In a 2006 interview for the Guardian newspaper [1], mathematician Marcus du Sautoy said the following: “[Mathematics] has beauty and romance. It’s not a boring place to be, the mathematical world. It’s an extraordinary place; it’s worth spending time there.” There are few places within the mathematical world where beauty is more clearly seen than in proofs. As the saying goes, mathematics is not a court of law; showing that something is true beyond reasonable doubt is simply not good enough. We need proofs. These proofs convince us that something is true, and more often than not these proofs contain beauty and romance.

Paul Erdős, the titan of 20th century mathematics, spoke of “The Book, in which God maintains the perfect proofs for mathematical theorems.” This book by Aigner and Ziegler, now in its fifth edition, seeks to pay homage to the late Paul Erdős by attempting to provide an approximation of “The Book.” Forty-four proofs across five different areas of mathematics are given, with beautiful illustrations by Karl Hofmann. The proofs themselves span from proofs relating to the infinity of primes (chapter 1) to answering the question “How often does one have to shuffle a deck of cards until it is random?” (chapter 30). Each chapter covers a single proof or multiple related proofs, and as one would expect follows a similar pattern: background, theorem, proof, references. Throughout, illustrations and figures are used to support the arguments in the main text; these can greatly help the readability of the proofs, especially for novices like me.

As previously stated, this is the fifth edition; in addition to some minor incremental changes, it adds four new chapters distributed throughout the book, so they may not be immediately obvious. These new chapters contain a proof for the spectral theorem (chapter 7); an answer to the question “Can the Borromean rings be built from three perfect circles?” in chapter 15 (not to give the game away, but the answer is a resounding no); the finite Kakeya problem (chapter 34); and finally a proof of Minc’s conjecture in chapter 37. It is wonderful that the book is not simply being updated but also added to, and I hope this continues in later editions.

So who should read this book? Well, Aigner and Ziegler state in the introduction that having a modest background in undergraduate mathematics should be sufficient. I agree that for the most part many of the proofs will be accessible for such readers, and indeed anyone with undergraduate knowledge of mathematics will almost certainly have come across some of these proofs before. Since the book was written at an accessible level, some proofs that people may expect to find in it have been omitted. This also means that the book can be enjoyed by a wider audience.

Overall, the book is a marvelous project and this new edition provides a good amount of fresh material. I found it particularly useful in solidifying my understanding of what makes a beautiful proof, and any reader with a solid mathematical background will see what Marcus du Sautoy meant when he said that mathematics has beauty and romance and is not a boring place to be.

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