The Natural Order : Nature's Numbers Chapter 1

We live in a universe of patterns. Every night the stars move in circles across the sky. The seasons cycle at yearly intervals. No two snowflakes are ever exactly the same, but they all have sixfold symmetry. Tigers and zebras are covered in patterns of stripes, leopards and hyenas are covered in patterns of spots. Intricate trains of waves march across the oceans; very similar trains of sand dunes march across the desert. Colored arcs of light adorn the sky in the form of rainbows, and a bright circular halo sometimes surrounds the moon on winter nights. Spherical drops of water fall from clouds.
Human mind and culture have developed a formal system of thought for recognizing, classifying, and exploiting patterns. We call it mathematics. By using mathematics to organize and systematize our ideas about patterns, we have discovered a great secret: nature's patterns are not just there to be admired, they are vital clues to the rules that govern natural processes. Four hundred years ago, the German astronomer Johannes Kepler wrote a small book, The Six-Cornered Snowflake, as a New Year's gift to his sponsor. In it he argued that snowflakes must be made by packing tiny identical units together. This was long before the theory that matter is made of atoms had become generally accepted. Kepler performed no experiments; he just thought very hard about various bits and pieces of common knowledge. His main evidence was the sixfold symmetry of snowflakes, which is a natural consequence of regular packing. If you place a large number of identical coins on a table and try to pack them as closely as possible, then you get a honeycomb arrangement, in which every coin-except those at the edges-is surrounded by six others, arranged in a perfect hexagon.



The regular nightly motion of the stars is also a clue, this time to the fact that the Earth rotates. Waves and dunes are clues to the rules that govern the flow of water, sand, and air. The tiger's stripes and the hyena's spots attest to mathematical regularities in biological growth and form. Rainbows tell us about the scattering of light, and indirectly confirm that raindrops are spheres. Lunar haloes are clues to the shape of ice crystals.

There is much beauty in nature's clues, and we can all recognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things. Mathematics is to nature as Sherlock Holmes is to evidence. When presented with a cigar butt, the great fictional detective could deduce the age, profession, and financial state of its owner. His partner, Dr. Watson, who was not as sensitive to su~h matters, could only look on in baffled admiration, until the master revealed his chain of impeccable logic. When presented with the evidence of hexagonal snowflakes, mathethematicians can deduce the atomic geometry of ice crystals. If you are a Watson, it is just as baffling a trick, but I want to show you what it is like if you are a Sherlock Holmes. Patterns possess utility as well as beauty. Once we have learned to recognize a background pattern, exceptions suddenly stand out. The desert stands still, but the lion moves. Against the circling background of stars, a small number of stars that move quite differently beg to be singled out for special attention. The Greeks called them planetes, meaning "wanderer," a term retained in our word "planet." It took a lot longer to understand the patterns of planetary motion than it did to work out why stars seem to move in nightly circles. One difficulty is that we are inside the Solar System, moving along with it, and things that look simple from outside often look much more complicated from inside. The planets were clues to the rules behind gravity and motion.

We are still learning to recognize new kinds of pattern. Only within the last thirty years has humanity become explicitly aware of the two types of pattern now known as fractals and chaos. Fractals are geometric shapes that repeat their structure on ever-finer scales, and I will say a little about them toward the end of this chapter; chaos is a kind of apparent randomness whose origins are entirely deterministic, and I will say a lot about that in chapter 8. Nature "knew about" these patterns billions of years ago, for clouds are fractal and weather is chaotic. It took humanity a while to catch up.

The simplest mathematical objects are numbers, and the simplest of nature's patterns are numerical. The phases of the moon make a complete cycle from new moon to full moon and back again every twenty-eight days. The year is three hundred and sixty-five days long-roughly. People have two legs, cats have four, insects have six, and spiders have eight. Starfish have five arms (or ten, eleven, even seventeen, depending on the species). Clover normally has three leaves: the superstition that a four-leaf clover is lucky reflects a deepseated belief that exceptions to patterns are special. A very curious pattern indeed occurs in the petals of flowers. In nearly all flowers, the number of petals is one of the numbers that occur in the strange sequence 3, 5, 8, 13, 21, 34, 55, 89. For instance, lilies have three petals, buttercups have five, many delphiniums have eight, marigolds have thirteen, asters have twenty-one, and most daisies have thirty-four, fifty-five, or eighty-nine. You don't find any other numbers anything like as often. There is a definite pattern to those numbers, but one that takes a little digging out: each number is obtained by adding the previous two numbers together. For example, 3 + 5 = 8, 5 + 8 = 13, and so on. The same numbers can be found in the spiral patterns of seeds in the head of a sunflower. This particular pattern was noticed many centuries ago and has been widely studied ever since, but a really satisfactory explanation was not given until 1993. It is to be found in chapter 9.

