What Is This?

a fractal,

it's the one-word answer to whether mathematics can be beautiful, originally created by Benoit Mandelbrot in 1980, the fractal is a visual representation of a simple equation: Z new=Z old 2+C,  the "C" value must lie within Mandelbrot's set, numbers that have both a real and imaginary component.

Bowdoin College's web programmer David Francis explains it well:

When you zoom in on a piece of the Mandelbrot set, you realize that that piece contains, and consists of, another Mandelbrot set. Zoom in again, and you see that that piece also contains and consists of another Mandelbrot set. Zoom in again. Same thing. In fact, you can zoom in forever and you will always see more Mandelbrot sets!

The Mandelbrot set doesn't iterate over these simple numbers. Instead it iterates over complex numbers. 

Complex numbers come in two parts: a real part and an imaginary part. The real part is easy to grasp. They are regular numbers that you know and love: 1, 0, -5, 4.534343, 232423432.4787865, -0.0000000000002, etc. The imaginary part of a complex number is a real number (like above) multiplied by a unique little number called i.

 for the Mandelbrot set, we start Z at zero...but our choice for C will be a complex number. Our exact choice for C is what determines the Mandelbrot set. The Mandelbrot set consists of all the choices for C we can find (where Z starts at zero and C is a complex number) so that the iterations never grow beyond the number 2, that is the mathematical definition of the Mandelbrot set,

and this is what it looks like as a GIF,

but if you want to really see the equation at work grab at least 2 cups of coffee and watch the video, but be warned it is long, very long as the pattern is infinitely variable and never ending.