Order-6 dodecahedral honeycombPerspective projection viewwithin Poincaré disk modelTypeHyperbolic regular honeycombParacompact uniform honeycombSchläfli symbol{5,3,6}{5,3}Coxeter diagram ↔ Cells{5,3} Facespentagon {5}Edge figurehexagon {6}Vertex figure triangular tilingDualOrder-5 hexagonal tiling honeycombCoxeter group, [5,3,6], [5,3]PropertiesRegular, quasiregular The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.
Symmetry A half symmetry construction exists as with alternately colored dodecahedral cells.
Images The model is cell-centered within the Poincaré disk model, with the viewpoint then placed at the origin.
The order-6 dodecahedral honeycomb is similar to the 2D hyperbolic infinite-order pentagonal tiling, {5,∞}, with pentagonal faces, and with vertices on the ideal surface.
Related polytopes and honeycombs The order-6 dodecahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
There are 15 uniform honeycombs in the [5,3,6] Coxeter group family, including this regular form, and its regular dual, the order-5 hexagonal tiling honeycomb.
The order-6 dodecahedral honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures:
It is also part of a sequence of regular polytopes and honeycombs with dodecahedral cells:
Rectified order-6 dodecahedral honeycomb Rectified order-6 dodecahedral honeycombTypeParacompact uniform honeycombSchläfli symbolsr{5,3,6}t1{5,3,6}Coxeter diagrams ↔ Cellsr{5,3} {3,6} Facestriangle {3}pentagon {5}Vertex figurehexagonal prismCoxeter groups, [5,3,6], [5,3]PropertiesVertex-transitive, edge-transitive The rectified order-6 dodecahedral honeycomb, t1{5,3,6} has icosidodecahedron and triangular tiling cells connected in a hexagonal prism vertex figure. Perspective projection view within Poincaré disk model
It is similar to the 2D hyperbolic pentaapeirogonal tiling, r{5,∞} with pentagon and apeirogonal faces.
Truncated order-6 dodecahedral honeycomb Truncated order-6 dodecahedral honeycombTypeParacompact uniform honeycombSchläfli symbolst{5,3,6}t0,1{5,3,6}Coxeter diagrams ↔ Cellst{5,3} {3,6} Facestriangle {3}decagon {10}Vertex figurehexagonal pyramidCoxeter groups, [5,3,6], [5,3]PropertiesVertex-transitive The truncated order-6 dodecahedral honeycomb, t0,1{5,3,6} has truncated dodecahedron and triangular tiling cells connected in a hexagonal pyramid vertex figure.
Bitruncated order-6 dodecahedral honeycomb
The bitruncated order-6 dodecahedral honeycomb is the same as the bitruncated order-5 hexagonal tiling honeycomb.
Cantellated order-6 dodecahedral honeycomb Cantellated order-6 dodecahedral honeycombTypeParacompact uniform honeycombSchläfli symbolsrr{5,3,6}t0,2{5,3,6}Coxeter diagrams ↔ Cellsrr{5,3} rr{6,3} {}x{6} Facestriangle {3}square {4}pentagon {5}hexagon {6}Vertex figurewedgeCoxeter groups, [5,3,6], [5,3]PropertiesVertex-transitive The cantellated order-6 dodecahedral honeycomb, t0,2{5,3,6}, has rhombicosidodecahedron, trihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure.
Cantitruncated order-6 dodecahedral honeycomb Cantitruncated order-6 dodecahedral honeycombTypeParacompact uniform honeycombSchläfli symbolstr{5,3,6}t0,1,2{5,3,6}Coxeter diagrams ↔ Cellstr{5,3} t{3,6} {}x{6} Facessquare {4}hexagon {6}decagon {10}Vertex figuremirrored sphenoidCoxeter groups, [5,3,6], [5,3]PropertiesVertex-transitive The cantitruncated order-6 dodecahedral honeycomb, t0,1,2{5,3,6} has truncated icosidodecahedron, hexagonal tiling, and hexagonal prism facets, with a mirrored sphenoid vertex figure.
Runcinated order-6 dodecahedral honeycomb
The runcinated order-6 dodecahedral honeycomb is the same as the runcinated order-5 hexagonal tiling honeycomb.
Runcitruncated order-6 dodecahedral honeycomb Runcitruncated order-6 dodecahedral honeycombTypeParacompact uniform honeycombSchläfli symbolst0,1,3{5,3,6}Coxeter diagramsCellst{5,3} rr{6,3} {}x{10} {}x{6} Facessquare {4}hexagon {6}decagon {10}Vertex figureisosceles-trapezoidal pyramidCoxeter groups, [5,3,6]PropertiesVertex-transitive The runcitruncated order-6 dodecahedral honeycomb, t0,1,3{5,3,6} has truncated dodecahedron, rhombitrihexagonal tiling, decagonal prism, and hexagonal prism facets, with an isosceles-trapezoidal pyramid vertex figure.
Runcicantellated order-6 dodecahedral honeycomb The runcicantellated order-6 dodecahedral honeycomb is the same as the runcitruncated order-5 hexagonal tiling honeycomb.
Omnitruncated order-6 dodecahedral honeycomb The omnitruncated order-6 dodecahedral honeycomb is the same as the omnitruncated order-5 hexagonal tiling honeycomb.
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