Fractals in nature

Ask how long a coastline is and you get an awkward answer: it depends on your ruler. Walk a pair of dividers set to 100 kilometers along the coast of Britain and you get one number. Set them to 10 kilometers and the dividers follow bays and headlands the long ruler stepped over, so the total comes out longer. At 1 kilometer they dip into coves and the total grows again. There is no point at which it settles down.

Lewis Fry Richardson, an English physicist interested in what makes wars break out, ran into this around 1950 while checking whether neighboring countries agreed on the length of their shared borders. They often did not, by large margins. He measured coasts and frontiers from maps at many ruler lengths and found that, plotted on logarithmic axes, length against ruler size made a straight line. His results were published in 1961, after his death. In 1967 Benoit Mandelbrot read that straight line as a dimension. In "How Long Is the Coast of Britain?" in Science he showed that Richardson's slope gives a number between 1 and 2 that says how wiggly a curve is: about 1.25 for the west coast of Britain, about 1.02 for the much smoother coast of South Africa.

The first scene of the walkthrough above does Richardson's experiment on a computer-made coast. Each time the ruler halves, the measured length grows by the same factor, which is exactly what a fractal dimension means.

Mountains are the same story one dimension up. A mountain range viewed from a plane has ridges, and the ridges have spurs, and the spurs have gullies. A close-up photograph of a rock face can be hard to tell from a distant photograph of a cliff unless something gives the scale away, which is why geologists put a hammer or a coin in the frame. Mandelbrot and his colleague Richard Voss built artificial landscapes from a random process called fractional Brownian motion and found that, with the right roughness setting, they looked convincingly like real terrain. That method is now the backbone of computer-generated landscapes, and Chapter 13 builds one you can control.

A tree is a recipe applied over and over: grow a stem, then split. The split is different every time in its details, but the rule is the same, and that is all it takes for the result to look self-similar. A single branch broken off a tree looks like a small tree. A twig looks like a small branch. Botanists describe the pattern with a few numbers: the angle between daughter branches, how much shorter each generation is, how many daughters there are, and how much the thickness drops at each fork.

Leonardo da Vinci noticed something about that last number. In his notebooks he wrote that all the branches of a tree at every stage of its height, when put together, are equal in thickness to the trunk below them. The usual modern reading is about cross-sectional area: at any fork, the areas of the daughter branches add up to the area of the parent. If a trunk of diameter d splits into branches of diameters d₁ and d₂, then d² = d₁² + d₂².

It is worth stating carefully what the rule is and isn't. It is an empirical observation that holds roughly for many species. It is not a law of physics, and measured exponents vary from tree to tree and species to species. Several explanations have been proposed. One is the "pipe model": each leaf is fed by its own bundle of conducting tissue running down to the roots, so the cross-section at any height is proportional to the leaves above it. Another, from the physicist Christophe Eloy in 2011, starts from wind: if every branch is built to be just as likely to snap in a strong wind as any other, the branch thicknesses that result obey Leonardo's rule with an exponent close to 2.

The tree grower below lets you set the exponent and see what it does. Below 2, daughters are fat for their parent and the crown looks heavy. Above 2, they are thin, and the tree looks like a winter silhouette drawn with a fine pen. The total length, the number of tips and the dimension of the branching skeleton are measured as you go.

Your body has a delivery problem. Oxygen has to get from the air to every cell, and the last step is diffusion: molecules drifting across a thin membrane. Diffusion is quick over a few millionths of a meter and hopelessly slow over a centimeter. So you need two things at once: an enormous area of thin membrane where air meets blood, and a network of pipes that brings air and blood within a whisker of every part of that area.

A branching tree solves both. Each split doubles the number of pipes while shrinking their size, so after n generations there are 2ⁿ ends spread through the volume. The human airway tree starts at the windpipe and, in Ewald Weibel's 1963 model, divides about 23 times before it reaches the air sacs, the alveoli. If every split were in two, 23 generations would give 2²³, about 8.4 million ends, and each end opens into clusters of tiny sacs. The total gas-exchange surface is measured in tens of square meters, folded into a chest a few liters in volume.

The reason a fractal-like design is so good at this is that its surface does not obey the usual rule for solid shapes. Scale up a ball and its volume grows faster than its surface, so big solid things have relatively little skin. A branching network has its surface spread through the volume like a sponge. Within its working range, adding generations adds surface roughly in proportion to the space filled, not to the outside of it. Blood vessels do the same from the other side: arteries branch into arterioles and then capillaries, and in most tissues almost every cell sits within a few cell-widths of one.

The third walkthrough scene shows the idealized version: an H-tree, where each generation is shorter by 1/√2 and turned through a right angle. A few generations in, the ends are spread evenly through a square, and every point of the square is close to one of them. That is a cartoon, not anatomy, but it is the right cartoon.

From the air, a river system looks like a tree lying on its side: thin streams join into creeks, creeks into rivers, rivers into a main stem. The resemblance is not a coincidence either. Rain falls everywhere on a landscape, and water has to be collected from every point and carried to one outlet. That is the lung's problem run backwards, and erosion solves it with the same kind of branching.

Hydrologists have counted these trees for eighty years. Robert Horton in 1945, and Arthur Strahler in a refined form in 1957, numbered the streams by order. A stream with no tributaries is order 1. Where two order-1 streams meet, they form an order-2 stream; where two order-2s meet, an order 3; and so on. A smaller stream joining a bigger one doesn't raise its order. Count the streams of each order and the counts fall off geometrically. The ratio between one order and the next, the bifurcation ratio, is typically between 3 and 5 in natural basins. Stream lengths and basin areas grow geometrically the other way. A fixed ratio from one scale to the next is the fingerprint of self-similarity.

A second regularity, Hack's law (1957), says that the length of the main stream grows with the area of its basin raised to a power of about 0.6. If basins were all the same shape at every size, the power would be 0.5, since length then grows as the square root of area. That it is larger means big basins are relatively longer and thinner than small ones, or that the main stream wiggles more over longer distances. Both are signs of a geometry that isn't simply scaled up.

In 1981 Thomas Witten and Leonard Sander described a model so simple it fits in a sentence. Put one sticky particle in the middle of a grid. Release another far away and let it wander at random, one step at a time, until it touches the first. It sticks. Release another. Repeat thousands of times.

You might expect a round blob. You get something else: a lacy, branching cluster with long arms and deep empty fjords. Its dimension, measured from how its mass grows with its radius, comes out close to 1.71 in two dimensions. That is more than a line and much less than a filled disk.

The reason is screening. A wandering particle is far more likely to bump into a tip that sticks out than to find its way down into a fjord, because to reach the fjord it has to pass all the tips around the entrance without touching any of them. So the tips grow, which makes them stick out further, which makes them catch even more. The fjords are starved. Any small bump becomes a branch, and every branch sprouts branches of its own. Nothing in the rule mentions branching; it falls out of the randomness.

What makes the model important rather than merely pretty is how many real processes it describes. Whenever growth is fed by something that diffuses, and the growing surface eats it as soon as it arrives, you get the same shapes. Zinc deposited from a solution onto an electrode grows in DLA-like leaves. Air pushed into a thin layer of oil between two glass plates (a Hele-Shaw cell) fingers out the same way, as does water forced into porous rock. The black, fern-like "dendrites" on the surface of some rocks, which are often mistaken for fossil plants, are mineral deposits that grew by diffusion through thin cracks. Some bacterial colonies grown on nutrient-poor plates branch the same way.

A lightning bolt is the most dramatic branching pattern most of us ever see, and it is DLA's close cousin. Before the main flash, a faint, branching channel of ionized air, the stepped leader, works its way down from the cloud in jumps of tens of meters. Each step goes where the electric field is strongest, and the field is strongest at the tips of the channel that already exists. That is the screening argument again, with electric field in place of wandering particles.

The same thing happens in solids. In 1777 the German physicist Georg Christoph Lichtenberg scattered fine powder over insulating plates he had charged with a spark, and the powder settled into branching, fern-like patterns that now carry his name. Lichtenberg figures appear today when high-voltage discharges are frozen into acrylic blocks, and sometimes as temporary branching marks on the skin of people struck by lightning.

