Geometry – Greek for “measuring the world” – is one of the oldest branches of mathematics and concerns shapes and their properties. In Shape: the Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else, Jordan Ellenberg, a mathematician at the University of Wisconsin, argues that what is often considered a head-spinning and irrelevant topic is anything but dull and can in fact shed light on many aspects of modern life. Ellenberg begins – following a foray into Euclid geometry – in topology and considers the deceptively taxing question of how many holes a humble straw has. He then delves into the theory of random walks and Markov chains, including how these concepts can be linked with the stock market or how “tree geometry” can be used to win certain games. Later chapters explore more recent topics such as artificial intelligence and – given that the book was mostly written during the COVID-19 pandemic – Ellenberg gives a stimulating analysis of “geometric processes” and how they can be applied to the spread of diseases. In the final chapter – which is rather taxing for the non-US reader – he examines the US voting system and how maths can tackle gerrymandering. Read more From geckos’ feet to Formula 1: how surface science underpins our world Shape is a hefty volume and feels too detailed in parts, especially on historical aspects where the narrative can sometimes veer off on a tangent from the maths being discussed. Yet it is worth persevering with, not only because Ellenberg is an entertaining guide, but also for the many fascinating mathematical insights that show geometry at its intriguing best. 2021 Allen Lane £20.00hb 480pp