Paulina Rowi艅ska (Picador (UK); Belknap Press (US, out 17 September))
Mathematics helps us understand the world, and in Paulina Rowi艅ska鈥檚 Mapmatics: How we navigate the world through numbers, the focus is on its indispensability in the making of visual representations of that world 鈥� maps.
As a mathematician, I already knew some of the theory, but it was a pleasant surprise to find this book full of fascinating examples and facts that I didn鈥檛 know, including real-world implications of ideas that might be assumed to lie squarely in the realm of the theoretical. Things like it being impossible to determine the exact length of a coastline because it keeps growing with the increasing precision of your measurements (or as Rowi艅ska puts it, if an ant and an elephant agree to meet for a drink 300 miles along the coast, they will almost certainly not end up in the same pub).
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Like most mathematicians, I was aware of this curious fractal quality of coastlines, but what I didn鈥檛 know is that legislators have come up against this as a genuine problem, in managing treaties that specify distance from the coast.
The book opens by looking at the challenge presented by Carl Friedrich Gauss鈥檚 Theorema Egregium (or Remarkable Theorem), which Rowi艅ska explains through the medium of pizza. This wonderful piece of maths gives us a kind of Heisenberg鈥檚 uncertainty principle of maps: a flat map of the spherical(ish) Earth can represent areas accurately, or distances accurately, but not both.
Maths can tell us what is possible and help us to achieve the goals of our map. There are many excellent illustrations in the book, so it may be churlish to wish for more, but it would have been helpful to see the Gall-Peters projection, which accurately shows the relative areas of continents, but distorts their shapes, alongside the Mercator projection, which does the opposite.
Rowi艅ska does a good job with her choice of topics, which provide a gateway to many intriguing ideas and offer several entertaining diversions. When talking about distance, for example, there is a discussion of Hamming codes, which are designed to detect errors when messages are transmitted across noisy channels. This doesn鈥檛 have much to do with maps, but I was willing to go along for the ride.
There were one or two curious omissions, however. Given the book鈥檚 cover features contour lines, it is strange they aren鈥檛 discussed, especially as the concept of isolines more generally is so powerful. I was also surprised that, though the book is strong on the inherent biases wrapped up in the historical choice of map design, there wasn鈥檛 much discussion of the longitude problem or why 0掳 longitude ended up running through Greenwich, London.
Mapmatics is entertaining throughout, with illuminating examples of a huge range of maps and their uses. On top of navigation, it covers how mathematical analysis of maps showing incidence of disease or crime has been used to stop outbreaks and catch criminals, as well as the uses and abuses of maps for political ends. A bureaucratic decision over boundaries can affect everything from how much funding your child鈥檚 school receives to who wins the next election. Mathematics can help detect bias in such decisions.
The chapters are loosely grouped around themes like scale and connectedness. Each is full of interesting applications of maths to related challenges, like the use of space-filling curves to make food deliveries more efficient. The writing style is appealing 鈥� playful but with a serious message 鈥� and Rowi艅ska clearly has a deep knowledge, both of maps and the maths behind them. An engaging look at a fascinating subject.
Sarah Hart is the professor of geometry at Gresham College in London
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