*Regular Polytopes: 3rd edn*. 1973*M. C. Escher: Art and Science*. 1986, with M. Emmer, Roger Penrose, M. I. Teuber*The Fifty-Nine Icosahedra: 3rd edn*. 1999, with Patrick Du Val, H. T. Flather, John F. Petrie

Polytopes are geometrical figures bounded by portions of lines, planes or hyperplanes.
In plane (two dimensional) geometry they are known as polygons
and comprise such figures as triangles, squares, pentagons, etc.
In solid (three dimensional) geometry they are known as polyhedra
and include such figures as tetrahedra (a type of pyramid), cubes, icosahedra and many more;
the possibilities, in fact, are infinite!
H. S. M. Coxeter’s book is the foremost book available on regular polyhedra,
incorporating not only the ancient Greek work on the subject,
but also the vast amount of information that has been accumulated on these since,
especially in the last hundred years.
The author, professor of Mathematics, University of Toronto,
has contributed much valuable work himself on polytopes and is a well known authority on them.

Professor Coxeter begins with the fundamental concepts of plane and solid geometry and then moves on to multi-dimensionality. Among the many subjects covered are Euler’s formula, rotation groups, star-polyhedra, truncation, forms, vectors, coordinates, kaleidoscopes, Petrie polygons, sections and projections, and star-polytopes. Each chapter ends with a historical summary showing when and how the information contained therein was discovered. Numerous figures and examples and the author’s lucid explanations also help to make the text readily comprehensible.

Although the study of polytopes does have some practical applications to mineralogy, architecture, linear programming and other areas, most people enjoy contemplating these figures simply because their symmetrical shapes have an aesthetic appeal. But whatever the reasons, anyone with an elementary knowledge of geometry and trigonometry will find this one of the best source books available on this fascinating study.

Unabridged, corrected republication of the second (1963) edition.