##
**Polyhedra.**
*(English)*
Zbl 0888.52012

Cambridge: Cambridge University Press. xiii, 451 p. (1997).

This is a mathematics book and a book about how mathematics is done. It explores how the study of polyhedra has developed the ways of people who have used them and thought about them over the ages, and how their ideas have evolved. The author finds a balance between covering the historical development of the theory surrounding polyhedra and representing a rigourous treatment of the mathematics involved. He places results in context to trace the development of the underlying ideas. Their influences on other subjects and their connections with them is shown both within mathematics and further afield. Only the three-dimensional geometric aspects of the subject are considered. With this book the author will give a help especially for students and such who want to study polyhedra and related subjects.

Throughout the course of the book the meaning of polyhedra is refined and altered at various times. The chapters of this book are a series of related essays, each of which explores a particular theme. They are arranged in approximate chronological order and are large independent units. Many of the results are accessible to non-specialists. The proofs are simply but fully explained. Many illustrations are included. At the end there is a large bibliography. Appendix I compares notions of some symmetry groups, in appendix II there is given a proof of the orbit-stabiliser theorem.

The titles of the 10 chapters of this book only are not sufficient to get a full impression of the content. Therefore, here are given some more remarks: Chapter 1 considers Egyptian, Babylonian, Chinese, and Greek mathematics. In this connection the notions incommensurability, dissection, and exhaustion are explained. Finally Hilbert’s third problem is considered. In Chapter 2 the Platonic solids are described and thus the notion regularity is introduced. Besides, honey combs and Archimedian solids belong to this chapter. Abstraction and existence are further notions which follow from these topics. The next Chapter 3 is devoted to mathematics and astronomy and also to perspectivity and perspective artists. Chapter 4 describes Kepler and his work. In Chapter 5 the author considers angles and their measurement. Then the Euler Formula and the Gauss-Bonnet Theorem are presented. Chapter 6: Equality, regidity and flexibility of polyhedra. Chapter 7: Stars, stellations, and skeletons. Chapter 8: Symmetry, shape, and structure. Chapter 9 is devoted to colouring, especially to the four-colour problem. The last Chapter 10 is dedicated to compound polyhedra, to regularity and symmetry of a polyhedron, to polyhedral metamorphosis, and to colour symmetries.

Throughout the course of the book the meaning of polyhedra is refined and altered at various times. The chapters of this book are a series of related essays, each of which explores a particular theme. They are arranged in approximate chronological order and are large independent units. Many of the results are accessible to non-specialists. The proofs are simply but fully explained. Many illustrations are included. At the end there is a large bibliography. Appendix I compares notions of some symmetry groups, in appendix II there is given a proof of the orbit-stabiliser theorem.

The titles of the 10 chapters of this book only are not sufficient to get a full impression of the content. Therefore, here are given some more remarks: Chapter 1 considers Egyptian, Babylonian, Chinese, and Greek mathematics. In this connection the notions incommensurability, dissection, and exhaustion are explained. Finally Hilbert’s third problem is considered. In Chapter 2 the Platonic solids are described and thus the notion regularity is introduced. Besides, honey combs and Archimedian solids belong to this chapter. Abstraction and existence are further notions which follow from these topics. The next Chapter 3 is devoted to mathematics and astronomy and also to perspectivity and perspective artists. Chapter 4 describes Kepler and his work. In Chapter 5 the author considers angles and their measurement. Then the Euler Formula and the Gauss-Bonnet Theorem are presented. Chapter 6: Equality, regidity and flexibility of polyhedra. Chapter 7: Stars, stellations, and skeletons. Chapter 8: Symmetry, shape, and structure. Chapter 9 is devoted to colouring, especially to the four-colour problem. The last Chapter 10 is dedicated to compound polyhedra, to regularity and symmetry of a polyhedron, to polyhedral metamorphosis, and to colour symmetries.

Reviewer: J.Böhm (Nerkewitz)

### MSC:

52B10 | Three-dimensional polytopes |

52-02 | Research exposition (monographs, survey articles) pertaining to convex and discrete geometry |

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

52-03 | History of convex and discrete geometry |

01-02 | Research exposition (monographs, survey articles) pertaining to history and biography |