Scale-isometric Polytopal Graphs in Hypercubes and Cubic Lattices: Polytopes in Hypercubes and Zn̳This 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
Preface | 1 |
from lattices Chapters 3 7 11 consider graphs in R Finally chapters | 2 |
Embedding of Fullerenes | 25 |
Regular Tilings and Honeycombs | 39 |
Truncation Capping and Chamfering | 53 |
Semiregular and Regularfaced npolytopes n 4 | 71 |
Polycycles and Other Chemically Relevant Graphs | 77 |
Plane Tilings | 83 |
Lattices Bilattices and Tiles | 107 |
Small Polyhedra | 115 |
Bifaced Polyhedra | 119 |
Special 1graphs | 137 |
Some Generalization of l₁embedding | 153 |
163 | |
171 | |
Uniform Partitions of 3space and Relatives | 99 |
Other editions - View all
Common terms and phrases
3-space 5-gonal adjacent alternating cuts Ambo(P amongst antiprisms Archimedean Archimedean tiling bifaced bifaced polyhedra bipartite called centrally symmetric chamfering chapter circuit combinatorial consider convex cuts cube cubic lattice cuboctahedron Delaunay partition Delaunay polytopes deltahedra denote diameter dimension distance-regular graphs dodecahedron dual Dürer's octahedron elongated embed embedded into H 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 l₁ l₁-embeddable l₁-graphs l₁-rigid Lemma M.Deza and M.I.Shtogrin metric mosaics n-polytope non-embeddable notation obtained oriented matroid path-metric pentagons planar plane graph Platonic solids polycycles polyhedron Prisms Proposition Pyr4 regular tilings regular-faced polyhedra respectively RhDo rhombicuboctahedron scale simple skeleton snub 24-cell snub cube snub disphenoid symmetry Table tetrahedron Theorem triangles triangular faces truncated twisted uniform partitions unique V₁ vectors vertex figure Voronoi partition Voronoi polytope zones zonohedra zonohedron zonotope