The man who made maths sexy

Forget about grinding school lessons, this is about unravelling the magic and mystery of the world itself

Ian Stewart, the professor of mathematics at Warwick University, reaches into a large top hat on the table beside him and pulls out a pair of toy rabbits. He puts them on the table and reaches in again, pulling out a second pair, and a pair of smaller ones - "baby rabbits", he announces. And again: this time two pairs of toy rabbits and one pair of babies.

As Stewart's hand reaches into the hat again, a disembodied voice speaks from underneath the table: "Hang on," it says, "give me some back." It is his assistant, who is helping rehearse a trick. Stewart pauses and declares: "Mathematics is the closest that humans get to true magic."

The rehearsal is taking place in the lecture theatre of the Royal Institution where Stewart will deliver five Christmas lectures to an audience of schoolchildren. This part of the first lecture deals with Fibonacci's ratio - the sequence 1, 1, 2, 3, 5, 8, 13, 21 - in which each new number is the sum of the preceding pair. This ratio crops up all over the natural world: in the breeding of rabbits and the growth of petals on a flower. Seeking out and explaining patterns like this is the sort of thing that makes Stewart beam. And he beams a lot. With his sprouting white hair and round features, he looks like a very happy, very tall gnome.

It's been a good year for him. The Royal Institution's Christmas lectures are 168 years old and Stewart will be only the second mathematician to deliver them. He has had three books published; and a business sideline derived from research will bring in thousands of pounds. Besides that, Stewart writes the "Mathematical games" column for the monthly magazine Scientific American - the same column that held him rapt as a teenager.

STEWART, 52, hasn't done badly for someone who built a career on a broken bone. When he was eight, growing up in Folkestone, he was a poor student of mathematics at Harcourt primary school. Then he broke his collar bone and, while he was confined to bed, his mother decided to take a hand in his education. She gave him a 400-question arithmetic test and, to her great surprise, Stewart got only four wrong answers. She thought, "There's nothing wrong with his maths." Stewart recalls that the problem was that he had been getting bored at school because the maths was too easy. When he got to Harvey grammar school he was put straight into the fast stream for maths.

Stewart dithered over physics or mathematics at university, but Cambridge offered him a place for maths and that made up his mind. When he graduated (with a First) he went to Warwick, where he was appointed a lecturer in 1984. He became a dedicated academic, disdainful of anything so populist as science books for a general public.

That view gradually changed, until his breakthrough in 1989 when he wrote Does God Play Dice?, a book examining the mathematics of chaos when it was still a new theory. Most people are now familiar with the idea that some systems are inherently unpredictable: even if you know everything about them when they start, you can't predict what they will do, even with ideal mathematical tools. The classic case is the weather, where tiny changes in conditions in one spot on earth can drastically alter conditions elsewhere.

Chaos theory grabbed the public imagination because home computers became commonly available in the 1980s. They allowed lovers of scientific games to do so much more than outline mathematical equations. They could use computer programs to enable them to see how chaos arises.

At the same time computer games came on the market which generated apparently real mountain scenery and coastlines using "fractals", along with the Mandelbrot set - those whirling, spiky patterns that look the same no matter how far you magnify or shrink them. (Mandelbrot images have become the standard decoration for compilations of rave music.)

"In the United States a big decline in the popularity of maths was reversed about 10 years ago," says Stewart. "Home computers grabbed the interest by putting maths in front of your eyes. Mandelbrot is a classic example: it has virtually no application, but has become an icon."

The past decade has demonstrated that mathematics are really interesting - once you have left school. Advanced mathematical concepts compete for our attention. This year there have been two books written about the 500- year search for a solution to Fermat's last theorem (which proposes that there are no whole numbers that satisfy the equation an + bn = cn if n is greater than 2). One of them has been in the bestseller list for months, which is a just reward for a problem whose eventual solution required years of solitary work by Andrew Wiles, a brilliant Cambridge mathematician.

Another recently published book, The Joy of Pi, contains a million digits of the number, woven around tales of the circular ratio's place in the Universe - from the Great Pyramid of Giza to testing modern computers.

But why has it taken so long for maths to become popular? Why don't we like maths at school yet love it when a professor like Stewart explains how the growth pattern of a sunflower's seed head fits the most precise formula, in which a difference of half a degree either way would mean extinction?

Stewart has a simple explanation. "Our minds aren't geared to doing arithmetic," he says. "There are psychological studies which show that humans aren't digital - our brains don't handle numbers digitally, as computers do. But in the modern world, getting arithmetic correct is important. If your bank manager says to you, 'You've got, oh, about pounds 1,300 in your account', that is not exact enough. Even when you're working things out with a calculator, you have to know what all the operators - plus, minus, divide, multiply - really mean, and what they do.

"We have had to divert the natural talents of the brain to these problems, and make it work more precisely than it was evolved to. I think the world's leading mathematicians are probably very bad at arithmetic. They have a feel for the whole of a problem, like a concert musician who doesn't have to think about how to play each note because he or she is concentrating on the overall performance.

"At school, however, the emphasis is on making sure that you can play each note correctly; that you get the right answer. The trouble is, after three or four hours of doing that, a child is too tired for the really interesting sorts of maths that they could handle. It's a relentless subject. But there's this enormous cultural need to be good at it." In a world of endowment or repayment mortgages, no-fee credit card with APR or low- interest card with annual fee, arithmetical skills may never have been more important.

