Order-5 hexagonal tiling honeycombPerspective projection viewfrom center of Poincaré disk modelTypeHyperbolic regular honeycombParacompact uniform honeycombSchläfli symbol{6,3,5}Coxeter-Dynkin diagrams ↔ Cells{6,3} Faceshexagon {6}Edge figurepentagon {5}Vertex figureicosahedronDualOrder-6 dodecahedral honeycombCoxeter group, [5,3,6]PropertiesRegular In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere, a flat plane in hyperbolic space that approaches a single ideal point at infinity.

The Schläfli symbol of the order-5 hexagonal tiling honeycomb is {6,3,5}. Since that of the hexagonal tiling is {6,3}, this honeycomb has five such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the icosahedron is {3,5}, the vertex figure of this honeycomb is an icosahedron. Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.Coxeter, The Beauty of Geometry, 1999, Chapter 10, Table III

Symmetry A lower-symmetry construction of index 120, [6,(3,5)*], exists with regular dodecahedral fundamental domains, and an icosahedral Coxeter-Dynkin diagram with 6 axial infinite-order (ultraparallel) branches.

Images The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact order-5 apeirogonal tiling, {∞,5}, with five apeirogonal faces meeting around every vertex.

Related polytopes and honeycombs The order-5 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

There are 15 uniform honeycombs in the [6,3,5] Coxeter group family, including this regular form, and its regular dual, the order-6 dodecahedral honeycomb.

The order-5 hexagonal tiling honeycomb has a related alternation honeycomb, represented by ↔ , with icosahedron and triangular tiling cells.

It is a part of sequence of regular hyperbolic honeycombs of the form {6,3,p}, with hexagonal tiling facets:

It is also part of a sequence of regular polychora and honeycombs with icosahedral vertex figures:

Rectified order-5 hexagonal tiling honeycomb Rectified order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolsr{6,3,5} or t1{6,3,5}Coxeter diagrams ↔ Cells{3,5} r{6,3} or h2{6,3} Facestriangle {3}hexagon {6}Vertex figurepentagonal prismCoxeter groups, [5,3,6], [5,3]PropertiesVertex-transitive, edge-transitive The rectified order-5 hexagonal tiling honeycomb, t1{6,3,5}, has icosahedron and trihexagonal tiling facets, with a pentagonal prism vertex figure.

It is similar to the 2D hyperbolic infinite-order square tiling, r{∞,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.

Truncated order-5 hexagonal tiling honeycomb Truncated order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolt{6,3,5} or t0,1{6,3,5}Coxeter diagramCells{3,5} t{6,3} Facestriangle {3}dodecagon {12}Vertex figurepentagonal pyramidCoxeter groups, [5,3,6]PropertiesVertex-transitive The truncated order-5 hexagonal tiling honeycomb, t0,1{6,3,5}, has icosahedron and truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure.

Bitruncated order-5 hexagonal tiling honeycomb Bitruncated order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbol2t{6,3,5} or t1,2{6,3,5}Coxeter diagram ↔ Cellst{3,6} t{3,5} Facespentagon {5}hexagon {6}Vertex figuredigonal disphenoidCoxeter groups, [5,3,6], [5,3]PropertiesVertex-transitive The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, has hexagonal tiling and truncated icosahedron facets, with a digonal disphenoid vertex figure.

Cantellated order-5 hexagonal tiling honeycomb Cantellated order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolrr{6,3,5} or t0,2{6,3,5}Coxeter diagramCellsr{3,5} rr{6,3} {}x{5} Facestriangle {3}square {4}pentagon {5}hexagon {6}Vertex figurewedgeCoxeter groups, [5,3,6]PropertiesVertex-transitive The cantellated order-5 hexagonal tiling honeycomb, t0,2{6,3,5}, has icosidodecahedron, rhombitrihexagonal tiling, and pentagonal prism facets, with a wedge vertex figure.

Cantitruncated order-5 hexagonal tiling honeycomb Cantitruncated order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symboltr{6,3,5} or t0,1,2{6,3,5}Coxeter diagramCellst{3,5} tr{6,3} {}x{5} Facessquare {4}pentagon {5}hexagon {6}dodecagon {12}Vertex figuremirrored sphenoidCoxeter groups, [5,3,6]PropertiesVertex-transitive

The cantitruncated order-5 hexagonal tiling honeycomb, t0,1,2{6,3,5}, has truncated icosahedron, truncated trihexagonal tiling, and pentagonal prism facets, with a mirrored sphenoid vertex figure.

