"How long is the coast of Britain?" is the question that launched mathematician Benoit Mandelbrot into a synthesis of ideas about fractional dimensions. How do you actually measure a coastline? One way would be to trace a line around the coastline on a map, measure the line, and then multiply by the scale of the map. That seems reasonable on the surface. But if you physically measured a coastline, the length you would get would depend on the scale of the ruler; the smaller the ruler, the greater the length. Every little inlet has its own little inlets, and these do not follow regular lines but are broken up by rocks and other natural features. By the time you start measuring the surface of the rocks and sand crystals at molecular scales, you will have convinced yourself that there is no precise answer to the question of how long a coastline is.
A coastline, in principle, is a line enclosing an irregular shape. Though an ordinary line drawn by a geometer may be one dimensional, most lines in nature have fractional dimensions somewhere between one and two (where two would be the dimensions of a plane). From this consideration, Mandelbrot introduced the concept of a fractal; a mathematical set whose fractional (or Hausdorff) dimension is greater than its topological dimension. It is conceivable to geometrically generate lines of infinite dimension that nevertheless are confined to a defined space, such as the Koch snowflake (see the image to the right). The middle sections of each line are replaced with an equilateral triangle—one line is erased and a new iteration begins. Typically, fractals demonstrate a self-similarity that is also scale invariant. Another example is the Mandelbrot set—typically rendered in psychedelic colors to display wild spiralling patterns, each of which contains mirror images of the set at smaller and smaller scales.
Mandelbrot popularized his ideas in a remarkable volume titled The Fractal Geometry of Nature. It was illustrated with remarkable pictures of islands, mountain ranges, continents and moonscapes that were actually created by computerized iterations of mathematical functions. Nature, it seems, is full of fractals. Her marvellous biological machines are also composed of elaborate geometries critical to their function.
Consider the human lungs. The respiratory membrane of the lungs- all the alveoli combined-- has a surface area of 750 square feet, about the size of a racquetball court. How is that possible? An iterative ramification occurs whereby the trachea branches to the primary bronchi, which branch to secondary bronchi, then to tertiary bronchi, to terminal bronchioles, to respiratory bronchioles, to alveolar ducts, to alveolar sacs, which finally end in the alveoli, formed by a single layer of epithelial cells. The end products are so delicate that they rely on surface tension to remain inflated. The respiratory surface is said to have a fractional dimension of 2.97.
Fractal patterns are found in everything from the shape of galaxies to the growth patterns of cauliflower, and Mandelbrot's ideas stimulated natural scientists as much as mathematicians.