The coastline of everything

The question sounds like a trick and is not. How long is the coast of Britain? Look it up and you get a number. Look it up somewhere else and you get a different number, and the gap is not a rounding error. Lewis Fry Richardson, a Quaker meteorologist who spent his retirement looking for the mathematics of war, noticed that Spain and Portugal could not agree on the length of their shared border. Each country's reference books had a figure, and the figures differed by nearly a quarter. He worked out why, and then he died, and the finding sat in a yearbook nobody read until Benoit Mandelbrot found it.

The ruler is the answer

Here is the why. Measure the coast with a ruler a hundred miles long and you skip every bay. Use a ten mile ruler and you catch the bays but skip the coves. Use a one mile ruler and you catch the coves and skip the rocks. Every time you shorten the ruler, the coast gets longer, and it does not settle down. There is no number the measurements are sneaking up on. Push the ruler down to a grain of sand and the coast of a small island is, for practical purposes, endless.

Mandelbrot's 1967 paper said this out loud in Science, and then said the thing that mattered: the rate at which the length grows as the ruler shrinks is itself a stable number. For the west coast of Britain it comes out around 1.25. A smooth line is one-dimensional. A filled square is two. The coast of Britain is one and a quarter. It is more than a line and less than a surface, and it is like that at every magnification you care to try. He needed a word for shapes like that, and in 1975 he made one up from the Latin for broken: fractal.

It is like that everywhere

Once you have the word you cannot stop seeing the thing. A fern is a stem with small ferns on it, each of which is a stem with smaller ferns on it. A head of Romanesco broccoli at the farm stand is a cone of cones of cones, and you can photograph it from a foot away or an inch away and not be able to tell which. Clouds, lightning, the crack in the sidewalk, the river seen from a plane, the roots under the oak, the branches in your lungs, which split twenty-odd times from the windpipe down and pack something like seventy square meters of surface into your chest. None of these is smooth at any scale. Smooth was the exception the whole time. Euclid's circles and lines were a beautiful idea about a world that does not actually contain any.

That is the part that belongs on this shelf. The Hermetic writers said as above, so below, and meant it as a theology. A fractal is as above, so below with the theology taken out and the arithmetic left in. The pattern at the big scale is the pattern at the small scale because the same rule made both.

The picture

Now the showpiece. Take a number, square it, add a constant, square the result, add the same constant, and keep going. For some constants the numbers stay small forever. For others they blow up. Color every constant that stays put black and every constant that escapes by how fast it escapes, and you have drawn the Mandelbrot set.

That is the entire recipe. One line. Gaston Julia worked out the mathematics of it in 1918, in a hospital, after a bullet took his nose in the first winter of the war, and he never saw a picture of what he had described because nobody could compute one. Sixty years later Mandelbrot had IBM's machines and a printer, and the first printouts came out so strange that he assumed the specks were dust on the paper. They were not dust. The boundary of the set is infinitely detailed. Zoom in on any point of the edge and you find spirals, seahorses, lightning, and, eventually, a small perfect copy of the whole set, which has its own edge, which has its own copies. It never stops, and all of it was in that one line of arithmetic the whole time, waiting for somebody to look.