The geometry of a rough world

How long is the coast?

Lewis Fry Richardson was a British meteorologist and a Quaker, a conscientious objector in the First World War, who spent much of his life asking an unfashionable question: what causes war? He gathered numbers on wars, and one of his ideas was that the chance of war between two countries might depend on how long their shared border is. To test it he needed the lengths of borders, and he found a problem. Atlases disagreed. The border between Spain and Portugal was 987 kilometers in one source and 1,214 in another. Between the Netherlands and Belgium the figures were 380 and 449. (Which country gave which number is reported both ways, so it is left unsaid here.) The disagreement was not carelessness. When he laid dividers of different openings along a map, he found that the shorter the divider, the longer the border, and the rate of growth was a regular one that differed from one frontier to the next. His results were published after his death, in 1961, in an appendix to the book Statistics of Deadly Quarrels.

In 1967 the mathematician Benoit Mandelbrot, then at IBM, read Richardson’s data differently. His paper in Science, “How long is the coast of Britain? Statistical self-similarity and fractional dimension” (5 May 1967), opens with a claim that sounds like a riddle: geographical curves, he wrote, are infinite or, rather, undefinable in length. The reason is that a coast has detail at every scale. A ruler of 100 kilometers skips every bay and peninsula smaller than itself. A ruler of 50 kilometers follows some of them, and the total grows. Mandelbrot proposed that the length depends on the ruler in a regular way, the way a power of the ruler’s length does, and that the exponent is a kind of dimension. From Richardson’s figures he quoted values of about 1.25 for the west coast of Britain, 1.15 for the German land frontier, 1.14 for the Portuguese frontier, 1.13 for the coast of Australia, and 1.02 for the coast of South Africa, which is smooth. A dimension of 1 is an ordinary line, and the rougher the coast, the more it exceeds 1.

Real coasts show the same effect in official statistics. The 1975 tabulation by the US agency NOAA (repeated in the Census Bureau’s statistical abstract) lists a “general coastline” for the United States of 12,383 miles, taken from small-scale maps, and a “tidal shoreline” of 88,633 miles, which includes bays, sounds, offshore islands, and tidal rivers measured on large-scale charts. It is the same country, and the second number is about seven times the first. Alaska alone is 6,640 miles of general coastline and 33,904 of shoreline. A 2021 NOAA tabulation at a still finer scale gives 221,923 miles, which the agency says is not an official figure. China’s official figures are about 18,000 kilometers of mainland coast and about 14,000 kilometers around its islands, together about 32,000. Each of those numbers is correct for its method, and none is “the” length.

Monsters, and a dimension

Mandelbrot’s riddle had a long mathematical prehistory, and its first chapters were told with some embarrassment. In 1872 Karl Weierstrass showed the Berlin Academy a function that is continuous everywhere and has no tangent anywhere, a curve with no smooth place on it. In 1883 Georg Cantor described a set built by removing the middle third of a line segment, then the middle thirds of what remains, forever: a dust of points that is nothing and not quite nothing. Giuseppe Peano drew a curve in 1890 that passes through every point of a square. In 1904 the Swedish mathematician Helge von Koch gave the simplest picture of all, the curve that bears his name. Wacław Sierpiński’s triangle followed in 1915, and in 1918 or 1919 Felix Hausdorff gave a precise meaning to a dimension that need not be a whole number. Gaston Julia and Pierre Fatou studied the strange sets produced by iterating a simple formula in the complex plane in the same years.

Mathematicians called these curves “pathological,” and sometimes worse. In a letter of 1893 Charles Hermite wrote that he turned away “with fear and horror” from functions with no derivatives. Henri Poincaré wrote, in an essay of 1906, that logic sometimes breeds monsters. They were treated as counterexamples, useful for showing what a theorem could not claim, and not as pictures of anything real.

The way into dimension is by scaling. Cut a line segment into three equal pieces and each piece is a copy of the whole at one third the size: three copies, scale factor three. Cut a square into nine squares and each is a copy at one third the width: nine copies, scale factor three. If N copies, each smaller by a factor s, rebuild the whole, the dimension is the number D for which N = s^D, that is, D = log N ÷ log s. A line gives D = 1 and a square gives D = 2. Now try the Koch curve. Each step replaces a segment by four segments, each a third as long, so N = 4 and s = 3, and D = log 4 ÷ log 3, which is about 1.26. The curve is more than a line and less than a surface. The Cantor dust, with two copies at a third the size, has dimension log 2 ÷ log 3, about 0.63, less than a line and more than a point. Sierpiński’s triangle, three copies at half the size, has dimension about 1.58.

