Hello, and welcome to our introduction to fractals. Actually, they don't need much introduction because they are all around us in nature. However, then, we will discuss what they are and the basics of how they work. So lets get started!
What is a fractal?
They can be many things, but to start and keep things simple, we will start with the basics from a math viewpoint.
In mathematics, for example we will describe some properties of the popular Mandelbrot fractal. Its given by the equation z = z^2 + c, where z is a complex number and c is a constant. Don’t be overwhelmed by this equation, it just means you calculate z by squaring z, (z times z) and adding a constant to it. There are some details though that are important to how this works, and how you can get from that simple equation to the complex and beautiful patterns that make fractals so amazing.
One key piece is that the number z is is a “complex number”. That doesn’t mean this is a complicated topic (but like any topic, it gets easier as you go), it just refers to the fact that a complex number can contain the number “i”, which is the square root of negative 1. The funny thing is, i is called an imaginary number, although there is nothing mysterious about it once you get used to the concept. But, to move on, the number z has two components to it. One “Real” part and one “imaginary” part. Now since a fractal is primarily plotted on a 2D graph, we use the “x” axis to map to the real component of z and the y axis to map to the imaginary component of z. Then, its easy to see that the constant c, is also a complex number and it represents the starting point on our 2D graph. Ok, so what next, well…
Now the interesting part. How can we go from a simple equation, to generating such an intricately detailed fractal pattern? The key is iteration. For example, you take the result of the equation, and that result becomes your new constant. Then you run the calculation again, but this time using the result of the previous calculation. Then you repeat this process many times. Now, what happens is, the next key to the process of generating the image. Depending on where you start this process on the 2D graph, the iterations will either diverge or converge. If they diverge, you get a “null” point on your graph. By keeping track of the number of iterations and if the point diverges, the fractal pattern will start to arise as you try different starting points on the graph. To get the beautiful colors, the number of iterations at each point can be assigned a color from a color pallet. That’s the key point to generating the beautiful images. Check back soon or sign up for our newsletter, and the next blog update will cover more details on the amazing properties of fractals. Thanks for reading!
