What Is a Fractal? Shapes That Repeat at Every Scale

The defining property is self-similarity: a part of the shape, magnified, resembles the whole. A fern frond is built of smaller fronds of the same form, a branch of a tree looks like a small tree, a river system's tributaries branch like the main channel, and each branching of the lungs repeats the pattern of the one before. In the strictest mathematical cases the resemblance is exact and continues without limit, so that magnifying any region forever produces the same structure; in nature it is approximate and runs out after a handful of levels, because a fern eventually reaches the size of a cell. The consequence is that these shapes have no characteristic scale. A photograph of a rock face without something familiar for size could be of a boulder or a cliff, and there is no way to tell from the shape alone.

Lewis Fry Richardson noticed in the 1950s, while looking at whether countries with longer shared borders fought more often, that published border lengths disagreed wildly, and that the measured length depended on the length of the ruler used. Measure Britain's coast in hundred-kilometre steps and you get one figure; measure in ten-kilometre steps and the ruler follows bays it previously skipped, giving a longer coast; measure in metres and it grows again. For a smooth curve this process converges on a fixed length, and for a coast it does not, which is the sense in which the question has no answer. Mandelbrot's insight was that the rate at which the measurement grows as the ruler shrinks is a meaningful number in its own right, a dimension between one and two, higher for a rugged coast than a smooth one, which is around 1.25 for Britain's west coast and close to 1.02 for South Africa's.

Most of the standard shapes are built by taking a simple rule and applying it forever, which is why they were once dismissed as monsters:

• •The Koch snowflake, from 1904: take a triangle, replace the middle third of each side with a triangular bump, repeat. The perimeter grows without limit while the area stays finite, so it is a curve of infinite length enclosing a bounded region
• •The Sierpinski triangle, from 1915: remove the middle triangle from a triangle, repeat on what remains, leaving a figure of zero area but infinite structure
• •Cantor's set, from 1883: remove the middle third of a line, repeat, ending with uncountably many points that occupy no length at all
• •The Menger sponge, the same idea in three dimensions, with infinite surface area and zero volume
• •Iterated function systems, which produce a convincing fern from four simple transformations applied at random, a result of Michael Barnsley's that showed how little information a complex natural shape can require

The image that made the subject public is generated from arithmetic so simple it can be written on a line. Take a complex number, square it, add the starting number, square the result, add the starting number again, and keep going; some starting points send the sequence racing off to infinity and others keep it bounded forever, and the set of the second kind, coloured black with the escape speed of the others shown in colour around it, is the Mandelbrot set. It was first plotted properly in 1980, and the boundary turned out to be endlessly intricate, containing distorted copies of the whole set at every depth, filaments, spirals and miniature versions connected by threads, all of it implicit in one line of arithmetic. The boundary has dimension exactly 2, proved in 1991, and whether the set is locally connected remains an open problem that serious mathematicians work on.

The applications are wider than the pretty pictures suggest. Antennas built in a repeating branched pattern work across many frequency bands at once in a small volume, which is why the chip inside a mobile handset carries one. Image compression exploited iterated systems in the 1990s, and the same mathematics underlies the terrain and vegetation generated in films and games, where a few rules produce a mountain range that would take an artist weeks. In medicine the branching of blood vessels, the surface of a tumour and the rhythm of a heartbeat are all characterised by their dimension, and a loss of complexity in a heart rate turns out to be a warning sign. In physics and geology the same measure describes turbulence, cloud edges, fracture surfaces and the distribution of galaxies, and in finance Mandelbrot's own later work argued that price movements are far rougher and more prone to extremes than the standard models assume, an argument that looked eccentric in 1997 and much less so in 2008.

The shapes named by Mandelbrot in 1975 are rough at every magnification and self-similar, so that a magnified part resembles the whole, which means they have no characteristic scale and no fixed length: a coastline measured with a shorter ruler simply gets longer. The rate of that growth defines a dimension that is not a whole number. Classic constructions such as the Koch snowflake and the Sierpinski triangle come from repeating one simple rule, the famous coloured set comes from one line of arithmetic, and the mathematics is used in antennas, computer graphics, medicine and the study of extreme market moves.