*Ed Rochead sent us this review of Aperiodipal Colin Beveridge’s latest pop maths book.*

This book is written to answer the question ‘when would you ever use maths in everyday life?’ It therefore focuses on applied maths, across a surprisingly wide breadth of applications. The book is organised into sections such as ‘the human world’, ‘the natural world’, ‘getting around’ and ‘the everyday’. Within each section there are approximately ten topics, for which the maths behind some facet of ‘everyday life’ is explained, with cheerful colour graphics and not shying away from using an equation where necessary.

As I obtained the book whilst on a visit to Bletchley Park, my attention is first drawn to the ‘technology’ section and the topic of Bitcoin, an application of cryptography. Bitcoin is a so-called *cryptocurrency* and world payment system, albeit one without a central bank, or similar, to regulate and guarantee it. To my surprise Bitcoin was only coined (sorry!) in 2008; the book describes it and the ‘blockchain’, along with the math behind Bitcoin. This rather pleasingly uses a graph that an A-Level student could visualise, \( y^2 = x^3 + 7\), alongside a delightfully named ‘hash function’, and requires ideas such as a line crossing a curve – again familiar to an A-Level, and probably a GCSE, mathematician – and introduces modular arithmetic, which probably is not. Bitcoin is a great example of a topic talked about in mainstream media which is totally dependent on mathematics.

As a Brit who also loves Barcelona, my attention was also drawn to the section on ‘the everyday’ and the topic of architecture. In this section, London’s St Mary Axe and Barcelona’s Sagrada Familia are used as examples of mathematics influencing architecture. The book uses St Mary Axe to illustrate some of the ways mathematics can make a structure cost-effective and reduce turbulence. The Sagrada Familia is used to introduce catenary and parabolic curves, using the hyperbolic cosine function and a quadratic equation respectively.

In the ‘human world’ section there is a fascinating topic called ‘cheating’ which looks at the mathematics behind detecting plagiarism, introducing techniques such as principal component analysis and stylometry. Stylometry is illustrated by comparing frequency analysis of words in both Shakespeare and Marlowe’s works. Later in the topic, we’re introduced to something that astounded me, Benford’s Law, which states that in a list of measurements in which the largest is at least 100 times bigger than the smallest (the size of the lakes in Michigan are used in the book), the first digit of the components of the list will be distributed according to the formula

\[\Pr(n) \sim \log\left(\frac{n+1}{n}\right) \]

This leads one to expect approximately 30% of such a list to begin with a 1, and only approximately 5% with a 9, which is contrary to the even spread that one imagines. Benford’s law has been used in forensic analysis, and one can only imagine its utility in detecting fraud and in areas such as ‘big data’. This law, astonishingly, is independent of the unit of measurement used, and the insight in the book leads one to find out more.

Throughout this book the illustrations are clear and are used to explain a range of applications of mathematics. Despite not requiring the reader to be aware of, and still less understand, the mathematical concepts described, I am confident that even knowledgeable mathematicians will learn something, and those who teach, tutor or merely communicate mathematics will find it invaluable in answering the question ‘when would you ever use maths in everyday life?’

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