Cromwell, Peter R.

Polyhedra

Cambridge University Press

Cambridge 1997

ISBN: 978-0-521-55432-9

#geometria

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[i][c] INDICE:

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 S symmetry type_{2n} | |||

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 |

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