What are fractals?
One way to define “fractal” is as a negation: a fractal is a set that does not look like a Euclidean object (point, line, plane, etc.) no matter how closely you look at it. Imagine focusing in on a smooth curve (imagine a piece of string in space)–if you look at any piece of it closely enough it eventually looks like a straight line (ignoring the fact that for a real piece of string it will soon look like a cylinder and eventually you will see the fibers, then the atoms, etc.). A fractal, like the Koch Snowflake, which is topologically one dimensional, never looks like a straight line, no matter how closely you look. There are indentations, like bays in a coastline; look closer and the bays have inlets, closer still the inlets have subinlets, and so on. Simple examples of fractals include Cantor sets (see [3.5], Sierpinski curves, the Mandelbrot set and (almost surely) the Lorenz attractor (see [2.12]). Fractals also approximately describe many real-world objects, such as clouds (see
