*From Christoffel Words to Markoff Numbers* presents the two primary approaches to Markov Theory in one text (Markoff is the French spelling). As Cassels did in his 1949 book, Retenauer presents Markoff theory both in its real approximation form and as minimal values of quadratic forms. Another recent book by Aigner,

*Markov's Theorem and 100 Years of the Uniqueness Conjecture,* covers the former but not the latter, though adding a number of results on the Frobenius uniqueness conjecture not seen in this text.

The first half of the book discusses these dual aspects of the theory, highlighting the connection through Christoffel words as defined geometrically in a two-dimensional vector space over a free monoid on two words \(\{a, b\}\). Essentially a lower Christoffel word of slope \(q/p\) for integers \(p>q>0\) is formed by considering a line segment of the same slope and a lattice path below the line segment such that the pair form a closed polygon containing no integer lattice points. An upper Christoffel word is defined in the same way for the path above the line segment, which is the reversal of the lower Christoffel word of the corresponding slope. The path linking this idea to both aspects of Markov’s theory is intriguing and makes the book an interesting read, winding through a number of mathematical fields along the way including group theory, number theory, and linear algebra.

The author notes that this link has been known since Frobenius (1913) but its relative absence in the recent literature motivated the writing of the book. The decision to present these in the form of sequences of real numbers rather than bi-infinite sequences makes this text more digestible for a novice than most monographs, though the mathematical background required would still preclude most undergraduate readers. The examples in the penultimate section of this part are particularly illuminating. There is also a nice history section at the end of this part which gives some context to the results presented, augmented by a lengthy further reading list in the bibliography for the book as a whole.

The second half of the book is devoted to more advanced work on Christoffel words. Included here is a tour through more recent developments on various related topics, including Lyndon words, palindromes, and Sturmian morphisms. In the last chapter, there is an interesting application of Christoffel words to music, concluding with a number of further reading references.

This book is a well-written introduction to a fascinating subtopic in mathematics that should be accessible to a graduate student or a research mathematician. Most subsections are a few pages in length, easily digestible in a brief time. The proofs are concise, but with enough detail to satisfy the reader.

Jonathan Bayless is an Associate Professor and the Chair of Mathematics at Husson University. His research interests lie mostly in analytic and combinatorial number theory.