Dice, dunes and daisies

From the laws of probability to deserts and the arrangement of petals in flowers, pattern is everywhere. In this extract from his new book, Ian Stewart, who has just won a top prize for the communication of science, explains the fascination of maths
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The Independent Culture
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 the same, but all have sixfold symmetry. Tigers and zebras are covered in patterns of stripes, leopards and hyenas in patterns of spots. Intricate trains of waves march across the oceans; similar trains of sand dunes march across the desert. Coloured arcs of light adorn the sky in the form of rainbows; a circular halo 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 recognising, classifying and exploiting patterns. We call it mathematics. By using it to organise and systematise our ideas about patterns, we have discovered a great secret: nature's patterns are not just there to be admired, but are vital clues to the rules that govern natural processes.

Some 450 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 up of atoms had become 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 pack them as closely as possible, 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 recognise 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 finances of its owner. His partner Dr Watson, not as sensitive to such 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, mathematicians 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.

OVER the centuries, the collective minds of mathematicians have created their own universe. I don't know where it is situated - I don't think there is a "where" in any meaningful sense of the word - but I assure you this mathematical universe seems real enough when you're in it. It is by necessity described in symbols and pictures, but the symbols are no more that world than musical notation is music. And, not despite its peculiarities but because of them, it has provided humans with their deepest insights into the world around them.

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 28 days. The year is 365 days long - roughly. People have two legs, cats have four, insects have six, spiders have eight. Starfish have five arms (or 10, 11, even 17, depending on species). Clover normally has three leaves: the superstition that four-leafed clover is lucky reflects the 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 13, asters 21, and most daisies have 34, 55 or 82. You don't find any other numbers anything like as often. Those numbers have a definite pattern, but one that takes a little digging out: each is obtained by adding the two previous ones 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.

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. It isn't 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. Io, 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 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?

As well as numerical patterns, there are geometric ones. Until recently, the 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 can be found in nature, though some are far more common, or more evident, than others. The rainbow is a collection of circles, one for each colour. 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, the human eye and spots 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 towards a beach in parallel ranks, spreading in a V-shape behind a moving boat, radiating outwards from an underwater earthquake. 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.

Nature's love of pattern extends into the animal kingdom. Animals and plants provide a happy hunting ground for the mathematically minded. Why do so many shells form spirals? Why are starfish equipped with asymmetric sets of arms? Why do many viruses assume regular geometric shapes, the most striking being that of an icosahedron - a regular solid formed from 20 equilateral triangles? Why is symmetry so often imperfect, disappearing when you look at fine detail such as the placing of the human heart or the two hemispheres of the human brain? Why are most of us right-handed - but not all?

In addition to patterns of form, there are patterns of movement. In the human walk, successive feet strike the ground in a regular rhythm: left- right-left-right-left-right. When a four-legged creature, such as a horse, 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, rolling across the hot sand and trying to keep as little of its body in contact with it as possible. 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 recognise - 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 recognisable 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 closely related to the physics of cloud formation. It is this: you can't tell what size a cloud is by looking at it. If you look at an elephant, you can: an elephant the size of a house would collapse under its own weight; one the size of a mouse would have absurdly thick legs. Clouds are not like this. A large cloud seen from far away, and a small cloud seen close up, could equally plausibly have been the other way round. They will be different in shape, but not in any manner that systematically depends upon 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 1km across look just like cloud patches 1,000km across. Again, this pattern is a clue. Clouds form when water undergoes a "phase transition" from vapour to liquid, and physicists now know that the same kind of scale invariance is associated with all phase transitions.

This "statistical self-similarity" extends to many other natural forms. A Swedish colleague who works on oilfield geology likes to show a slide of a friend standing up in a boat, leaning nonchalantly against a shelf of rock that comes up to his armpit. The photo is entirely convincing, and it is clear the boat must have been moored at the edge of a rocky gully about 2m deep. In fact, the rocky shelf is the side of a distant fjord, 1,000m high. The main problem for the photographer was to get both the foreground figure and the distant landscape in convincing focus.

Nobody even tries to play that kind of trick with an elephant. But you can play it with many of nature's shapes, including mountains, river networks, trees, and possibly the way matter is distributed throughout the universe. In the term made famous by Benoit Mandelbrot, they are all "fractals". A new science of irregularity, fractal geometry, has sprung up in the last 15 years. The dynamic process that causes them is known as chaos.

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 new- found understanding of nature's secret regularities is being used to steer artificial satellites to new destinations using far less fuel than thought possible, to help avoid wear on the wheels of locomotives, 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 understanding of the universe in which we live, and of our own place in it.

! Extracted from `Nature's Numbers' by Ian Stewart (Weidenfeld and Nicholson pounds 9.99). This month the author won the Royal Society Michael Faraday Award, given to the scientist who has done most to further the public understanding of science.


10^10^10^10^46 (where, to avoid a printer's nightmare, ^ indicates `forming a power'): This is the number of prime numbers you must list before there is a discernible change in their pattern. The sequence begins 2, 3, 5, 7, 11, 13, 17, 19 and goes on forever. Apart from 2, all primes are odd and fall into two groupings - those that are one less than a multiple of four (such as 3, 7, 11, 19) and those that are one more (5, 13, 17). If you run along the first sequence of primes above, there are more numbers (four) in the first category than there are in the second (three); this persists up to at least a trillion. When the numbers get really big, the second category goes into the lead. In full, the number would have a very large number of 0s; if all the matter in the universe were turned into paper, and a zero inscribed on every electron, there wouldn't be enough of them to hold even a tiny fraction of the necessary zeros.

Fibonacci numbers The numbers of petals in plants display mathematical regularities. They form the beginning of the Fibonacci series, in which each number is the sum of the two preceding it. In giant sunflowers, the florets are arranged in two intersecting families of spirals, one winding clockwise, the other counterclockwise. In some species the clockwise spirals number 34, the counterclockwise 55. Both are Fibonacci numbers. The precise number depends on the species, but you often get 34 and 55, or 55 and 89, or even 89 and 144 - the next Fibonnaci number. Pineapples have eight rows of scales sloping left, 13 sloping to the right. These, too, are Fibonacci numbers.

2 The square root of 2 isn't exactly representable as a fraction. That is, if you multiply any fraction by itself, you won't get 2 exactly. You can get very close. For example, the square of 17/12 is 289/144, and if only it were 288/144 you'd get 2. But it isn't, and you don't.


Titles in the series include River Out of Eden by Richard Dawkins and The Origin of Humankind by Richard Leakey. To order Nature's Numbers: Discovering Order and Pattern in the Universe by Ian Stewart, or any of the others, telephone 01903 732596 quoting your Access or Visa number, or write (cheque made payable to `The Orion Publishing Group Ltd') to: Littlehampton Book Services, 14 Eldon Way, Lineside Industrial Estate, Littlehampton, W Sussex BN17 7HE. All titles cost pounds 9.99.