Mandelbrot had been born in Warsaw in 1924. His family moved to Paris in 1936 and, with the war, to the small town of Tulle in 1939, and he studied at the École Polytechnique under Julia and Paul Lévy, then at Caltech, and received his doctorate in Paris in 1952. After a year at the Institute for Advanced Study, as John von Neumann’s last postdoctoral student, he joined IBM’s Thomas J. Watson Research Center in 1958 and stayed for 35 years; he taught at Yale from 1987. At IBM he worked on problems no one else had a name for: noise on telephone lines, the flooding of the Nile, the prices of cotton. He read widely in forgotten corners of mathematics. In Gleick’s book Chaos he says he started looking in “the trash cans of science,” and the pathological curves were there. In 1975 he coined the word fractal, from the Latin fractus, “broken” (the story is that he found it in his son’s Latin dictionary), in a French book, and in 1982 he published The Fractal Geometry of Nature, which opens by saying that clouds are not spheres and mountains are not cones. He received the Wolf Prize in Physics in 1993, and in 2003 the Japan Prize, shared with James Yorke. He died in Cambridge, Massachusetts, in 2010.

The image that made fractals famous, the Mandelbrot set, is built from a one-line rule, z → z² + c, started from z = 0: a point c belongs to the set if the numbers it generates stay bounded. The first computer pictures of it were drawn by Robert Brooks and Peter Matelski in 1978, and Mandelbrot made his own at IBM on 1 March 1980. In 1982 Adrien Douady and John Hubbard proved that the set is a single connected piece, something Mandelbrot had asked about. Mandelbrot conjectured that the boundary of the set has dimension 2, and Mitsuhiro Shishikura proved it in 1998. Whether the set is “locally connected” is still unknown.

Where nature is fractal, and where it stops

It is tempting to say that everything in nature is fractal, and Mandelbrot himself encouraged the thought. The honest version is narrower. A real object is rough over a limited range of scales, from the size of the whole down to some small size at which the structure ends. In 1998 David Avnir and colleagues asked in Science whether the geometry of nature is fractal, and they looked at published papers that claimed a fractal pattern. In their follow-up letter, they reported that the claimed scaling, in 96 papers, spanned only about one and a half orders of magnitude on average, a factor of about 30 between the smallest and largest size. That is a long way from the infinite self-similarity of the Koch curve. A 2021 analysis of the coast of Australia likewise found fractal scaling only down to a lower limit of about one kilometer.

Within its range, fractal geometry does describe real things well, and the examples are beautiful. A Romanesco cauliflower is built from cones within cones. A 2021 study in Science by Azpeitia and colleagues explained why: the flower buds of the plant fail to open into flowers but keep a memory of the flowering state, so each bud makes more buds, in nested spirals over several scales (they stop where the buds become too small to form). River networks follow regular branching laws found by Robert Horton in 1945 and Arthur Strahler in 1952, with each stream typically joined by three to five smaller ones, and fractal dimensions of around 1.5 to 1.7 have been estimated. The branching tubes of the human lung divide about 23 times on average, ending in some 480 million air sacs (estimates range from 274 to 790 million). The surface area of those sacs is often said to be the size of a tennis court. The number is actually uncertain: published estimates run from 23 to 180 square meters, the physiologist who did the classic measurements gave 150 to 180 in 1980 and 130 in 2009, and a recent review calls the tennis-court comparison “not factually based,” since a singles court is 196 square meters. Course 15 tells how the branching of such networks bears on how metabolism scales with body size.

The lecture offers one more example: the saw-toothed joints between the bones of a mammal’s skull. These sutures are indeed intricate. Engineers have shown that the geometry of a suture affects how well it carries and absorbs loads, and in a 1990 study of goat skulls the sutures absorbed 16 to 100 percent more energy per unit of volume than bone did. Researchers have also measured the fractal dimension of human and rodent sutures, with values above 1 and up to 1.4 or more. Whether a higher dimension means better protection is not established, and a 2021 modeling study found that adding a third level of complexity did not improve stiffness or strength over two.