Patterns in Nature

Why the Natural World Looks the Way It Does

The University of Chicago Press

Contents

SYMMETRY

Why your left is like your right (and why it's different)

What is a pattern, anyway? We usually think of it as something that repeats again and again. The math of symmetry can describe what this repetition may look like, as well as why some shapes seem more orderly and organized than others. That's why symmetry is the fundamental scientific "language" of pattern and form. Symmetry describes how things may look unchanged when they are reflected in a mirror, or rotated, or moved. But our intuitions about symmetry can be deceptive. In general, shape and form in nature arise not from the "building up" of symmetry, but from the breaking of perfect symmetry — that is, from the disintegration of complete, boring uniformity, where everything looks the same, everywhere. The key question is therefore: why isn't everything uniform? How and why does symmetry break?

People dreamed of an ordered universe even in ancient times — perhaps especially then, when they were more vulnerable to the random whims of nature. "God, wishing that all things should be good, and so far as possible nothing be imperfect," wrote the Greek philosopher Plato in the fourth century BCE, "reduced the visible universe from disorder to order, as he judged that order was in every way better." Plato imagined a universe created using geometric principles, based on ideas about harmony, proportion, and symmetry. It is a vision that has resonated strongly ever since. Symmetry is one of the key concepts that modern physicists use to understand the world, and they believe its deepest laws will show this feature.

What exactly are these properties of symmetry and pattern that we find in nature, and where do they come from? The best way to understand symmetry is as a property of an object or structure that allows us to change it in some way while leaving it looking just the same as it was before. Think of a sphere: you can rotate it any way you like, and you'd never know: it appears unchanged. Or think of the grid of lines on a piece of graph paper. If you move the paper exactly one grid square's width in a direction parallel to the lines, the grid is superimposed on how it looked at the outset.

These are both symmetries, but of different kinds. The sphere has so-called rotational symmetry, meaning that its appearance is unchanged by rotation. The graph paper has (ignoring the edges) translational symmetry: a "translation" here means a movement in a particular direction. The sphere in fact has perfect rotational symmetry, meaning that it is symmetrical for any angle of rotation. Imagine instead a soccer ball made from hexagonal and pentagonal patches sewn together: only certain rotation angles will superimpose the hexagons and pentagons exactly on their initial positions.

Another kind of symmetry is reflection, which is really just what it sounds like. If you put a mirror upright on the graph paper, the reflection in the mirror looks just like the piece of the sheet that lies behind it. This is exactly true only if the plane of the mirror is placed in just the right position: it has to run either along one of the grid lines or exactly at the halfway point of a square, so that the half-squares you can see and the other halves in the mirror reflection look like a full square. There's another place you can put the mirror, too: exactly along the diagonals of the squares, at an angle of 45° to the grid lines. So this is another of the pattern's "planes of symmetry." If the angle is any different from 45°, the reflection doesn't superimpose exactly on the original grid that it hides: that isn't a true plane of symmetry.

Mathematicians call these rotations, reflections, and translations "symmetry operations" — movements that don't alter the appearance of the object. A plus sign and a square have the same symmetry: they have an identical set of operations that leaves them looking unchanged. A square grid, meanwhile, has a different set of operations from a hexagonal grid such as a bee's honeycomb or chicken wire.

Bodies

One of the most common kinds of symmetry that we see in the natural world is called bilateral symmetry. An object with this symmetry looks unchanged if a mirror passes cleanly through its middle. To put it another way, the object has a left side and a right side that are mirror images of each other. This, of course, is a characteristic of the human body, although little random quirks and accidents of our life history make the symmetry imperfect. There's some evidence that people whose faces are more symmetrical are deemed more attractive on average, and it has also been claimed that other animals with bilaterally symmetrical bodies have more mates the more symmetrical they are.

Bilateral symmetry seems almost to be the default shape for animals. Fish, mammals, insects, and birds all share this attribute. Why is that? One possibility is that bilateral symmetry makes it easier to move in a specific direction: think of the streamlined gliding of a fish, compared with the awkward wriggling of a starfish. Or perhaps a bilateral body meant that such creatures could develop a spine and central nervous system, which has advantages in terms of organizing nerves into a brain. Even starfish have evolved from bilateral ancestors, and in fact their larvae are still bilateral: starfish only develop their fivefold symmetrical bodies as they mature into adults. This kind of shape, which can be superimposed on itself by rotating through a particular angle around one axis, is said to be "radially symmetrical."