Numerology is the easiest-and consequently the most dangerous-method for finding patterns. It is easy because anybody can do it, and dangerous for the same reason. The difficulty lies in distinguishing significant numerical patterns from accidental ones. Here's a case in point. Kepler was fascinated with mathematical patterns in nature, and he devoted much of his life to looking for them in the behavior of the planets. He devised a simple and tidy theory for the existence of precisely six planets (in his time only Mercury, Venus, Earth, Mars, Jupiter, and Saturn were known). He also discovered a very strange pattern relating the orbital period of a planet-the time it takes to go once around the Sun-to its distance from the Sun. Recall that the square of a number is what you get when you multiply it by itself: for example, the square of 4 is 4 x 4 = 16. Similarly, the cube is what you get when you multiply it by itself twice: for example, the cube of 4 is 4 x 4 x 4 = 64. Kepler found that if you take the cube of the distance of any planet from the Sun and divide it by the square of its orbital period, you always get the same number. It was not an especially elegant number, but it was the same for all six planets.

Which of these numerological observations is the more significant? The verdict of posterity is that it is the second one, the complicated and rather arbitrary calculation with squares and cubes. This numerical pattern was one of the key steps toward Isaac Newton's theory of gravity, which has explained all sorts of puzzles about the motion of stars and planets. In contrast, Kepler's neat, tidy theory for the number of planets has been buried without trace. For a start, it must be wrong, because we now know of nine planets, not six. There could be even more, farther out from the Sun, and small enough and faint enough to be undetectable. But more important, we no longer expect to find a neat, tidy theory for the number of planets. We think that the Solar System condensed from a cloud of gas surrounding the Sun, and the number of planets presumably depended on the amount of matter in the gas cloud, how it was distributed, and how fast and in what directions it was moving. An equally plausible gas cloud could have given us eight planets, or eleven; the number is accidental, depending on the initial conditions of the gas cloud, rather than universal, reflecting a general law of nature.

The big problem with numerological pattern-seeking is that it generates millions of accidentals for each universal. Nor is it always obvious which is which. For example, there are three stars, roughly equally spaced and in a straight line, in the belt of the constellation Orion. Is that a clue to a significant law of nature? Here's a similar question. 10, Europa, and Ganymede are three of Jupiter's larger satellites. They orbit the planet in, respectively, 1.77, 3.55, and 7.16 days. Each of these numbers is almost exactly twice the previous one. Is that a significant pattern? Three stars in a row, in terms of position; three satellites "in a row" in terms of orbital period. Which pattern, if either, is an important clue? I'll leave you to think about that for the moment and return to it in the next chapter.

In addition to numerical patterns, there are geometric ones. In fact this book really ought to have been called Nature's Numbers and Shapes. I have two excuses. First, the title sounds better without the "and shapes." Second, mathematical shapes can always be reduced to numbers-which is how computers handle graphics. Each tiny dot in the picture is stored and manipulated as a pair of numbers: how far the dot is along the screen from right to left, and how far up it is from the bottom. These two numbers are called the coordinates of the dot. A general shape is a collection of dots, and can be represented as a list of pairs of numbers. However, it is often better to think of shapes as shapes, because that makes use of our powerful and intuitive visual capabilities, whereas complicated lists of numbers are best reserved for our weaker and more laborious symbolic abilities.

Until recently, the main shapes that appealed to mathematicians were very simple ones: triangles, squares, pentagons, hexagons, circles, ellipses, spirals, cubes, spheres, cones, and so on. All of these shapes can be found in nature, although some are far more common, or more evident, than others. The rainbow, for example, is a collection of circles, one for each color. We don't normally see the entire circle, just an arc; but rainbows seen from the air can be complete circles. You also see circles in the ripples on a pond, in the the human eye, and on butterflies' wings.

Talking of ripples, the flow of fluids provides an inexhaustible supply of nature's patterns. There are waves of many different kinds-surging toward a beach in parallel ranks, spreading in a V-shape behind a moving boat, radiating outward from an underwater earthquake. Most waves are gregarious creatures, but some-such as the tidal bore that sweeps up a river as the energy of the incoming tide becomes confined to a tight channel-are solitary. There are swirling spiral whirlpools and tiny vortices. And there is the apparently structureless, random frothing of turbulent flow, one of the great enigmas of mathematics and physics. There are similar patterns in the atmosphere, too, the most dramatic being the vast spiral of a hurricane as seen by an orbiting astronaut.

There are also wave patterns on land. The most strikingly mathematical landscapes on Earth are to be found in the great ergs, or sand oceans, of the Arabian and Sahara deserts. Even when the wind blows steadily in a fixed direction, sand dunes form. The simplest pattern is that of transverse dunes, which-just like ocean waves-line up in parallel straight rows at right angles to the prevailing wind direction. Sometimes the rows themselves become wavy, in which case they are called barchanoid ridges; sometimes they break up into innumerable shield-shaped barchan dunes. If the sand is slightly moist, and there is a little vegetation to bind it together, then you may find parabolic dunes-shaped like a U, with the rounded end pointing in the direction of the wind. These sometimes occur in clusters, and they resemble the teeth of a rake. If the wind direction is variable, other forms become possible. For example, clusters of star-shaped dunes can form, each having several irregular arms radiating from a central peak. They arrange themselves in a random pattern of spots.