BUT OUR growing fascination with the natural world means we are interested in maths if we can be shown how it affects the plants and animals we see. Stewart says: "I'm really interested in dynamic pattern forming." The maths of this is very beautiful. Take the apparently humble sunflower whose petal seeds are produced centrally and then move radially outward. The seeds appear to form overlapping spirals. But the sunflower can only grow one seed at a time. So how does it manage this?

The key, Stewart shows, is that each new seed grows at an angle of 137.5 degrees around the stem from the last one. This means that the new seeds do not interfere with old ones, which are needed to help growth. Called the "golden angle", it's the smallest angle which will ensure that growth is optimised. If petal growth began half a degree earlier or later, the petals would grow in "spokes", meaning that older ones would get in the way of new ones, stunting growth while leaving the seed head uncovered.

Quite where in the plant's DNA such instructions reside is still a mystery; but Stewart is fascinated by similar questions, such as the patterns of stripes on tigers and spots on leopards or giraffes. He admits that he has come a long way from the callow youth who received his degree in maths. "In those days I would have thought that Fermat's last theorem was a bit too concrete. Now, I love that sort of stuff."

Which leads to one final question: what use is Fermat's last theorem? Stewart beams. "A physicist did say to me that it could be useful in quantum mechanics, but the real answer is, 'Absolutely none'. Maths on the frontier tends to be five steps removed from things you'd find in your kitchen. Chaos theory was unusual, that was just two steps away. But there's always something that will fascinate you, even if it isn't useful as such."

With that, Professor Stewart returns to the more prosaic magic of extracting rabbits from hats.

Follow the sun: the pattern of seeds on the head of a sunflower is not random, yet only requires one rule to describe. Each seed starts growing at the centre of the circular head, at an angle of 137.5 degrees - the "golden angle" - around from its predecessor. Computer programs can easily show that only this angle guarantees maximum coverage of the head. Varying that starting angle by even half a degree will soon leave the seeds lined up in spokes, exposing the flower's head and limiting its potential growth. It's a pattern that reappears throughout nature, from sea shells to animal populations

Baby blues: at first sight the patterning on giraffes isn't symmetrical: left and right sides don't match. Yet the way their polygonal markings develop is described by "Turing equations". These suggest that the patterning is laid down by chemicals in the womb turned into pigment during gestation. In the very early embryo, consisting of a few cells, those points are evenly distributed, but as it develops some parts grow faster, creating irregular, yet contiguous polygons. The same process happens with zebras: the difference between species' markings can be pinpointed to an eight- hour difference in the time chemicals are deposited in the womb

Mathematical teasers

Some of these problems may take longer than others. A pen, pad of paper, calculator and patience are recommended.

1) Being a lover of superhardboiled eggs, you want to boil one for exactly 20 minutes. You have boiling water but no timepieces apart from two egg- timers - a 17-minute one and a 14-minute one. What do you do?

2) A drawer contains some red and some blue socks. If you pull out two at random, the probability that both are red is exactly 1/2. What is the smallest number of socks the drawer can contain? [Hint: it's a small drawer.]

3) A drawer contains some red socks and an even number of blue socks. If you pull out two socks at random, the probability that both are red is exactly 1/2. What is the smallest number of socks this drawer can contain?

4) Four people share 25 bananas. Mary got the most, Henry the fewest, and Mary got twice as many as Henry. How many did Viviek and Lucy get between them? (No broken ones allowed.)

5) I'm walking across a cricket ground with a bucket when it starts raining. If I run, does the water accumulate in the bucket faster, slower, or at the same rate? (Even running, the bucket stays upright, and there's no wind.)

6) I have some cannonballs in my armoury. I can pile them in a square pyramid - one on top, four underneath, nine below that, and so on. Or I can lay them in a single, filled-in square. So how many cannonballs have I got? How many on the side of the square? And how many layers in the pyramid?

7) I start with 17 cauliflowers and take away all but five. How many do I have?

8) Indiana Jones discovers two sacred stone cubes. Astonishingly, each measures a whole number of centimetres. The difference in their volume is exactly 819 cubic centimetres. What are their sizes?

9) I drop a tennis ball from the top of a 180-metre tower. Whenever it hits the ground, it bounces to exactly 1/10th of its previous height. How far up and down does it travel before finally coming to rest, an infinite time later? (Hint: it's a whole number.)

10) Fermat's last theorem proposes that if a, b and c are whole numbers, and n>2, then the equation an + bn = cn has no solutions. After years of work, Andrew Wiles proved how many solutions there really are. Is it (i) zero (ii) not zero, but a finite number (iii) an infinite number?

Prof Ian Stewart writes the 'Mathematical games' column in 'Scientific American'.

ANSWERS

(1) Turn both egg-timers on end. When the 14-minute timer runs out, put the egg in the boiling water. When the 17-minute timer runs out, three minutes later, turn it over to time the remaining 17 minutes. (2) 4: 3 red, 1 blue. (3) 21: 15 red, 6 blue. (4) 13. The bananas are divided as 8, 7, 6 and 4; Mary and Henry got 12. (5) It makes no difference to the rate: the water has to pass through the same surface area - the bucket's opening. (6) 4,900 - the square of 70, and the sum of the first 24 perfect squares. So the square is 70 per side, and the pyramid has 24 layers. This is the only number that works! (7) Five - you took away the other 12. (8) 8cm and 11cm per side, respectively. (1331 - 512 = 819) (9) 220 metres. It's 180 x (1 + 2/10 +2/100 + 2/1000...) which is an infinite series: 180 x 1.2222... The latter number is the decimal version of 11/9, and 180 x 11/9 = 220. (10) Andrew Wiles proved there are no solutions to Fermat's last theorem.