The sky's no limit
Peel back the edges of today's world, almost anywhere, and just beneath the surface you will find mathematics, often very sophisticated, and a lot of it brand new.
You pick up the telephone and dial a friend in Australia. It feels like talking face to face, but an amazing amount happens to your voice, and your friend's, as the words travel from one end of the globe to the other. The sound waves created by your voice are turned into electric signals, chopped into tiny segments, interleaved with a hundred other conversations, beamed along optical fibres, transmitted from ground stations to orbiting satellites, then disentangled and turned back into sounds.
All of these steps rely on mathematics: communication theory, coding theory - even celestial mechanics, which lets engineers calculate where the satellite will be.
Too technological? Go into the garden and look at your raspberries. No mathematics there, surely? But there is. Raspberries are disease-prone - which is why they are imported from Scotland, where the cold keeps pests at bay. The plant-breeders perform all sorts of experiments to develop the best plants, and they analyse the results using statistics.
Why is mathematics so all-pervasive? It is the ultimate in technology transfer - it lets us transfer ideas, not just bits of equipment, from one problem to another. Its versatility stems directly from its generality. The number "two" applies equally well to two plants, two planes, or two plants; and the same goes for the rest of mathematics. It is the logical structure, not the interpretation, that counts.
Where does new mathematics come from? Sometimes it comes from a direct attack on a practical problem, but surprisingly often it does not. Mathematical research is about building a mental tool kit, and sometimes it makes more sense to worry about tool-building than tool-using. Take knot theory. At first sight it is hard to imagine anything more pointless than a theory of knots. Maybe back in the days of sailing ships . . . but today we only use knots to tie shoelaces and parcels. But mathematicians' knots are not limited to pieces of string. They can be knots in DNA strands or knots in the fabric of space time. Indeed they can be knots in anything, and they behave in the same manner whatever their physical interpretation.
In 1984 a New Zealander named Vaughan Jones invented a mathematical formula associated with any knot, and thereby revolutionised knot theory. Quite incidentally, his idea turned out to be useful for studying DNA, because the DNA molecule is like a double-stranded rope. If you cut the rope and join its ends, you get tangled knots. Jones's idea lets you work out which knots, and that tells biologists useful things about DNA chemistry.
Another new idea is that of a fractal, a highly intricate geometric shape determined by simple mathematical rules. Until recently the most obvious application of fractals was as images on psychedelic T-shirts and posters. However, about 10 years ago a mathematician called Michael Barnsley realised that fractal images can be "compressed" (represented by surprisingly small amounts of data). Instead of the complex shape, just think of the simple rule that generates it.
Unable to convince industry that his novel and unorthodox idea was any good, he founded his own company. Today it is worth millions of dollars, because fractal image compression can be used for the efficient storage of high-quality images on CD-Roms, and it can also be used to increase the number of television channels carried by a communications satellite.
A close relative of fractals is chaos, which first came to public attention in the Seventies. The name does not imply random disorder; it refers to apparent disorder with hidden patterns. Chaos theory now forms the basis of a new industrial technique, an automated quality control device for spring wire. The FRACMAT machine, as it is called, was invented by engineers at the Spring Research and Manufacturer's Association in Sheffield, with the help of mathematicians at the University of Warwick. It winds a very long test spring around a metal rod, measures the spacings of successive coils and then uses chaos theory to analyse the hidden patterns that occur in those numbers. The patterns predict whether or not the wire can successfully be formed into springs on a conventional coiling machine. Cost to the taxpayer: Pounds 76,000. Potential savings to UK industry: Pounds 12 million to Pounds 18 million a year. Not bad!
Not all new mathematics is related to applications. Three and a half centuries ago Pierre de Fermat noted that two perfect squares can add up to another square, for example 9+16=25. But he could not find two cubes that would add up to a cube, and the same went for fourth powers, fifth powers, and so on. He claimed to have a proof that no such numbers existed, a statement that became known as Fermat's Last Theorem.
After centuries of effort it was finally proved in 1994 by Andrew Wiles in a tour de force of significant new mathematics. There are no practical uses for Fermat's Last Theorem - yet. But it would be foolish to bet on things staying that way. For the past few centuries everybody's candidate for the least applicable area of pure mathematics has been the one to which Fermat's Last Theorem belongs, which is called number theory. The subject was intellectually deep, extremely interesting, and totally useless. In the past 20 years, however, the growth of digital electronics has suddenly turned number theory into a very practical area indeed. It forms the basis of error-free transmission of messages on the Internet. It underlies the structure of secure codes, which let people buy goods over the Internet with their credit cards. It is essential to a technique that has mapped the surface of Venus using Earth-based radar. It even plays a role in the acoustic design of new concert halls.
We are living in the Golden Age of mathematics, and every year brings marvellous new ideas, some of obvious practical benefit, some significant because of their impact within mathematics itself. For example in 1988 Zhihong Xia showed that five point masses, moving according to Newtonian gravitation, can be arranged in such a manner that they propel themselves to infinity in a finite period of time. The importance of this result is not interstellar flight, but an improved understanding of "singularities" (nasty behaviour) in gravitating systems.
Today's mathematics is so full of novel ideas that even its raw materials often run counter to everyday intuition. There are curved spaces with four dimensions - or a million. There are geometries in which parallel lines do not exist, and other geometries in which there are infinitely many parallels to a given line, all passing through the same point. There are shapes so bizarre that they do not have meaningful volumes; indeed in theory you could cut a solid sphere up into six such shapes and reassemble the pieces to form two spheres, each the same size as the original - a result so counter intuitive that it is known as the Banach-Tarski paradox.
Are these ideas good for anything? They surely are. Curved spaces lie at the heart of Einstein's theory of gravity, but they also lie within your heart: they describe the geometry of the heart's muscle fibres. In order to understand the million different commodities and prices that make up the UK economy you must grapple with a million-dimensional space. The most obvious geometry that lacks parallels is formed by great circles on the sphere, which determine the quickest route by air from any city to any other; but there is a far less obvious application of non-Euclidean geometry to the design of telephone networks, where it determines how much of the network can be connected to everything else. So whenever you feel the force of gravity holding you to the ground, listen to your heart thumping, hand over your money in the supermarket check-out, fly to the Costa del Sol for a holiday, or telephone that friend in Australia, bear in mind that just below the surface it all depends on mathematics.
But what about the Banach-Tarski paradox? Unfortunately, the pieces required are infinitely intricate, and until we learn how to subdivide atoms the vision of making a quick billion by repeatedly doubling a sphere of solid gold cannot be realised. If the Banach-Tarski paradox has any practical value, you will have to seek it by thinking laterally, not literally.
The theme of this article is developed in depth in Ian Stewart's new book From Here to Infinity, published by the Oxford University Press Ian Stewart is Professor of Mathematics at Warwick University