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.
Author(s): Peter R. Cromwell
Publisher: Cambridge University Press
Year: 1997
Language: English
Commentary: Reupped, Bookmarked, Added Metadata, Numbered Pages, Deskewed, OCR (clearscan) , Smaller file size, ~ 50% Less
Pages: 451
City: Cambridge
Tags: Mathematics, Geometry, General
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
