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 them 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.

Table of Contents:
I. POLYGONS AND POLYHEDRA 1*1 Regular polygons 1*2 Polyhedra 1*3 The five Platonic Solids 1*4 Graphs and maps 1*5 "A voyage round the world" 1*6 Euler's Formula 1*7 Regular maps 1*8 Configurations 1*9 Historical remarks II. REGULAR AND QUASI-REGULAR SOLIDS 2*1 Regular polyhedra 2*2 Reciprocation 2*3 Quasi-regular polyhedra 2*4 Radii and angles 2*5 Descartes' Formula 2*6 Petrie polygons 2*7 The rhombic dodecahedron and triacontahedron 2*8 Zonohedra 2*9 Historical remarks III. ROTATION GROUPS 3*1 Congruent transformations 3*2 Transformations in general 3*3 Groups 3*4 Symmetry opperations 3*5 The polyhedral groups 3*6 The five regular compounds 3*7 Coordinates for the vertices of the regular and quasi-regular solids 3*8 The complete enumeration of finite rotation groups 3*9 Historical remarks IV. TESSELLATIONS AND HONEYCOMBS 4*1 The three regular tessellations 4*2 The quasi-regular and rhombic tessellations 4*3 Rotation groups in two dimensions 4*4 Coordinates for the vertices 4*5 Lines of symmetry 4*6 Space filled with cubes 4*7 Other honeycombs 4*8 Proportional numbers of elements 4*9 Historical remarks V. THE KALEIDOSCOPE 5*1 "Reflections in one or two planes, or lines, or points" 5*2 Reflections in three or four lines 5*3 The fundamental region and generating relations 5*4 Reflections in three concurrent planes 5*5 "Reflections in four, five, or six planes" 5*6 Representation by graphs 5*7 Wythoff's construction 5*8 Pappus's observation concerning reciprocal regular polyhedra 5*9 The Petrie polygon and central symmetry 5*x Historical remarks VI. STAR-POLYHEDRA 6*1 Star-polygons 6*2 Stellating the Platonic solids 6*3 Faceting the Platonic solids 6*4 The general regular polyhedron 6*5 A digression on Riemann surfaces 6*6 Ismorphism 6*7 Are there only nine regular polyhedra? 6*8 Scwarz's triangles 6*9 Historical remarks VII. ORDINARY POLYTOPES IN HIGHER SPACE 7*1 Dimensional analogy 7*2 "Pyramids, dipyramids, and prisms" 7*3 The general sphere 7*4 Polytopes and honeycombs 7*5 Regularity 7*6 The symmetry group of the general regular polytope 7*7 Schafli's criterion 7*8 The enumeration of possible regular figures 7*9 The characteristic simplex 7*10 Historical remarks VIII. TRUNCATION 8*1 The simple truncations of the genral regular polytope 8*2 "Cesaro's construction for {3, 4, 3}" 8*3 Coherent indexing 8*4 "The snub {3, 4, 3}" 8*5 "Gosset's construction for {3, 3, 5}" 8*6 "Partial truncation, or alternation" 8*7 Cartesian coordinates 8*8 Metrical properties 8*9 Historical remarks IX. POINCARE'S PROOF OF EULER'S FORMULA 9*1 Euler's Formula as generalized by Schlafli 9*2 Incidence matrices 9*3 The algebra of k-chains 9*4 Linear dependence and rank 9*5 The k-circuits 9*6 The bounding k-circuits 9*7 The condition for simple-connectivity 9*8 The analogous formula for a honeycomb 9*9 Polytopes which do not satisfy Euler's Formula X. "FORMS, VECTORS, AND COORDINATES" 10*1 Real quadratic forms 10*2 Forms with non-positive product terms 10*3 A criterion for semidefiniteness 10*4 Covariant and contravariant bases for a vector space 10*5 Affine coordinates and reciprocal lattices 10*6 The general reflection 10*7 Normal coordinates 10*8 The simplex determined by n + 1 dependent vectors 10*9 Historical remarks XI. THE GENERALIZED KALEIDOSCOPE 11*1 Discrete groups generated by reflectins 11*2 Proof that the fundamental region is a simplex 11*3 Representation by graphs 11*4 "Semidefinite forms, Euclidean simplexes, and infinite groups" 11*5 "Definite forms, spherical simplexes, and finite groups" 11*6 Wythoff's construction 11*7 Regular figures and their truncations 11*8 "Gosset's figures in six, seven, and eight dimensions" 11*9 Weyl's formula for the order of the largest finite subgroup of an infinite discrete group generated by reflections 11*x Historical remarks XII. THE GENERALIZED PETRIE POLYGON 12*1 Orthogonal transformations 12*2 Congruent transformations 12*3 The product of n reflections 12*4 "The Petrie polygon of {p, q, ... , w}" 12*5 The central inversion 12*6 The number of reflections 12*7 A necklace of tetrahedral beads 12*8 A rational expression for h/g in four dimensions 12*9 Historical remarks XIII. SECTIONS AND PROJECTIONS 13*1 The principal sections of the regular polytopes 13*2 Orthogonal projection onto a hyperplane 13*3 "Plane projections an,ssn,?n" 13*4 New coordinates for an and ssn 13*5 "The dodecagonal projection of {3, 4, 3}" 13*6 "The triacontagonal projection of {3, 3, 5}" 13*7 Eutactic stars 13*8 Shadows of measure polytopes 13*9 Historical remarks XIV. STAR-POLYTOPES 14*1 The notion of a star-polytope 14*2 "Stellating {5, 3, 3}" 14*3 Systematic faceting 14*4 The general regular polytope in four dimensions 14*5 A trigonometrical lemma 14*6 Van Oss's criterion 14*7 The Petrie polygon criterion 14*8 Computation of density 14*9 Complete enumeration of regular star-polytopes and honeycombs 14*x Historical remarks Epilogue Definitions of symbols Table I: Regular polytopes Table II: Regular honeycombs Table III: Schwarz's triangles Table IV: Fundamental regions for irreducible groups generated by reflections Table V: The distribution of vertices of four-dimensional polytopes in parallel solid sections Table VI: The derivation of four-dimensional star-polytopes and compounds by faceting the convex regular polytopes Table VII: Regular compunds in four dimensions Table VIII: The number of regular polytopes and honeycombs Bibliography Index