Polyhedra

Description

Polyhedra have cropped up in many different guises throughout recorded history. Ancient manuscripts from Egypt and China relate ideas concerning the calculation of the volumes of polyhedra, while the Greek tradition of geometry gave us the construction of the regular polyhedra or Platonic solids. In modern times, polyhedra and their symmetries have been cast in a new light by combinatorics and group theory. This book comprehensively documents the many and varied ways that polyhedra have come to the fore throughout the development of mathematics. The author strikes a balance between covering the historical development of the theory surrounding polyhedra, and presenting a rigorous treatment of the mathematics involved. It is attractively illustrated with dozens of diagrams to illustrate ideas that might otherwise prove difficult to grasp. Historians of mathematics as well as to those more interested in the mathematics itself, will find this unique book fascinating. Front Page Full Title Page Library of Congress Cataloguing in Publication data Contents 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 Indivisible, 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 the regular polyhedra What is regularity? Bending the rules The Archimedean solids Polyhedra with regular faces Figure 2.29 The families of convex 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'I-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 woodcrafts Piero della Francesca Luca Pacioli Albrecht Durer Wenzel 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 Archimedean 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's proof 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's conjecture Cayley's formula Cauchy's enumeration of star polyhedra Face-stellation Stellations of the 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 Figure 8 .28. A decision tree to determine the symmetry type of a polyhedron Groups of symmetries Crystallography and the development of symmetry 9 Counting, Colouring and Computing Colouring the Platonic solids 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 Appendix I Appendix II Sources of Quotations Introduction Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Bibliography Introduction Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Name Index Subject Index