I can explain fractal geometry.
Okay. Take any old fuction... like this.
y = x^2 + c
Where c is a constant. Now, we are going to solve y where x is a complex number.
A complex number is one which comprises of two components - a real number (one that can be expressed with decimals, fractions, whatever) and an imaginary numbers. Imaginary numbers are all multiples of i (i being the square root of -1).
Now, once we solve y, we input that into x, and iterate the function - perform it again. If we used real numbers instead of complex numbers, the results of the iteration would always climb to infinity forever. However, because complex numbers are wierd, some of them do not follow this behaviour. They bounce back and forth between different numbers, return to their original complex number, even get smaller.
Now to graph this with fractal geometry, we do the following. For all those numbers that perform as expected, and get bigger to infinity forever, paint them a certain colour based on their escape velocity - say, bright colours for numbers that escape to infinity fastest, and darker colours for numbers that escape to infinity, but slower.
All the numbers that do the wierd thing, we paint black.
Then, what we get is a graph - a very wierd shape - and using y = x^2 + c, we get what is called the Mandelbrot set. The trick is with fractal geometry, is that if we magnify into any area, we will eventually find the exact same Mandelbrot set again. It recurs on itself. And that's fractal geometry...