Animals first acquired bilaterally symmetrical body designs at least half a billion years ago, and branches of the animal kingdom that don't share this form show that the alternatives tend to have more permissive symmetries, or none at all. There are, for example, the sponges and corals, which, with their tubular, branched, or crinkly funguslike shapes, might easily be mistaken for sea plants. There are tentacled anemones, which often have an approximate radial symmetry: they have an obvious top and bottom, but from the side they look the same from any angle.

Symmetry-breaking and patterning

All kinds of systems and processes, involving both living and nonliving objects, can spontaneously find their way into more or less orderly and patterned states: they can self-organize. There is no longer any reason to appeal to some divine plan to explain this, and there is nothing mysterious about it — but that need not diminish our sense of wonder and appreciation when we see it happen. Without any blueprint or guidance, molecules, particles, grains, rocks, fluids, and living tissues can arrange themselves into regular, sometimes geometrical patterns. The laws of nature seem capable of delivering "order for free." Patterns appear in systems even though we can't find any prescription for them in the fundamental rules that govern how their individual components behave. In this case, the patterns and ordering are said to be emergent: they are a property of the whole system, not deducible by looking reductively at the separate parts.

Symmetry is at the root of understanding how such patterns appear. Because in everyday terms we associate patterns with symmetry — think of the designs on wallpaper or Persian rugs, for example — we might be inclined to imagine that the spontaneous appearance of a pattern in nature involves the spontaneous generation of symmetry. In fact, the opposite is true. Pattern comes from the (partial) destruction of symmetry.

The most symmetrical thing you can imagine is something that you can rotate, reflect, or translate any which way and yet it still looks the same. That's true if the thing is perfectly uniform. So to get pattern from something that is initially unpatterned and uniform involves reducing the symmetry: it is what scientists call a process of symmetry-breaking, which is nature's way of turning things that are initially the same into things that are different. The more symmetry that gets broken, the more subtle and elaborate the pattern.

Randomness might seem the opposite of uniformity, but the two can be equivalent: a random structure is perfectly symmetrical and uniform on average, which means that it too recognizes no "special" directions in space. In the natural world, perfect uniformity or randomness are surprisingly hard to find, at least at the everyday scale. Put yourself on a seashore. The sky is scattered with clouds, perhaps patterned into rows or feathery cirrus. The sea's surface is wrinkled into waves that arrive on the shore with a distinctive pulse. There are plants around the shore, each with its own characteristic shape of flower and leaf. The sand at the water's edge is grooved with ripples, and strewn with the delicate whorls of shells. All around there is shape and form: diverse, yes, but far from random, far from uniform. Symmetry is being broken, again and again.

Cause and effect

In the natural world, when symmetry breaks we often have no cause to have anticipated it. Here's what I mean. If we turn a random pile of bricks into the regular pattern of a wall, it's because we have laid each brick in place. The uniform symmetry of a sheet of paper is broken in making a paper airplane because we folded it that way. In other words, the symmetry gets broken by some force — our moving hand — that compels it to break that way. It's obvious where the symmetry-breaking came from: we put it there.

Compare this to a droplet falling onto the still surface of water. At the start it is perfectly circularly symmetrical: it looks the same in any direction parallel to the surface. But then the splash develops a rim that breaks up into a series of little points, a crown that spits out little droplets from its tips. The rim no longer has its circular symmetry, but has acquired a lower grade of radial symmetry, like a starfish, in which some directions are distinct from others. The process of splashing spontaneously lowered the droplet's own symmetry.

In this book we will see many more examples of such symmetry-breaking. A smooth layer of water breaks up into cells of top-to-bottom circulation when it is uniformly heated from below. A block of material that is shrinking in all directions splits into a network of cracks. Spiraling patterns form in a perfectly mixed solution of chemicals. This is how so many natural patterns form: as if by magic out of a featureless landscape.

We can see it happen, too, on a spider's web. The web itself is a gorgeous natural pattern, but it's not a spontaneous one: the spider makes it the hard way, as we would, by stringing each thread in its place. However, catch the web while the early morning dew is still out and you'll find it beautifully decorated with tiny beads of water hung out like rows of pearls. Did the spider put them there? Not at all — they have been self-organized as dew condensed to coat the silk threads. A thin column of water like this is unstable: it will develop a waviness that pinches it off into little beads, each one regularly spaced where the peaks of the wave were.

