Math Behind the Beauty
We think of mountains and other objects in the real world as having three dimensions. In Euclidean geometry we assign values to an object's length, height and width, and we calculate attributes like area, volume and circumference based on those values. But most objects are not uniform; mountains, for example, have jagged edges. Fractal geometry enables us to more accurately define and measure the complexity of a shape by quantifying how rough its surface is. The jagged edges of that mountain can be expressed mathematically: Enter the fractal dimension, which by definition is larger than or equal to an object's Euclidean (or topological) dimension (D => DT).
A relatively simple way for measuring this is called the box-counting (or Minkowski-Bouligand Dimension) method. To try it, place a fractal on a piece of grid paper. The larger the fractal and more detailed the grid paper, the more accurate the dimension calculation will be.
D = log N / log (1/h)
In this formula, D is the dimension, N is the number of grid boxes that contain some part of the fractal inside, and h is the number of grid blocks the fractals spans on the graph paper [source: Fractals Unleashed]. However, while this method is simple and approachable, it's not always the most accurate.
One of the more standard methods to measure fractals is to use the Hausdorff Dimension, which is D = log N / log s, where N is the number of parts a fractal produces from each segment, and s is the size of each new part compared to the original segment. It looks simple, but depending on the fractal, this can get complicated pretty quickly.
You can produce an infinite variety of fractals just by changing a few of the initial conditions of an equation; this is where chaos theory comes in. On the surface, chaos theory sounds like something completely unpredictable, but fractal geometry is about finding the order in what initially appears to be chaotic. Start counting the multitude of ways you can change those initial equation conditions and you'll quickly understand why there are an infinite number of fractals.
You won't be cleaning the floor with the Menger Sponge though, so what good are fractals anyway?