You must have seen this kind of unique wallpapers with colorful abstract patterns and intricate, complex designs. These are fractals. Fractals are mathematically generated geometric patterns that repeat themselves infinitely at smaller scales. The following two examples will give you an idea of how mathematics is related to patterns and how it occurs infinitely.
Fractals are never ending and can be zoomed in on infinitely to reveal more and more intricate patterns. When you zoom in even more, you will notice some of the shapes that you saw in the beginning reappearing, but in large numbers. It's just mind boggling.
The fact is, nature is full of fractals. One good example is the snow flake. Another example is the coastline. How...? If you measure the coastline with a mile long ruler you will get a certain finite length. But if you take foot long ruler and measure every small turns and twists the coastline makes, you will find that the coastline is actually longer than you thought. Now reduce the scale to 1 centimeter, and the same coastline becomes even more longer. If you keep reducing the scale further and further, ultimately, you will find that you can't measure the coastline because it's infinitely long! That's how fractals are.
The famous Mandelbrot set.
A closer view of the Mandelbrot set. Yup, it comes from the same fractal
To be honest, I have no idea by what process the mathematical formulas are transformed into fractals. So how am I going to teach you how to create fractals if I'm as clueless as you? Thankfully, there are lots of fractal generators that can generate fractals for you. All you have to do is get one and play around the equations, colors and zoom level and you can generate your own beautiful patterns. If you are a mathematical genius, you can create fractal patterns giving it shapes as you wish. If you are not, the only thing you can do is blindly tweak the various parameters to generate patterns. Fortunately, almost all fractals are a beauty and you will end up creating lots of cool designs. The trick is to use the zoom feature. As I said, fractals are infinitely zoom-able, you can zoom on and on to get more and more patterns. Besides, different parts of the fractals can produce different patterns. Change formula, zoom, change colors, zoom more ... just go wild!
The software that I used is FractalForge. You can also try ChaosPro which is similar to FractalForge or checkout this big list. I used FractalForge because it's relatively easy to use. It also comes with lots of sample fractals (found under the folder "images") that you can use as a base to generate your own fractals. Go on, make your own wallpapers.