Octahedron-hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbol{(3,4,3,6)} or {(6,3,4,3)}Coxeter diagrams or Cells{3,4} {6,3} r{6,3} Facestriangular {3}square {4}hexagon {6}Vertex figurerhombicuboctahedronCoxeter group[(6,3,4,3)]PropertiesVertex-transitive, edge-transitive In the geometry of hyperbolic 3-space, the octahedron-hexagonal tiling honeycomb is a paracompact uniform honeycomb, constructed from octahedron, hexagonal tiling, and trihexagonal tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
Symmetry A lower symmetry form, index 6, of this honeycomb can be constructed with [(6,3,4,3*)] symmetry, represented by a trigonal trapezohedron fundamental domain, and a Coxeter diagram .
Related honeycombs Cyclotruncated octahedral-hexagonal tiling honeycomb Cyclotruncated octahedral-hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolct{(3,4,3,6)} or ct{(3,6,3,4)}Coxeter diagrams or Cells{6,3} {4,3} t{3,4} Facestriangular {3}square {4}hexagon {6}Vertex figuretriangular antiprismCoxeter group[(6,3,4,3)]PropertiesVertex-transitive The cyclotruncated octahedral-hexagonal tiling honeycomb is a compact uniform honeycomb, constructed from hexagonal tiling, cube, and truncated octahedron cells, in a triangular antiprism vertex figure. It has a Coxeter diagram .
Symmetry A radial subgroup symmetry, index 6, of this honeycomb can be constructed with [(4,3,6,3*)], represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram .
See also Convex uniform honeycombs in hyperbolic space List of regular polytopes
References Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213) Jeffrey R. Weeks The Shape of Space, 2nd edition (Chapter 16-17: Geometries on Three-manifolds I, II) Norman Johnson Uniform Polytopes, Manuscript N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups