(Image: lrargerich; / Luis Argerich)
We live in a world of three-dimensional objects bounded by two-dimensional surfaces and outlined by one-dimensional lines. All in all, a comforting, intelligible, whole-number sort of world.
Or do we? As the mathematician Benoit Mandelbrot pointed out in his 1982 book The Fractal Geometry of Nature, clouds are not spheres, mountains are not cones and coastlines are not circles. The dimensions of the raw, rough real world do not, it turns out, come in tidy integers.
Imagine, for example, tracing the delicate outline of a snowflake. As you zoom in, you find yourself following an ever more intricate pattern, and the closer you get, the longer the line you trace becomes. Your drawing is still a line, but its crinkles embrace far more of the space on the page than a straight line. And yet a line, however hard it squirms, can never be more than a 1D object. Or can it?
Welcome to fractal dimensions, the irregular landscapes between the familiar worlds of one, two and three dimensions. Fractal dimensions are not the same as the left-right, back-front and up-down directions we’re used to, but they are intimately related: they describe how much more space a complex object fills as you look it at finer scales and measure more of its detail (see diagram).
It’s not just about snowflakes. Many natural objects have fractal geometry: river networks, branching lightning, clouds, broccoli. You might even claim to live in a fractal landscape, more or less so depending on where you are in the world. The rugged coastline of Great Britain, for example, differs in…
