Alternated hexagonal tiling honeycombTypeParacompact uniform honeycombSemiregular honeycombSchläfli symbolsh{6,3,3}s{3,6,3}2s{6,3,6}2s{6,3}s{3[3,3]}Coxeter diagrams ↔ ↔ ↔ ↔ Cells{3,3} {3} Facestriangle {3}Vertex figure truncated tetrahedronCoxeter groups, [3,3]1/2 , [6,3,3]1/2 , [3,6,3]1/2 , [6,3,6]1/2 , [6,3]1/2 , [3[3,3]]PropertiesVertex-transitive, edge-transitive, quasiregular In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3}, or , is a semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron vertex figure. It is named after its construction, as an alternation of a hexagonal tiling honeycomb.
Symmetry constructions
It has five alternated constructions from reflectional Coxeter groups all with four mirrors and only the first being regular: [6,3,3], [3,6,3], [6,3,6], [6,3] and [3[3,3]] , having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are , , , and , representing different types (colors) of hexagonal tilings in the Wythoff construction.
Related honeycombs The alternated hexagonal tiling honeycomb has 3 related forms: the cantic hexagonal tiling honeycomb, ; the runcic hexagonal tiling honeycomb, ; and the runcicantic hexagonal tiling honeycomb, .
Cantic hexagonal tiling honeycomb Cantic hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolsh2{6,3,3}Coxeter diagrams ↔ Cellsr{3,3} t{3,3} h2{6,3} Facestriangle {3}hexagon {6}Vertex figurewedgeCoxeter groups, [3,3]PropertiesVertex-transitive The cantic hexagonal tiling honeycomb, h2{6,3,3}, or , is composed of octahedron, truncated tetrahedron, and trihexagonal tiling facets, with a wedge vertex figure.
Runcic hexagonal tiling honeycomb Runcic hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolsh3{6,3,3}Coxeter diagrams ↔ Cells{3,3} {}x{3} rr{3,3} {3} Facestriangle {3}square {4}hexagon {6}Vertex figuretriangular cupolaCoxeter groups, [3,3]PropertiesVertex-transitive The runcic hexagonal tiling honeycomb, h3{6,3,3}, or , has tetrahedron, triangular prism, cuboctahedron, and triangular tiling facets, with a triangular cupola vertex figure.
Runcicantic hexagonal tiling honeycomb Runcicantic hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolsh2,3{6,3,3}Coxeter diagrams ↔ Cellst{3,3} {}x{3} tr{3,3} h2{6,3} Facestriangle {3}square {4}hexagon {6}Vertex figurerectangular pyramidCoxeter groups, [3,3]PropertiesVertex-transitive The runcicantic hexagonal tiling honeycomb, h2,3{6,3,3}, or , has truncated tetrahedron, triangular prism, truncated octahedron, and trihexagonal tiling facets, with a rectangular pyramid vertex figure.
See also Convex uniform honeycombs in hyperbolic space Regular tessellations of hyperbolic 3-space Paracompact uniform honeycombs Semiregular honeycomb Hexagonal tiling honeycomb
References
Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) The Beauty of Geometry: Twelve Essays (1999), Dover Publications, , (Chapter 10, Regular Honeycombs in Hyperbolic Space ) Table III Jeffrey R. Weeks The Shape of Space, 2nd edition (Chapters 16–17: Geometries on Three-manifolds I, II) N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H3: p130.