The text under review is a translation and expansion of a classic work on convex polyhedra, first published in Russian in 1950, then translated into German in 1958. It has been updated by V. A. Zalgaller, and fitted with supplements by Yu. A. Volkov and L. A. Shor. It is a testimony to Alexandrov's depth of insight and care in exposition that this monograph is still the basic text in the subject, and that it finally has an English translation (by Dairbekov, Kutateladze, and Sossinsky) more than fifty years after its writing.
What is most compelling about this book is the choice of method — a polyhedron is a geometric object of some simplicity, accessible by ideas from synthetic geometry and elementary topology, and demonstrating anew the power of these methods. The basic question of the book is to establish which data associated to a polyhedron determine it up to Euclidean equivalence. In two dimensions, for example, the conditions a + b > c, a + c > b and b + c > a determine the lengths of the sides of a triangle, which is unique up to isometry of the plane. The first such uniqueness theorem for polyhedra was given by Cauchy in 1813: two closed convex polyhedra composed of the same number of equal similarly situated faces are congruent (via a Euclidean motion of three-space).
The combinatorial data that describe a polyhedron are given by a development. A development is a collection of polygons, together with a prescription for gluing them along their sides. The rules for gluing include the geometric assumptions that sides can be glued only if they have the same lengths, and that a side is glued to at most one other side. Such data also carry topological implications and conditions for convexity follow.
To restore spatial geometry Alexandrov introduced the intrinsic metric of a development. Distances are measured along the faces of the polyhedron and so notions such as geodesics are defined. Viewing a closed, convex polyhedron as a sort of surface, the analogue of the Gauss map is a prescription of face directions. In this setting, Alexandrov develops and proves Minkowski's theorem, which states that two polyhedra with pairwise parallel faces of equal area are, in fact, parallel translates of each other.
The failure of uniqueness for polyhedra with boundary and a given development is the notion of a flexible polyhedron. Alexandrov proved conditions equivalent to the existence of flexible polyhedra from which his rigidity theorems follow.
The original editions of this book included conjectures and problems, many of which were solved in the ensuing decades. Before his death, Alexandrov, with the assistance of Zalgaller, prepared footnotes for this edition bringing it up to date and including an expanded bibliography. The richness and beauty of the mathematics of polyhedra is the main gift of this book. It is bound to influence another fifty years of research on this subject.
John McCleary is Professor of Mathematics at Vassar College.