardioid’, a circular structure on the left of the cardioid called the ‘bulb’ and other smaller circles attached around the cardioid called the ‘primary bulbs’. Surrounding the Mandelbrot set are intricate patterns or myriad ‘decorations’.

The main cardioid basically consists all those values for c which when iterated in

with the seed of iteration as 0, remains fixed at one particular value. For example, c = 0, as we had seen previously.

The bulb, on the left, consists all those values for c which when iterated in

with the seed of iteration as 0, bounces between 2 values or is a 2-cycle. For example, c = -1, as we had seen previously.

The primary bulbs similarly consists all those values for c which when iterated in

with the seed of iteration as 0, are 2, 3, 4, 5….. -cycles.

Given below are the zoomed in image of the 3,4,5,7-cycle primary bulbs. It becomes even more interesting if you count the spokes attached to the primary bulb. You would notice that the 3-cycle bulb consists of 3 spokes, the 4-cycle bulb consists of 4 spokes, the 5-cycle bulb consists of 5 spokes, the 7-cycle bulb consists of 7 spokes and so on.

Thus, it is truly unbelievable that a very simple equation such as

can create something so complex and beautiful. After all, it is just plotting the graph of a quadratic equation on a complex plane by iterating it several times. Simple processes like this has paved the way to a complete new area of geometry, fractals. Are there other such interesting images or fractals similar to the Mandelbrot set? If so, what do you think is the factor that is changing for the fractal to change? The equation? The seed value? or something else?

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Picture credits: https://plus.maths.org/content/unveiling-mandelbrot-set