Circles, Spheres and Spherical Geometry by Hiroshi Maehara and Horst Martini surveys a wide array of results about its titular topics. Beginning with key results about inversion and stereographic projection, the narrative proceeds to results on circles in the plane (e.g., Steiner cycles, the Koebe-Adreev-Thurston theorem), spherical geometry (geodesics, trigonometry, properties of polygons), spherical configurations (the problem of thirteen balls),  geometric probability (random polygons, intersection graphs of spherical caps), higher-dimensional spheres (Moser’s Paradox, volumes), and assorted generalizations (Casey’s Theorem, Cayley-Menger Matrices).

The coverage is an eclectic survey of results about spheres, but it works because the techniques require minimal background to develop, and the story is held together with thoughtful and extensive notes in every chapter that fill in historical and technical details, suggest digressions and references, and situate the content in the context of contemporary research. There are 842 references, and the notes are indexed along with the main text. Each chapter includes several exercises of varying difficulty with selected solutions.

The book is written for a graduate audience although there are no specific graduate-level content prerequisites. Proofs are clear and concise, aided by a modest but sufficient number of thoughtful black-and-white illustrations. The coverage is broad, and the techniques employed vary considerably, so even the recommended prerequisites of linear algebra, advanced calculus, and probability theory are only necessary for a few isolated chapters. The topics are mostly self-contained, so the reader is free to skip around.

One gets a broad sense of what kinds of questions we can ask about spheres in general and how the results and tools vary as we refine them. For example, there is a proof that at most thirteen unit spheres can be simultaneously tangent to a central sphere. The approach is to reduce the problem to a question in spherical geometry by looking at the points of tangency of the central sphere. This all comes after extended analysis of tangent configurations of planar circles, which are far less rigid. This sub-story nicely showcases the interplay of planar, spherical, and three-dimensional geometries, which have different techniques and are infrequently considered in much depth in the same book. But we also see results that do generalize, such as how the Cayley-Menger matrix can be used to generalize Ptolemy’s Theorem classifying cyclic quadrilaterals both to a stronger planar result and to higher dimensions. These results can be found in other books, but this development highlights the mathematical and historical connections among them.

The book could be used on its own for an innovative graduate course or seminar, or as a supplement to a variety of geometry and geometry-adjacent courses. Outside of the classroom, its broad and unique coverage of classical and modern results about some of the core objects in geometry, along with plenty of direction on how to pursue them further, will interest emerging and practiced geometers alike.

Bill Wood is a professor of mathematics at The University of Northern Iowa.