From Scientific American:
Do you ever wonder what mathematicians study—and why? Most of what they do is complex and difficult to understand, but fractal art might give us a glimpse. Mathematicians study fractals, which are naturally occurring figures used in many branches of science and technology. But you don’t have to be a mathematician to appreciate their beauty. Did you know you can also create one? In this activity you will get to take out some paint to make artwork—and discover how common fractals are.
Fractals are geometric figures. They are difficult to define formally but their features and beauty make them accessible and intriguing.
One feature is self-similarity, which describes how fractals have patterns that recur at different scales. In other words, when you zoom in, you will find a smaller version of a pattern you had seen initially. When you zoom in some more, you will find an even smaller version of that pattern, and so on. This seems to go on infinitely.
The way fractals scale is another feature that sets them apart from traditional geometric figures such as lines, squares and cubes. When you double the length of a line segment, the original line segment will fit twice (or 21) in it, making a line one-dimensional. If you double the length of the sides of a square, the original square will fit four (or 22) times in the new, bigger square, making a square two-dimensional. Do the same on a cube, and the original cube will fit eight (or 23) times into the new figure, making the cube three-dimensional. Apply this operation on a fractal and the number of times the original fractal fits into the bigger one could be three or five or any other number that is not a whole power of two. This is characteristic for fractals. …