Mapmatics - Paulina Rowińska ***

Popular mathematics can be hard to make engaging. Though some topics (such as infinity or zero) can be made interesting in isolation, usually it's best if it can be tied to something more concrete, and what Paulina Rowińska does here is to bring us the story of maps and the the maths behind them. Although Rowińska starts with Mercator and other early projections, it's not really a history of mapping - for example, there is no mention of Roger Bacon's description of using coordinates for mapping - instead the focus is the twin mathematical bases of mapping, geometry and trigonometry before moving onto other maths connections from fractals and operational research to Bayes' theorem.

We start with the nature of a curved world and the compromises that need to be made to translate a 3D surface onto a sheet of paper - compromises that are rarely stated and make a huge difference to the look of the map. This is mostly very engaging, except when it spends too long on geometry and trigonometry. Then there's a dive into fractals, based on Richardson's observations that country border lengths often vary as seen from either side of the border and Mandelbrot's formative 'How long is the coast of Britain?', straying into fractal dimensions. We then move onto the way maps need not be spatial representation - the classic example being the London tube map. Things get even more abstract as we move from maps to graphs (the node and connector type, not charts) and some well known mapping problems like travelling salesman and the four colour theorem. US gerrymandering gets its own chapter, as does Snow's cholera map and other such lifesaving mapping, before finally looking at what can just about be called mapping in terms of identifying the internal structure of the planet.

There are some great stories in here, but for me, unfortunately, once you've got over the genuinely interesting stuff about the difficulties of representing 3D geometry on a 2D map, a lot of the early mathematical basis is, frankly, a bit dull. It's no surprise that geometry and trigonometry figure large (the words do, after all, mean 'earth measuring' and 'triangle measuring'), but I always found them the most tedious aspect of maths. Mostly Rowińska avoids using too many mathematical formulations, but they do creep in quite regularly here. Later on we do get to more interesting mathematical areas such as topology and graph theory, but in these case the reverse happens: the maths isn't given enough depth to really get a grip of it - we might have been better with fewer topics and more detail once past the basics of projection.

In the first section, there's quite a lot about how the Mercator projection makes southern countries look smaller than they are in area, and northern states bigger, which some observers apparently take as a sort of colonial put down. This seems bizarre, as the point of the maps was initially navigation, but also it seems perfectly reasonable that early map makers would have seen things from their own country as a starting point. I presume map makers in the same period from southern countries would have seen things from their own viewpoint too, but this isn't explored.

It was particularly disappointed by the relative lack of illustrations, which I would have thought were essential for a book about maps. There are some, but, for example, when talking about the genuine limitations of Mercator and how other projections allow different types of information to be taken from the map, there are far too few illustrations to show us what those different projections would look like.

I liked what this book is trying to do, but I'm afraid I didn't particularly enjoy reading it.

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Review by Brian Clegg - See all Brian's online articles or subscribe to a weekly email free here