Alex Bellos in Numberland

Last week I went to a book launch, put on by the publishers Bloomsbury (whom I have heard of – they publish a couple of things I have mentioned here, Riddley Walker and Logicomix) in a bookshop in the back end of Notting Hill.

They were launching two books by two men who had been at school together. The one that had led to my invitation was Alex’s Adventures in Numberland, by Alex Bellos. (We had been in touch about one of the pictures in the book, an aerial photo of an experiment at Rothamsted laid out as a complete Latin square, designed by Rosemary Bailey, which I had used in talks and a paper about Sudoku.) The author told me that in the USA it has the title Here’s Looking at Euclid, but market research suggested that that lovely title wouldn’t work in Britain; a sad reflection on the low status of geometry in our school curricula.

At a book launch, you don’t have to buy the book, but it is on sale if the impulse strikes you. I bought a copy, but resisted the temptation to have the author sign it.

The first problem: is it a book of recreational mathematics, to be filed with Martin Gardner and Flatland and such like? The first chapter convinced me that it wasn’t, but left me wondering whether it had to be filed with Georges Ifrah, Stanislaus Dehaene and Brian Butterworth. But it’s not that either. It really is mathematics (real mathematics) for the lay person.

I hope this doesn’t sound patronising, but I think Alex has done an excellent job with the book. He has a foot in both camps, having a degree in mathematics and philosophy from Oxford (but too young for me to have taught him – how old I feel!) and works as a journalist. He endeared himself to me early on by a story showing that almost all journalists are innumerate. (Ben Goldacre’s Bad Science, which I read recently, suggests that they actually have much worse faults.) But he manages not to talk down to his audience, while including a number of things that I didn’t know, such as Poincaré measuring the weight of his daily baguette.

Some other goodies from the book: how Yorkshire shepherds count (his list agrees with the song by Jake Thackray); why business card origami is abhorrent to the Japanese; a mnemonic for the digits of π which is a remarkable modernist pastiche of Poe’s The Raven; and why we most commonly use x as the name of a variable. We meet fanatics, cranks, anthropologists and holy men as well as a few mathematicians (including Persi Diaconis, Gordon Royle and Neil Sloane).

So: buy it, and read it!

Here is an example of how he manages to avoid complication and remain mathematically honest. He has just explained the construction where you take a rectangle whose sides are in the golden ratio and dissect it into squares, and then draw quarter-circles in the squares. He says:

The curve is an approximation of a logarithmic spiral.

A true logarithmic spiral will pass through the same corners of the same squares, yet it will wind itself smoothly, unlike the curve in the diagram, which will have small jumps in curvature where the quarter-circle sections meet.

He goes on to tell that Jakob Bernoulli “asked to have [a logarithmic spiral] engraved on his tombstone, but the sculptor engraved an Archimedean spiral by mistake.”

There were only two places where I caught him out.

• He describes the sliding-block “Fifteen Puzzle” and explains how mathematicians proved that only half of the possible starting positions can lead to a solution, and then goes on to say, “The Fifteen puzzle remains the only international craze in which the puzzle does not always have a solution”. Then he goes on to the Rubik cube. With this puzzle, only one-twelfth of the possible starting positions lead to a solution; if you take it apart (as most people who own one have probably done) and re-assemble it, the probability of producing an impossible puzzle is eleven-twelfths. If your Rubik cube is one of those with pictures on the faces of the small cubes, things are even worse: only one-forty-eighth of the starting positions are solvable.
• In the penultimate paragraph of the book, describing infinite cardinals, he makes two claims: first, that the number of curves in the plane is larger than c, the cardinality of the continuum (this is false if curves must be continuous; there are only c of them); and second, that nobody has been able to come up with a larger set (Cantor proved that the set of subsets of any set is larger than the original set – perhaps Alex meant a larger “naturally-occurring” set).

And here is a point where I think he missed a trick. He explains (very clearly) the notion of “regression to the mean”, according to which children of very tall parents tend to be shorter than their parents, and children of very short parents tend to be taller than their parents. This does not mean, as it might seem to, that the human race will end up of uniform height; for it is also true that parents of very tall children tend to be shorter than their children. There is a very simple geometric derivation.

It has to do with correlation, as Alex says. Imagine that we plot correlated variables such as heights of parents and children. If the correlation were an exact relation, the points will lie on a line. In general, if there is some correlation, we would expect them to be distributed in an elliptical cloud with its major axis at an acute angle to the diagonal, as shown in the figure. In the case of parents’ and children’s heights, it will be at 45 degrees to the axis. This is the red ellipse in the figure.

Selecting parents of given height and measuring the average height of their children corresponds to taking vertical sections of the ellipse, working out their mid-points, and plotting these; we get a line whose slope is less than that of the ellipse. This is shown in green in the first figure. Similarly, selecting children of given height and measuring their parents corresponds to taking horizontal sections; the line we get will have slope greater than that of the ellipse. This is shown in blue in the second figure.

So regression to the mean is explained by simple geometry! (In my diagram, the ellipse is twice as long as it is wide; the slopes of the regression lines are 3/5 and 5/3 respectively.)