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.