Learning biology by the numbers

IT’S ‘maths week,’ the festival of numbers.

Mathematics may be the language of physics and chemistry but biology seems immune to its charms; no E = mc² equations or neat, straight-line graphs govern the messy world of living things. At a deeper level, arithmetic and geometry still rule the biological roost, as a famous Italian mathematician showed.

Leonardo of Pisa, better known as Fibonacci, was born in 1217. The son of a merchant, he accompanied his father on business trips to the Arab world, where he became acquainted with the number system the local traders used. The old Roman numerals, the Is Vs Xs Ls and Cs, were still used in Europe. The North Africans had more elegant ones, the familiar 0 to 9 characters we use today, and the superiority of which Fibonacci soon recognised. Try multiplying 18 (XVIII) by 7 (VII) using Roman numerals. The stroke of genius in the Arabian system was a symbol for absence, the 0. This allowed numbers to be stacked in 10s, 100s, 1000s and manipulated accordingly.

We are so used to these symbols that we forget the zero is an extraordinary invention. The notion that you can treat nought as just another number must be one of the greatest intellectual breakthroughs of all time. Credit for the discovery goes to an anonymous Indian of the ninth century. The zero first appeared on a temple inscription dating from that time. Our numbers are frequently referred to as ‘Arabic’ but it’s more correct to call them ‘Hindu-Arabic’.

When Fibonacci introduced the system to his native Italy, feathers were ruffled. Accountants who made their living doing calculations with the cumbersome old system were out of a job. Tempers flared and the use of Hindu-Arabic numerals was outlawed for a time in Florence, then the banking capital of Europe.

Fibonacci is also remembered for a sequence of numbers which bear his name. He developed it to solve a problem about the growth of rabbit populations. It’s easier to explain when applied to bees.

You and I have two parents, four grandparents, eight great-grandparents, etc, but things are different for male bees.

Worker bees, all females, come from fertilised eggs. Drones develop from unfertilised ones. While a worker has two parents, a drone has only one, its mother. He has just two grandparents. At the great-grandparent level, there are three progenitors, the parents of his mother and the single parent of her father.

He has five great-great grandparents and so on. The sequence of forebears is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 etc. The essential feature of the series is that each number is the sum of the two previous ones.

But the Fibonacci sequence describes more than the ancestry of bees and rabbits; it underlies a host of natural processes. The numbers of petals on flowers often follow its pattern.

Science teacher Terry Flanagan tells me that lilies have three petals, buttercups have five, delphiniums eight, marigolds and ragwort 13, asters 21 and daises 34, 55 or 89.

All are Fibonacci numbers. Similar patterns are found among leaves and grasses, although explaining why these patterns arise is more difficult than for the family tree of bees.

But there’s a deeper mystery here. Divide each number, in turn, by the one preceding it and you get a sequence which converges to 1.618. A precise figure can’t be given, because, like ‘pie’, this is an irrational number, one with decimals recurring to infinity. Extraordinarily, the ratio is the famous ‘golden mean’ beloved of architects and artists, ancient and modern.

It’s the relationship of the height to the width of the Parthenon facade, a harmony which seems to resonate deeply within us. That it has such appeal should be no surprise; echoes of the golden mean are found in the proportions of limbs and the branching of veins and nerves.

Some have suggested that the golden mean can be found even in some of the DNA configurations of the human genome.