Polytopes and SymmetryConvex polytopes are the analogues in space of any dimension of convex plane polygons and of convex polyhedra in ordinary space. This book describes a fresh approach to the classification of these objects according to their symmetry properties, based on ideas of topology and transformation group theory. Although there is considerable agreement with traditional treatments, a number of new concepts emerge that present classical ideas in a quite new way. For example, the family of regular convex polytopes is extended to the family of 'perfect polytopes'. Thus the familiar set of five Platonic polyhedra is replaced by the less familiar set of nine perfect polyhedra. Among the many unsolved problems that arise, that of finding all perfect polytopes, and more generally all perfect convex bodies, is perhaps the most attractive. This book will be of value to specialists and graduate students in pure mathematics, especially those studying symmetry theory, convex bodies, and polytopes. |
Contents
7 Symmetry groups | 54 |
8 Deficiency and perfection | 55 |
9 The rectangular sum | 56 |
Polygons | 58 |
2 Possible symmetry groups | 60 |
3 Symmetry types | 61 |
4 Deficiency | 62 |
5 Triangles | 64 |
3 Vertexregular polyhedra | 79 |
4 The nonprismatic groups | 84 |
5 The prismatic groups | 93 |
6 Faceregular polyhedra | 96 |
7 Edgeregular polyhedra | 97 |
8 Perfect polyhedra | 100 |
Concluding Remarks | 101 |
Bibliography | 104 |
6 Quadrilaterals | 68 |
7 The 1skeleton | 71 |
Polyhedra | 74 |
2 Cuboids | 75 |
Index of symbols | 107 |
| 109 | |
| 110 | |
Common terms and phrases
action affine hull affine planes Archimedean solids called combinatorial automorphism combinatorial equivalence combinatorial invariants Combinatorial structure combinatorial types combinatorially regular compact complete conjugacy classes construction conv convex bodies cube cuboctahedron cuboids deficiency defined denote the set dimension dodecahedron duality edge-regular edges element Euclidean space example Exercise face-lattice face-regular faces facets finite subgroup Fo(P follows Hence homeomorphic icosidodecahedron isomorphism isosceles label lattice Lie group line-segments linear decomposition linear subspace m-gons manifold matrix metric space n-polytope nodes nonprismatic Normal polytopes octahedron orbit types orthogonal Platonic solids polarity polygon polyhedra polyhedron prisms quadrilaterals rectangle rectangular product rectangular sum regular polytopes sequence shown in Figure Sim(n similarity classes simplicial singleton space of polytopes subsets symmetry equivalence symmetry group symmetry type structure Theorem theory topological transformation group triangles vert vertex Vertex-regular polyhedra vertices