| | | Preface |
| | | Acknowledgements |
| | | Introduction |
| | | Polyhedra in architecture |
| | | Polyhedra in art |
| | | Polyhedra in ornament |
| | | Polyhedra in nature |
| | | Polyhedra in cartography |
| | | Polyhedra in philosophy and literature |
| | | About this book |
| | | The inclusion of proofs |
| | | Approaches to the book |
| | | Basic concepts |
| | | Making models |
| | 1. | Indivisibile, Inexpressible and Unavoidable |
| | | Castles of eternity |
| | | Egyptian geometry |
| | | Babylonian geometry |
| | | Chinese geometry |
| | | A common origin for oriental mathematics |
| | | Greek mathematics and the discovery of incommensurability |
| | | The nature of space |
| | | Democritus' dilemma |
| | | Liu Hui on the volume of a pyramid |
| | | Eudoxus' method of exhaustion |
| | | Hilbert's third problem |
| | 2. | Rules and Regularity |
| | | The Platonic solids |
| | | The mathematical paradigm |
| | | Abstraction |
| | | Primitive objects and unproved theorems |
| | | The problem of existence |
| | | Constructing the Platonic solids |
| | | The discovery of regular polyhedra |
| | | What is regularity? |
| | | Bending the rules |
| | | The Archimedean solids |
| | | Polyhedra with regular faces |
| | 3. | Decline and Rebirth of Polyhedral Geometry |
| | | The Alexandrians |
| | | Mathematics and astronomy |
| | | Heron of Alexandria |
| | | Pappus of Alexandria |
| | | Plato's heritage |
| | | The decline of geometry |
| | | The rise of Islam |
| | | Thabit ibn Qurra |
| | | Abu'l-Wafa |
| | | Europe rediscovers the classics |
| | | Optics |
| | | Campanus' sphere |
| | | Collecting and spreading the classics |
| | | The restoration of the Elements |
| | | A new way of seeing |
| | | Perspective |
| | | Early perspective artists |
| | | Leon Battista Alberti |
| | | Paolo Uccello |
| | | Polyhedra in wodcrafts |
| | | Piero della Francesca |
| | | Luca Pacioli |
| | | Albrecht Dürer |
| | | Wenzeln Jamnitzer |
| | | Perspective and astronomy |
| | | Polyhedra revived |
| | 4. | Fantasy, Harmony and Uniformity |
| | | A biographical sketch |
| | | A mystery unravelled |
| | | The structure of the universe |
| | | Fitting things together |
| | | Rhombic polyhedra |
| | | The Archhimedean solids |
| | | Star polygons and star polyhedra |
| | | Semisolid polyhedra |
| | | Uniform polyhedra |
| | 5. | Surfaces, Solids and Spheres |
| | | Plane angles, solid angles, and their measurement |
| | | Descartes' theorem |
| | | The announcement of Euler's formula |
| | | The naming of parts |
| | | Consequences of Euler's formula |
| | | Euler's proof |
| | | Legendre's proof |
| | | Cauchy's proof |
| | | Exceptions which prove the rule |
| | | What is a polyhedron? |
| | | Von Staudt'sproof |
| | | Complementary viewpoints |
| | | The Gauss-Bonnet theorem |
| | 6. | Equality, Rigidity and Flexibility |
| | | Disputed foundations |
| | | Stereo-isomerism and congruence |
| | | Cauchy's rigidity theorem |
| | | Cauchy's early career |
| | | Steinitz' lemma |
| | | Rotating rings and flexible frameworks |
| | | Are all polyhedra rigid? |
| | | The Connelly sphere |
| | | Further developments |
| | | When are polyhedra equal? |
| | 7. | Stars, Stellations and Skeletons |
| | | Generalised polygons |
| | | Poinsot's star polyhedra |
| | | Poinsot'sconjecture |
| | | Cayley's formula |
| | | Cauchy's enumeration of star polyhedra |
| | | Face-stellation |
| | | Stellations of icosahedron |
| | | Bertrand's enumeration of star polyhedra |
| | | Regular skeletons |
| | 8. | Symmetry, Shape and Structure |
| | | What do we mean by symmetry? |
| | | Rotation symmetry |
| | | Systems of rotational symmetry |
| | | How many systems of rotational symmetry are there? |
| | | Reflection symmetry |
| | | Prismatic symmetry types |
| | | Compound symmetry and the S2n symmetry type |
| | | Cubic symmetry types |
| | | Icosahedral symmetry types |
| | | Determining the correct symmetry type |
| | | Groups of symmetries |
| | | Crystallography and the development of symmetry |
| | 9. | Counting, Colouring and Computing |
| | | Colouring the Platonic |
| | | How many colourings are there? |
| | | A counting theorem |
| | | Applications of the counting theorem |
| | | Proper colourings |
| | | How many colours are necessary? |
| | | The four-colour problem |
| | | What is proof? |
| | 10. | Combination, Transformation and Decoration |
| | | Making symmetrical compounds |
| | | Symmetry breaking and symmetry completion |
| | | Are there any regular compounds? |
| | | Regularity and symmetry |
| | | Transitivity |
| | | Polyhedral metamorphosis |
| | | The space of vertex-transitive convex polyhedra |
| | | Totally transitive polyhedra |
| | | Symmetrical colourings |
| | | Colour symmetries |
| | | Perfect colourings |
| | | The solution of fifth degree equations |
| | I. | Appendix I |
| | II. | Appendix II |
| | | Sources of Quotations |
| | | Bibliography |
| | | Name Index |
| | | Subject Index |
