This monograph identifies polytopes that are "combinatorially R1-embeddable", within interesting lists of polytopal graphs, i.e. such that corresponding polytopes are either prominent mathematically (regular partitions, root lattices, uniform polytopes and so on), or applicable in chemistry (fullerenes, polycycles, etc.). The embeddability, if any, provides applications to chemical graphs and, in the first case, it gives new combinatorial perspective to "R2-prominent" affine polytopal objects. The lists of polytopal graphs in the book come from broad areas of geometry, crystallography and graph theory. The book concentrates on such concise and, as much as possible, independent definitions. The scale-isometric embeddability — the main unifying question, to which those lists are subjected — is presented with the minimum of technicalities.
Author(s): Michel Deza, Viatcheslav Grishukhin, Mikhail Shtogrin
Year: 2004
Language: English
Pages: 188
Scale-Isometric Polytopal Graphs in Hypercubes and Cubic Lattices: Polytopes in Hypercubes and Zn......Page 4
Preface......Page 6
Contents......Page 8
1.1 Graphs......Page 12
1.2 Embeddings of graphs......Page 14
1.3 Embedding of plane graphs......Page 21
1.4 Types of regularity of polytopes and tilings......Page 25
1.5 Operations on polytopes......Page 28
1.6 Voronoi and Delaunay partitions......Page 29
1.7 Infinite graphs......Page 30
2. An Example: Embedding of Fullerenes......Page 36
2.1 Embeddability of fullerenes and their duals......Page 37
2.3 Katsura model for vesicles cells versus embeddable dual fullerenes......Page 41
3.1 Regular tilings and honeycombs......Page 46
3.2 The planar case......Page 47
3.4 The case of dimension d 3......Page 51
4.1 Semi-regular polyhedra......Page 54
4.2 Moscow, Globe and Web graphs......Page 56
4.3 Stellated k-gons, cupolas and antiwebs......Page 59
4.4 Capped antiprisms and columns of antiprisms......Page 61
5.1 Truncations of regular partitions......Page 64
5.2 Partial truncations and cappings of Platonic solids......Page 65
5.3 Chamfering of Platonic solids......Page 70
6. 92 Regular-faced (not Semi-regular) Polyhedra......Page 74
7.1 Semi-regular (not regular) n-polytopes......Page 82
7.3 Archimedean 4-polytopes......Page 83
7.4 The embedding of the snub 24-cell......Page 84
8.1 (r,q)-polycycles......Page 86
8.2 Quasi-(r, 3)-polycycles......Page 88
8.3 Coordination polyhedra and metallopolyhedra......Page 91
9.1 58 embeddable mosaics......Page 94
9.2 Other special plane tilings......Page 98
9.3 Face-regular bifaced plane tilings......Page 100
10. Uniform Partitions of 3-space and Relatives......Page 110
10.1 28 uniform partitions......Page 111
10.2 Other special partitions......Page 114
11.1 Irreducible root lattices......Page 118
11.2 The case of dimension 3......Page 119
11.3 Dicings......Page 121
Voronoi polytopes......Page 122
Delaunay polytopes......Page 123
12.1 Polyhedra with at most seven faces......Page 126
12.2 Simple polyhedra with at most eight faces......Page 127
13. Bifaced Polyhedra......Page 130
13.1 Goldberg’s medial polyhedra......Page 131
13.2 Face-regular bifaced polyhedra......Page 134
13.3 Constructions of bifaced polyhedra......Page 136
13.4 Polyhedra 3n and 4n......Page 137
13.6 Polyhedra ocn (octahedrites)......Page 140
14.1 Equicut l1-graphs......Page 148
Small bipartite polyhedral graphs......Page 156
Zonotopes......Page 157
The tope graphs of oriented matroids......Page 158
15.1 Quasi-embedding......Page 164
15.3 Polytopal hypermetrics......Page 168
15.4 Simplicial n-manifolds......Page 171
Bibliography......Page 174
Index......Page 182