Symmetry offers a useful way of thinking about pattern and shape, but even apparently irregular, totally unsymmetrical objects can have a hidden order that mathematics can reveal. Take a pebble. How would you describe its shape? It is sort of round, like a sphere but not quite. A perfect sphere is easy to define mathematically: it has the same amount of curvature everywhere on its surface. But for a pebble, the curvature differs slightly from place to place, and from pebble to pebble. There is a range of curvatures, and the general "pebble shape" can be described by a graph showing the relative amounts of different degrees of curvature in a selection of many pebbles.

Unlike a sphere, pebbles often have some parts that are concave rather than convex: dimples, not bulges. (Potatoes have similar shapes, and the concave parts are the bits that are hard to peel.) Mathematically, these parts are said to have negative curvature. So the graph showing the distribution of curvature of a pebble reaches into negative as well as positive values. But for any collection of pebbles, the overall graphs of curvature have the same shape! Each individual shape differs, but on average there is a single "pebble shape" described by the curvature distribution. Math reveals the common forms underlying the apparent diversity.

FRACTALS

Why mountains look like molehills

If we look at an aerial photograph of a jagged coastline, without a scale bar we can't be sure if we're seeing a stretch that is a mile long, or ten miles, or even a hundred. This indistinguishable appearance at different scales of magnification is a property called fractal. It is a remarkably common feature of nature: think of the fluffy edge of a cloud, or the way a twig from a branch tip mimics the shape of the whole tree, or the repeated branching of the lung's passageways. In fact, fractals have even been called the geometry of nature. Many natural fractals look disorderly when you first see them: there's no exact symmetry in a tree or a mountain profile. But the fractal property discloses a kind of "hidden logic" to the pattern: there's a hierarchical repetition of the same general form at decreasing scales. What are the processes that create this logic? And why is it useful for living organisms?

Natural patterns with a symmetrical form, like the bee's honeycomb, surprise and delight us precisely because they are rather rare. Nature doesn't often display such strict order and regularity. Whether it is the spindly filigree of a naked tree's silhouette in winter or the rugged jumble of a mountainous skyline, what we find in the wild seems more often to have a lot of unpredictability and disorder to it.

Yet these structures have a hidden kind of pattern, too. The logic of the shape or form only becomes fully evident when we try to describe it mathematically, but we already intuit a sort of organization, even without this specialized knowledge. There is surely something pleasing and entrancing about the branching shape of a tree that we wouldn't discover in a totally random arrangement of parts. It's not hard to see where this magic ingredient lies. The shape of a tree is complicated, and we can't easily describe it in the same way as we might describe a square or a hexagon. But we can give a very concise description if we focus instead on the process that produces the shape. A tree shape might be said to be "a trunk that keeps branching."

This description is what scientists would call an algorithm — an instruction for making a structure or, more generally, a process that has to be carried out to get what you're after. The reason why a tree's shape "feels" pleasing rather than incomprehensibly complicated is, I would argue, that we sense the simplicity of the algorithm needed to make it.

Even minor changes to the algorithm will produce a wide range of different treelike shapes. If the branching angle is small and the branches stay straight, we'll end up with a poplarlike network. If the branching angle is wider and the branches can bend and twist, the result is more like an oak.

Looked at this way, an object that seems at first to be geometrically very complex, compared to a cone or a cube, is revealed to have an underlying simplicity. How can we describe this geometry mathematically? A tree has no real symmetry at all in the sense explained in the previous chapter: you can't rotate or reflect it in any way to produce a shape that looks identical. We might be tempted to conclude that geometry has, in fact, nothing to say about this pattern.

But it does. We just need a different kind of geometry. It is called fractal geometry, and it has been said to be "the geometry of nature."

The key to fractal geometry lies with the algorithmic approach to the forms it produces. What the "tree algorithm" is really saying is: keep making the same kind of structure (in this case, a branching junction) again and again at ever-smaller scales. Because of this repetition at different scales, a small part of a tree can resemble the whole thing. Break off the end of a branch and you have something that looks rather like a miniature tree. If you imagine continuing the branching steps without end, then you'd not really be able to tell, simply by looking at a fragment of the shape, how big it was: whether you were seeing the whole tree, or a yard-long branch segment, or a tip no longer than your thumb.

This kind of structure that repeats again and again at smaller scales is said to be "self-similar." Fractals are always self-similar. Their structure is "hierarchical," which means that it is patterned over a succession of different size scales: the trunk of a tree represents one level of the hierarchy, the main branches constitute the next level, and so on.