In 1984 Lucian Niemeyer, Luciano Pietronero and Hans Wiesmann turned this into a model, the dielectric breakdown model. Solve for the electric potential around the existing discharge, then add one new piece of channel at a spot on its edge, chosen with probability proportional to the local field raised to a power η. With η = 1 the model is mathematically equivalent to DLA and grows patterns of dimension about 1.7, in good agreement with their measurements of real surface discharges. A smaller η gives fatter, denser patterns; a larger η gives sparse, almost linear ones, more like a single lightning channel.

Snowflakes are where many people first notice self-similarity: six arms, each with side branches, each side branch with smaller side branches. The sixfold symmetry comes from the hexagonal arrangement of water molecules in ice. The branching comes from an instability that William Mullins and Robert Sekerka analyzed in 1964 for crystals growing from a melt. A crystal growing from vapor is fed by water molecules diffusing toward it, and any bump on its surface sticks out into slightly richer air, so it grows faster. That is the DLA mechanism on a hexagonal crystal. The Japanese physicist Ukichiro Nakaya, who grew the first artificial snow crystals in the 1930s, showed that the shape depends on temperature and humidity: plates, columns, needles or the familiar branched stars.

A snowflake has only a few levels of branching, though, three at most and usually fewer, before the branches are too small to split again. Frost on a window is often richer: it grows across a surface for longer, and its feathery patterns can show branching over a wider range of sizes.

Romanesco, a variety of cauliflower, is the most famous fractal vegetable. Its head is a spiral of cones, each cone a spiral of smaller cones, each of those a spiral of smaller ones still, repeated over several levels you can see with the naked eye. The spirals themselves follow the same arrangement as sunflower seeds, with successive buds placed about 137.5° apart, the golden angle. A 2021 study in Science traced the self-similar cones to the plant's flower-making genes. The plant repeatedly starts to make flowers, fails, and makes a new shoot instead; each of those shoots does the same, faster, producing cones on cones.

Clouds were among the first natural objects to have a fractal dimension measured over a truly huge range of sizes. In 1982 Shaun Lovejoy compared the area and perimeter of rain and cloud regions in satellite and radar images. For smooth shapes, perimeter grows as the square root of area. For clouds it grew faster, consistent with a boundary dimension of about 1.35, and the same relation held from about one kilometer to about a thousand, three factors of ten. That is unusually wide for a natural object.

A mathematical fractal has detail all the way down. Nature never does. Every natural fractal has an inner cutoff, below which the pattern changes character, and an outer cutoff, above which it ends. A coastline's roughness eventually gives way to individual pebbles, then sand grains, then molecules. A tree stops branching at the twig, and its outermost size is the tree. The lung's airways stop at the air sacs. A DLA cluster in a laboratory is limited below by the size of a single deposited particle and above by the size of the dish.

That matters when people claim to have found a fractal. A dimension is a slope on a log–log plot, and almost any curved set of points looks roughly straight if you look at a short enough piece of it. In 1998 David Avnir and colleagues surveyed experimental papers in Physical Review journals that reported fractal dimensions. Most of the claimed straight lines spanned only about one to two factors of ten, often less. Their point was not that nature isn't fractal, but that "fractal" is a statement about a range, and a short range is weak evidence.

A careful measurement does three things. It says over which range of scales the slope holds. It shows the plot, including where it bends. And it checks the dimension against a mechanism: DLA predicts about 1.7, Murray's law predicts the cube rule, a river network should show Horton's geometric ratios. Instrument 2 above follows the same discipline in miniature: it fits only the part of the cluster that has finished growing, and tells you which radii it used.

The cutoffs are also where the interesting biology and physics live. The inner cutoff of the lung, the size of an air sac, is set by how far oxygen can diffuse. The outer cutoff of a tree is set by how tall a column of water a plant can pull up and how much weight its trunk can bear. The fractal range is what the process does when nothing else interferes; the ends of the range tell you what finally does.