BUCHBESPRECHUNG – BOOK REVIEW https://doi.org/10.1007/s00591-023-00345-2 Mathematische Semesterberichte Colin Adams: The Tiling Book: An Introduction to the Mathematical Theory of Tilings American Mathematical Society, Providence, RI 2022, 298 Seiten, ISBN 978-1-4704-6897-2/hbk; 978-1-4704-7101-9/ebbok Dirk Frettlöh The mathematical theory of tilings has three ingredients that make it attractive to a wide readership: Many difficult problems are easy to formulate. Many mathematical subfields meet here, such as algebra, topology, discrete geometry, and computability theory. Practically no other mathematical topic can be illustrated so beautifully. This book offers all these ingredients and combines them into a well-rounded whole. The standard work on the mathematical theory of tilings in the Euclidean plane is Tilings and Patterns by Branko Grünbaum and G.C. Shephard, published in 1987. It’s goal was to be encyclopedic; it should contain everything that was known at that time about the mathematics of tilings of the Euclidean plane. The present book doesn’t want to be encyclopedic. It is a “ best of” Tilings and Patterns , but in color (lots of color!). In addition to tilings of the Euclidean plane, Colin Adams also addresses (very briefly in each case) tilings in other geometries in his book: spherical, hyperbolic, three-dimensional Euclidean and non-Euclidean manifolds. As in Tilings and Patterns , there is also a chapter on aperiodic tilings, of which the most prominent representatives are probably the Penrose Tilings. Here there have been far-reaching discoveries and results since Tilings and Patterns , but only a few of these have made it into this book. In fact, this book contains few topics that are not already in Grünbaum and Shephard's book. Among these few are: the complete list of convex pentagons that tile the plane; the Kari-Culik Wang tiles; the Taylor-Socolar prototile; records on Heesch numbers; and Conway's orbifold notation. The merit of this book, therefore, lies not in its depth or novelty, but in its clear and accessible treatment of the subject and its imaginative arrangement. In a zeroth chapter, all the basic mathematical concepts needed (such as countability, groups, homeomorphism, isometry, linear mapping) are explained simply and clearly, yet precisely. In later chapters, all concepts and results are explained with playful ease, yet practically everything is also neatly proved. The author's creative streak shines through again and again in the many color illustrations. Those who know other books by Colin Adams know what is meant. This makes this book a valuable and fruitful read, especially for the non-mathematically educated reader - if the fact that the book is written in English does not deter him. The author does not shy away from demanding and long proofs in some places - not without explicitly warning against them in the preface. Among them are, for example, the Extension Theorem, the Periodicity Theorem or the proofs of the aperiodicity of the Robinson prototiles or the Taylor-Socolar prototile. For me, the clarity and readability of these proofs are the remarkable contributions in this book. However, I am sure everyone sees this differently. Overall, I see four audiences to whom I can recommend the book: 1) Interested laymen will see here how mathematical de fi nitions, derivations, and proofs are developed starting from intuitive concepts. 2) Teachers and instructors who want to conduct student projects, in the context of a student academy or project weeks or the like, will fi nd more than ample material here, from intermediate to university level. There are a great many interesting exercises as well as suggestions for projects, many in the running text and some collected separately in an appendix. Both the assignments and the projects range from challenging puzzles to open- ended research problems. 3) University instructors, for whom the book can serve as a template for a one-semester lecture. It appears that the book was designed from such a lecture. However, there is considerably more material than can be covered in one semester, so a selection must be made. Suggestions for doing so can be found in the preface. 4) Mathematicians who do not want to miss out on mathematical content while on vacation, but are looking for a light, colorful, and interesting read. Beware, the theory of tilings is much further developed than described in this book. However, the open problems are indeed all open to my knowledge, except the one on page 193.