This may sound like an exaggerated title, but once you’ll finish reading this article, and you’ll know everything about the Mandelbrot set, this definition won’t seem that extravagant.

First off, here is a gif that will show you exactly what the Mandelbrot set is all about, and how stunning it is:

As you can see, the set repeats variations of itself infinitely. This is a very fancy gif. Why stop here? Let’s find out the math behind this incredible discovery.

The Mandelbrot Set

The Mandelbrot set is a very famous assortment of complex numbers named after mathematician Benoit Mandelbrot. Wikipedia’s definition of the Mandelbrot set is: 

“The Mandelbrot set is the set of complex numbers for which the function  Fc (z) = z2 does not diverge when iterated from z=0“

When first glancing at this, unless you have a solid background in mathematics, this may seem like a group of letters and numbers glued together at random…

However, there is no need to panic, as we’ll go through each definition thoroughly. By the end of this article, you’ll be able to interpret Wikipedia’s definition effortlessly. 

Complex Numbers

First of all, we must take into consideration a number line: the visual representation of… numbers on a line:

We are representing “natural numbers“. These include positive numbers, negative numbers, decimals, and so on. 

On the line, we can also position things like squared roots, for example, the squared root of 1, which is 1, and so on. 

However, what we cannot position on the line is the squared root of negative numbers. This is because when you square negative numbers, the final result will always be positive. 

For example:

2^2 = 4

-2^2 = 4

This means the squared root of, for example, -1 is unobtainable. However, mathematicians have established that the square root of -1 does exist, and its value is i.

The letter i stands for “imaginary” and its values are not included on the number line. However, if we draw a perpendicular axis things change.

Instead of a number line, we now have a complex number plane, which is the combination of natural and imaginary numbers. Every number on the number plane is technically the combination of a natural and imaginary number.

How do we know what numbers belong to the Mandelbrot set? Let’s figure out what functions are first.

Functions

A function is a relationship between two given numbers. Functions apply a predetermined set of rules to a set of numbers. Let’s take an example:

As you can see, we use x as a placeholder for any number. An input of 2 would result in an output of 4, an input of 4 would result in an output of 16, and so on. 

Verifying if a number belongs to the set

To verify if a number belongs to the Mandelbrot set, we use the function we mentioned at the beginning of the article:

First, we assign z a value of 0 (that’s the rule for the function to correctly work), next, we take any complex number (c), for example, 1, and add it to z. We then take the result and insert it into the function, iterating it. If the function keeps returning non-consistent values, then we know the complex number we picked does not belong to the Mandelbrot value.

However, with certain numbers such as -1 and start iterating them, we will notice a certain pattern repeating itself: the results of the function will always be either 0 or 1. This means -1 is part of the Mandelbrot set

Conclusion

The Mandelbrot set is one of modern mathematic’s greatest discoveries. It is amazing to think about how much it took us to find it.

We had to invent computers, graphical processing software, and so on. This shows just how little we know about our universe, and how much more there is to discover