Fractals are objects that have fractional dimension.
Wow, isn't that a lot
Here is how this idea came up. If you try to measure a non-exotic
object like a randomly-flopped garden hose with a ruler, the number you get
for the length will depend on the length of the ruler.
A six foot ruler
will give only a crude approximation to the length of the hose. A six inch
ruler "flip-walked" along the hose does a better job.
As you use smaller
and smaller rulers you converge on (get closer and closer to) the actual
length of the hose.
With a fractal object, like a coastline, the smaller
you make your ruler, the longer the coastline gets. This is because smaller
and smaller rulers measure smaller and smaller jigs and jags in the
coastline and fractals objects have jigs and jags on all scales. They do
not start to look smooth as you magnify them.
In other words, a fractal is a mathematical object that is self-similar and chaotic. Fractals are infinitely complex: the closer you
look the more detail you see. Most fractals are generated by a relatively simple equation where the results are fed
back into the equation until it grows larger than a certain boundary. Most fractals are just a graph of an equation
using complex numbers. The real line is the X-axis. The imaginary number line is the Y-axis. The point is fed into
the equation such as the one for the Mandelbrot set.