## PolyhedraPolyhedra 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 |

435 | |

439 | |

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### Common terms and phrases

angle sum antiprism Archimedean solids axes axis base Cauchy centre Chapter congruent constructed contains convex polyhedron crystals cub-octahedron cube definition deltahedron described dihedral angles dissection edges Elements equal equations equilateral Euclid Euler's formula example face-planes face-transitive flexible polyhedron four geometry Greek H. S. M. Coxeter hexagonal icosahedral icosahedron inscribed interior angles Kepler kernel kind labelled lemma mathematicians mathematics mirror plane number of colourings number of faces number of sides number of vertices objects octahedron Pacioli pattern pentagon pentagram perspective plane angles Platonic solids Poinsot prism problem produce proof properly coloured properties pyramid regular polygons regular polyhedra regular solids result rhomb-cub-octahedron rhombic right angles rotational symmetry shown in Figure solid angle space sphere spherical polygon square star polygons star polyhedra structure surface symmetry group symmetry operation symmetry type tetrahedron theorem triangles triangular faces truncated uncoloured vertex figures vertex-transitive volume yang-ma