Nature's love of stripes and spots extends into the animal kingdom, with tigers and leopards, zebras and giraffes. The shapes and patterns of animals and plants are a happy hunting ground for the mathematically minded. Why, for example, do so many shells form spirals? Why are starfish equipped with a symmetric set of arms? Why do many viruses assume regular geometric shapes, the most striking being that of an icosahedron-a regular solid formed from twenty equilateral triangles? Why are so many animals bilaterally symmetric? Why is that symmetry so often imperfect, disappearing when you look at the detail, such as the position of the human heart or the differences between the two hemispheres of the human brain? Why are most of us right-handed, but not all of us?

In addition to patterns of form, there are patterns of movement. In the human walk, the feet strike the ground in a regular rhythm: left-right-Ieft-right-Ieft-right. When a four-legged creature-a horse, say-walks, there is a more complex but equally rhythmic pattern. This prevalence of pattern in locomotion extends to the scuttling of insects, the flight of birds, the pulsations of jellyfish, and the wavelike movements of fish, worms, and snakes. The sidewinder, a desert snake, moves rather like a single coil of a helical spring, thrusting its body forward in a series of S-shaped curves, in an attempt to minimize its contact with the hot sand. And tiny bacteria propel themselves along using microscopic helical tails, which rotate rigidly, like a ship's screw.

Finally, there is another category of natural pattern-one that has captured human imagination only very recently, but dramatically. This comprises patterns that we have only just learned to recognize-patterns that exist where we thought everything was random and formless. For instance, think about the shape of a cloud. It is true that meteorologists classify clouds into several different morphological groups-cirrus, stratus, cumulus, and so on-but these are very general types of form, not recognizable geometric shapes of a conventional mathematical kind. You do not see spherical clouds, or cubical clouds, or icosahedral clouds. Clouds are wispy, formless, fuzzy clumps. Yet there is a very distinctive pattern to clouds, a kind of symmetry, which is closely related to the physics of cloud formation. Basically, it is this: you can't tell what size a cloud is by looking at it. If you look at an elephant, you can tell roughly how big it is: an elephant the size of a house would collapse under its own weight, and one the size of a mouse would have legs that are uselessly thick. Clouds are not like this at all. A large cloud seen from far away and a small cloud seen close up could equally plausibly have been the other way around. They will be different in shape, of course, but not in any manner that systematically depends on size.

This "scale independence" of the shapes of clouds has been verified experimentally for cloud patches whose sizes vary by a factor of a thousand. Cloud patches a kilometer across look just like cloud patches a thousand kilometers across. Again, this pattern is a clue. Clouds form when water undergoes a "phase transition" from vapor to liquid, and physicists have discovered that the same kind of scale invariance is associated with all phase transitions. Indeed, this statistical self-similarity, as it is called, extends to many other natural forms. A Swedish colleague who works on oil-field geology likes to show a slide of one of his friends standing up in a boat and leaning nonchalantly against a shelf of rock that comes up to about his armpit. The photo is entirely convincing, and it is clear that the boat must have been moored at the edge of a rocky gully about two meters deep. In fact, the rocky shelf is the side of a distant fjord, some thousand meters high. The main problem for the photographer was to get both the foreground figure and the distant landscape in convincing focus.

Nobody would try to play that kind of trick with an elephant. However, you can play it with many of nature's shapes, including mountains, river networks, trees, and very possibly the way that matter is distributed throughout the entire universe. In the term made famous by the mathematician Benoit Mandelbrot, they are all fractals. A new science of irregularity- fractal geometry-has sprung up within the last fifteen years. I'm not going to say much about fractals, but the dynamic process that causes them, known as chaos, will be prominently featured.

Thanks to the development of new mathematical theories, these more elusive of nature's patterns are beginning to reveal their secrets. Already we are seeing a practical impact as well as an intellectual one. Our newfound understanding of nature's secret regularities is being used to steer artificial satellites to new destinations with far less fuel than anybody had thought possible, to help avoid wear on the wheels of locomotives and other rolling stock, to improve the effectiveness of heart pacemakers, to manage forests and fisheries, even to make more efficient dishwashers. But most important of all, it is giving us a deeper vision of the universe in which we live, and of our own place in it

Chapter 1 : The Natural Order
Chapter 2 : What Mathematics is For
Chapter 3 : What Mathematics is About
Chapter 4 : The Constants of Change
Chapter 5 : From Violins to Videos