Scale-isometric Polytopal Graphs In Hypercubes And Cubic Lattices: Polytopes In Hypercubes And ZnThis monograph identifies polytopes that are “combinatorially ℓ1-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 “ℓ2-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. |
Contents
| 1 | |
Embedding of Fullerenes | 25 |
3 Regular Tilings and Honeycombs | 35 |
4 Semiregular Polyhedra and Relatives of Prisms and Antiprisms | 43 |
5 Truncation Capping and Chamfering | 53 |
6 92 Regularfaced not Semiregular Polyhedra | 63 |
7 Semiregular and Regularfaced npolytopes n 4 | 71 |
8 Polycycles and Other Chemically Relevant Graphs | 75 |
10 Uniform Partitions of 3space and Relatives | 99 |
11 Lattices Bilattices and Tiles | 107 |
12 Small Polyhedra | 115 |
13 Bifaced Polyhedra | 119 |
14 Special l1graphs | 137 |
15 Some Generalization of l1embedding | 153 |
Bibliography | 163 |
| 171 | |
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Common terms and phrases
1-embeddable 1-graphs 1-rigid 3-space 5-gonal adjacent alternating cuts amongst antiprisms APrism.4 Archimedean Archimedean tiling bifaced bifaced polyhedra bipartite called capped centrally symmetric Cham(Y3 chamfering chapter circuit combinatorial consider cube cubic lattice cuboctahedron cut S,S Delaunay partition Delaunay polytopes deltahedra denote diameter dimension dodecahedron dual Dürer's octahedron elongated embed embeddable equicut graph example extreme hypermetric face F facets finite fullerenes graph G half-cube hexagons hypercube i-capped icosahedron induced subgraph infinite isometric subgraph Johnson graph Lemma metric moreover mosaics n-dimensional n-polytope non-embeddable notation obtained opposite oriented matroid path-metric pentagons planar plane graph Platonic solids polycycle polyhedron Prism Prism.8 Proposition Pyra regular tilings regular-faced polyhedra respectively rhombicuboctahedron scale semi-regular polyhedra simple skeleton snub 24-cell snub cube snub disphenoid symmetry Table tetrahedron Theorem triangles triangular faces truncated twisted uniform partitions unique vectors vertex figure Voronoi partition Voronoi polytope zonohedra zonohedron zonotope