Scale-isometric Polytopal Graphs in Hypercubes and Cubic Lattices: Polytopes in Hypercubes and Zn̳
This 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.
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Graphs and their Scaleisometric Embedding
from lattices Chapters 3 7 11 consider graphs in Rn Finally chapters
Embedding of Fullerenes
Regular Tilings and Honeycombs
Semiregular Polyhedra and Relatives of Prisms and Antiprisms
Truncation Capping and Chamfering
Semiregular and Regularfaced npolytopes n 4
Uniform Partitions of 3space and Relatives
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3-space 5-gonal adjacent alternating cuts Ambo(P amongst antiprisms APrismm Archimedean Archimedean tiling bifaced bifaced polyhedra bipartite called centrally symmetric chamfered chapter circuit combinatorial consider convex cuts corresponding cube cubic lattice cuboctahedron Delaunay partition Delaunay polytopes deltahedra denote diameter dimension dodecahedron dual elongated embed embeddable equicut graph example extreme hypermetric face F facets finite fullerenes graph G half-cube hexagons honeycombs hypercube i-capped i-embeddable i-graphs i-rigid icosahedron induced subgraph infinite isometric subgraph Johnson graph Lemma metric moreover mosaics n-dimensional n-polytope non-embeddable notation obtained octahedron oriented matroid path-metric pentagons planar plane graph Platonic solids polycycles polyhedron Prism3 Proposition regular tilings regular-faced polyhedra respectively RhDo rhombicuboctahedron scale semi-regular 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 zones zonohedra zonohedron zonotope