Maths Moves in Mysterious Ways

The shapes and patterns of Nature have some surprising explanations

Cosmos Science

More than 1,000 years ago — in the ninth century to be precise — Baghdad had the greatest scientific academy in the world. The Abbasid Caliphs founded a “House of Wisdom” that attracted great minds from far and wide, and it is from this period that we have the terms algebra and algorithm. Both come from the same scholar, Abu Ja’far Mohammed ibn Musa Al-Khwarizmi, whose book Kitab al-jabr w’al-muqabala, gave us the term al-jabr — algebra in modern English — and his name, al-Khwarizmi, yielded the modern word algorithm, a method for undertaking a recurring sequence of mathematical steps.

This inspired work did not come from nowhere. Ancient Greek texts were translated into Arabic, one of the first being Euclid’s Elements, the greatest geometry text ever written, and when Europeans in the Middle Ages accessed Greek mathematics and astronomy it was via Arabic translations. For example, Adelard of Bath travelled to Spain to acquire an Arabic version of Euclid, which in 1120 he translated into Latin. A translation from Greek directly into Latin did not appear until the European Renaissance nearly 400 years later. Meanwhile, other Arabic manuscripts had been translated, such as al-Khwarizmi’s text on algebra, by Robert of Chester in 1145.

The early Islamic world delighted in mathematics and its application to astronomy. A mathematical understanding of the universe was not in any way a threat to Islam, nor indeed to Christianity, until the Vatican felt threatened by Galileo’s assertion that the earth was not the centre of the universe. The story of his conflict with the Church and the forced recantation of his conclusions is the subject of Bertolt Brecht’s wonderful play The Life of Galileo. It is also well told in Mario Livio’s recent book Is God a Mathematician? (Simon & Schuster, 2009), which also recounts other applications of mathematics to the natural world.

Certainly, mathematics is the language of physics and astronomy, and disputes by religious fundamentalists about the conclusions of radio-carbon dating or astrophysical evidence of the early universe carry no great weight with anyone else. But the same is not quite true of evolution and biology, where mathematics is seen to play no role, and fundamentalists regard a detailed and well-founded theory as being little more than an opinion, often expressed by ungodly persons who would happily expunge the divine hand from the reality of creation. It turns out, however, that mathematical methods can explain a great deal about the shapes and patterns of nature, and in a series of three books (Shapes, Flows and Branches) published this year (OUP), Philip Ball gives us some very interesting food for thought. His work is based on D’Arcy Wentworth Thompson’s 1917 classic On Growth and Form — now revised in paperback by Dover Publications — which tells us, for example, the reason why honeycombs have hexagonal cells. It is not a “Darwinian” explanation that some bees made square ones, or triangular ones, and could not compete with the “hexagonal” bees, which used less material and were therefore more efficient. D’Arcy Thompson had no patience with that. No, it is a simple mathematical fact that when like-sized coins are packed on a table top, they naturally fall into a hexagonal pattern. Replace the coins by cylindrical cavities, press them together and you have a honeycomb. Simple. Nature takes the natural solution.

Now this may not seem to need mathematics, because it can be determined by experiment, but mathematics can show the inevitable result of such experiments. More generally, mathematics determines patterns, and Ball covers a great many other interesting features of the world, some using very sophisticated applications of mathematical principals. For example the volume on branches describes the angles at which a branch will emerge from a tree stem and how small tree-like structures appear in rocks, looking just like fossils of ancient plants, and forming similar patterns to those of river basins. Branching also accounts for the cracks that form when a material expands and then contracts. For example, the hairline cracks that weave across old paintings — known to art dealers as craquelure — vary according to the materials used and the ambient environment, giving a useful indication of authenticity to an expert.

Ball’s volume on flow discusses not just the flow of water but of solid particles such as sand, from the rippling dunes of the Sahara to the dunes in the very different environment of Mars. The mathematics of flow, often known as fluid dynamics, has its artistic side and the author starts by giving a lovely example of Leonardo da Vinci’s artistry where the flow of water is so beautifully and accurately treated. He later brings in the vexed question of turbulence, well illustrated in some late paintings of van Gogh. In the natural world, Jupiter’s famous red spot is a glorious example that has existed for more than 300 years and is larger than the surface of the earth. Turbulence is very difficult to model mathematically. As Sir Horace Lamb said in 1932 in an address to the British Association for the Advancement of Science, “When I die and go to heaven, there are two matters on which I hope for enlightenment. One is quantum electrodynamics and the other is the turbulent motion of fluids. About the former, I am rather optimistic.”

Though turbulence may be too difficult to model mathematically with present methods, flow is certainly susceptible to calculation, and mathematics has been applied to the flow of motor traffic on roads. It can also be used to model crowds of people moving in relatively confined spaces, which under the worst conditions can lead to something remarkably similar to turbulence. A frightening example of this is the annual pilgrimage to Mecca where masses of people have got into an uncontrolled swirl that has proved fatal. After 300 died in January 2006, the authorities brought in European experts who had studied traffic flow and devised methods to successfully avoid repeating the tragedy. 

The Arab world, which once learned from the Greeks and taught the Europeans, now learns from Europe and America, where mathematical and scientific creativity flourish because of an openness to new ideas.