Polyhedra

Front Cover
Cambridge University Press, Jul 22, 1999 - Mathematics - 451 pages
Polyhedra have cropped up in many different guises throughout recorded history. In modern times, polyhedra and their symmetries have been cast in a new light by combinatorics an d 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 those more interested in the mathematics itself, will find this unique book fascinating.
 

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Contents

Indivisible Inexpressible and Unavoidable
13
Egyptian geometry
15
Babylonian geometry
19
Chinese geometry
20
A common origin for oriental mathematics
24
Greek mathematics and the discovery of incommensurability
25
The nature of space
29
Democritus dilemma
31
What is a polyhedron?
201
Von Staudts proof
206
Complementary viewpoints
210
The GaussBonnet theorem
211
Equality Rigidity and Flexibility
215
Disputed foundations
216
Stereoisomerism and congruence
221
Cauchys rigidity theorem
224

Liu Hui on the volume of a pyramid
34
Eudoxus method of exhaustion
37
Hilberts third problem
40
Rules and Regularity
47
The mathematical paradigm
54
Primitive objects and unproved theorems
55
The problem of existence
57
Constructing the Platonic solids
62
The discovery of the regular polyhedra
66
What is regularity?
70
Bending the rules
75
Polyhedra with regular faces
82
Decline and Rebirth of Polyhedral Geometry
91
Heron of Alexandria
94
Pappus of Alexandria
95
Platos heritage
96
The decline of geometry
97
The rise of Islam
98
Thabit ibn Qurra
99
AbulWafa
100
Optics
101
Campanus sphere
102
Collecting and spreading the classics
103
The restoration of the Elements
104
A new way of seeing
105
Perspective
107
Early perspective artists
108
Leon Battista Albert
110
Paolo Uccello
111
Polyhedra in woodcrafts
112
Piero della Francesca
114
Luca Pacioli
118
Albrecht Durer
122
Wenzel Jamnitzer
124
Perspective and astronomy
128
Polyhedra revived
132
Fantasy Harmony and Uniformity
135
A mystery unravelled
138
The structure of the universe
144
Fitting things together
145
Rhombic polyhedra
147
The Archimedean solids
152
Star polygons and star polyhedra
164
Semisolid polyhedra
169
Uniform polyhedra
171
Surfaces Solids and Spheres
177
Plane angles solid angles and their measurement
179
Descartes theorem
183
The announcement of Eulers formula
185
The naming of parts
187
Consequences of Eulers formula
189
Eulers proof
193
Legendres proof
194
Cauchys proof
196
Exceptions which prove the rule
198
Cauchys early career
229
Steinitz lemma
231
Rotating rings and flexible frameworks
233
Are all polyhedra rigid?
236
The Connelly sphere
239
Further developments
240
When are polyhedra equal?
243
Stars Stellations and Skeletons
245
Poinsots star polyhedra
247
Poinsots conjecture
252
Cayleys formula
253
Cauchys enumeration of star polyhedra
255
Facestellation
259
Stellations of the icosahedron
263
Bertrands enumeration of star polyhedra
277
Regular skeletons
278
Symmetry Shape and Structure
285
Systems of rotational symmetry
288
How many systems of rotational symmetry are there?
293
Reflection symmetry
296
Prismatic symmetry types
297
Compound symmetry and the S2n symmetry type
301
Cubic symmetry types
304
Icosahedral symmetry types
307
Determining the correct symmetry type
308
Groups of symmetries
310
Crystallography and the development of symmetry
314
Counting Colouring and Computing
323
Colouring the Platonic solids
324
How many colourings are there?
326
A counting theorem
327
Applications of the counting theorem
330
Proper colourings
333
How many colours are necessary?
343
The fourcolour problem
344
What is proof?
350
Combination Transformation and Decoration
355
Are there any regular compounds?
361
Regularity and symmetry
362
Transitivity
363
Polyhedral metamorphosis
369
The space of vertextransitive convex polyhedra
372
Totally transitive polyhedra
377
Symmetrical colourings
390
Colour symmetries
393
Perfect colourings
396
The solution of fifth degree equations
398
Appendix I
402
Appendix II
404
Sources of Quotations
407
Bibliography
412
Name Index
435
Subject Index
439
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