Runcinated order-5 hexagonal tiling honeycomb Runcinated order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolt0,3{6,3,5}Coxeter diagramCells{6,3} {5,3} {}x{6} {}x{5} Facessquare {4}pentagon {5}hexagon {6}Vertex figureirregular triangular antiprismCoxeter groups, [5,3,6]PropertiesVertex-transitive The runcinated order-5 hexagonal tiling honeycomb, t0,3{6,3,5}, has dodecahedron, hexagonal tiling, pentagonal prism, and hexagonal prism facets, with an irregular triangular antiprism vertex figure.

Runcitruncated order-5 hexagonal tiling honeycomb Runcitruncated order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolt0,1,3{6,3,5}Coxeter diagramCellst{6,3} rr{5,3} {}x{5} {}x{12} Facestriangle {3}square {4}pentagon {5}dodecagon {12}Vertex figureisosceles-trapezoidal pyramidCoxeter groups, [5,3,6]PropertiesVertex-transitive The runcitruncated order-5 hexagonal tiling honeycomb, t0,1,3{6,3,5}, has truncated hexagonal tiling, rhombicosidodecahedron, pentagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

Runcicantellated order-5 hexagonal tiling honeycomb

The runcicantellated order-5 hexagonal tiling honeycomb is the same as the runcitruncated order-6 dodecahedral honeycomb.

Omnitruncated order-5 hexagonal tiling honeycomb Omnitruncated order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolt0,1,2,3{6,3,5}Coxeter diagramCellstr{6,3} tr{5,3} {}x{10} {}x{12} Facessquare {4}hexagon {6}decagon {10}dodecagon {12}Vertex figureirregular tetrahedronCoxeter groups, [5,3,6]PropertiesVertex-transitive The omnitruncated order-5 hexagonal tiling honeycomb, t0,1,2,3{6,3,5}, has truncated trihexagonal tiling, truncated icosidodecahedron, decagonal prism, and dodecagonal prism facets, with an irregular tetrahedral vertex figure.

Alternated order-5 hexagonal tiling honeycomb Alternated order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSemiregular honeycombSchläfli symbolh{6,3,5}Coxeter diagram ↔ Cells{3} {3,5} Facestriangle {3}Vertex figuretruncated icosahedronCoxeter groups, [5,3]PropertiesVertex-transitive, edge-transitive, quasiregular The alternated order-5 hexagonal tiling honeycomb, h{6,3,5}, ↔ , has triangular tiling and icosahedron facets, with a truncated icosahedron vertex figure. It is a quasiregular honeycomb.

Cantic order-5 hexagonal tiling honeycomb Cantic order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolh2{6,3,5}Coxeter diagram ↔ Cellsh2{6,3} t{3,5} r{5,3} Facestriangle {3}pentagon {5}hexagon {6}Vertex figuretriangular prismCoxeter groups, [5,3]PropertiesVertex-transitive The cantic order-5 hexagonal tiling honeycomb, h2{6,3,5}, ↔ , has trihexagonal tiling, truncated icosahedron, and icosidodecahedron facets, with a triangular prism vertex figure.

Runcic order-5 hexagonal tiling honeycomb Runcic order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolh3{6,3,5}Coxeter diagram ↔ Cells{3} rr{5,3} {5,3} {}x{3} Facestriangle {3}square {4}pentagon {5}Vertex figuretriangular cupolaCoxeter groups, [5,3]PropertiesVertex-transitive The runcic order-5 hexagonal tiling honeycomb, h3{6,3,5}, ↔ , has triangular tiling, rhombicosidodecahedron, dodecahedron, and triangular prism facets, with a triangular cupola vertex figure.

Runcicantic order-5 hexagonal tiling honeycomb Runcicantic order-5 hexagonal tiling honeycombTypeParacompact uniform honeycombSchläfli symbolh2,3{6,3,5}Coxeter diagram ↔ Cellsh2{6,3} tr{5,3} t{5,3} {}x{3} Facestriangle {3}square {4}hexagon {6}decagon {10}Vertex figurerectangular pyramidCoxeter groups, [5,3]PropertiesVertex-transitive The runcicantic order-5 hexagonal tiling honeycomb, h2,3{6,3,5}, ↔ , has trihexagonal tiling, truncated icosidodecahedron, truncated dodecahedron, and triangular prism facets, with a rectangular pyramid vertex figure.

See also Convex uniform honeycombs in hyperbolic space Regular tessellations of hyperbolic 3-space Paracompact uniform honeycombs

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, (1999), Dover Publications, , , (Chapter 10, Regular Honeycombs in Hyperbolic Space ), Table III 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