Order-5 cubic honeycombPoincaré disk modelsTypeHyperbolic regular honeycombUniform hyperbolic honeycombSchläfli symbolCoxeter diagramCells (cube)Faces (square)Edge figure (pentagon)Vertex figureicosahedronCoxeter group, [4,3,5]DualOrder-4 dodecahedral honeycombPropertiesRegular
In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol it has five cubes around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.
Description
One cell, centered in Poincare ball modelMain cellsCells with extended edges to ideal boundary
Symmetry It has a radical subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: [4,(3,5)*], index 120.
Related polytopes and honeycombs
The order-5 cubic honeycomb has a related alternated honeycomb, ↔ , with icosahedron and tetrahedron cells.
The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space:
There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the order-5 cubic honeycomb as the regular form:
The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures.
It is also in a sequence of regular polychora and honeycombs with cubic cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb.
Rectified order-5 cubic honeycomb Rectified order-5 cubic honeycombTypeUniform honeycombs in hyperbolic spaceSchläfli symbolr{4,3,5} or 2r{5,3,4}2r{5,31,1}Coxeter diagram ↔ Cellsr{4,3} {3,5} Facestriangle {3}square {4}Vertex figurepentagonal prismCoxeter group, [4,3,5], [5,31,1]PropertiesVertex-transitive, edge-transitive The rectified order-5 cubic honeycomb, , has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure.
Related honeycomb
There are four rectified compact regular honeycombs:
Truncated order-5 cubic honeycomb Truncated order-5 cubic honeycombTypeUniform honeycombs in hyperbolic spaceSchläfli symbolt{4,3,5}Coxeter diagramCellst{4,3} {3,5} Facestriangle {3}octagon {8}Vertex figurepentagonal pyramidCoxeter group, [4,3,5]PropertiesVertex-transitive The truncated order-5 cubic honeycomb, , has truncated cube and icosahedron cells, with a pentagonal pyramid vertex figure.
It can be seen as analogous to the 2D hyperbolic truncated order-5 square tiling, t{4,5}, with truncated square and pentagonal faces:
It is similar to the Euclidean (order-4) truncated cubic honeycomb, t{4,3,4}, which has octahedral cells at the truncated vertices.
Related honeycombs
Bitruncated order-5 cubic honeycomb
The bitruncated order-5 cubic honeycomb is the same as the bitruncated order-4 dodecahedral honeycomb.
Cantellated order-5 cubic honeycomb Cantellated order-5 cubic honeycombTypeUniform honeycombs in hyperbolic spaceSchläfli symbolrr{4,3,5}Coxeter diagramCellsrr{4,3} r{3,5} {}x{5} Facestriangle {3}square {4}pentagon {5}Vertex figurewedgeCoxeter group, [4,3,5]PropertiesVertex-transitive The cantellated order-5 cubic honeycomb, , has rhombicuboctahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.
Related honeycombs It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr{4,3,4}:
Cantitruncated order-5 cubic honeycomb Cantitruncated order-5 cubic honeycombTypeUniform honeycombs in hyperbolic spaceSchläfli symboltr{4,3,5}Coxeter diagramCellstr{4,3} t{3,5} {}x{5} Facessquare {4}pentagon {5}hexagon {6}octagon {8}Vertex figuremirrored sphenoidCoxeter group, [4,3,5]PropertiesVertex-transitive The cantitruncated order-5 cubic honeycomb, , has truncated cuboctahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure.
Related honeycombs It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr{4,3,4}:
Runcinated order-5 cubic honeycomb Runcinated order-5 cubic honeycombTypeUniform honeycombs in hyperbolic spaceSemiregular honeycombSchläfli symbolt0,3{4,3,5}Coxeter diagramCells{4,3} {5,3} {}x{5} Facessquare {4}pentagon {5}Vertex figureirregular triangular antiprismCoxeter group, [4,3,5]PropertiesVertex-transitive The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb , has cube, dodecahedron, and pentagonal prism cells, with an irregular triangular antiprism vertex figure.
It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr{4,5}, with square and pentagonal faces:
Related honeycombs It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t0,3{4,3,4}:
Runcitruncated order-5 cubic honeycomb Runctruncated order-5 cubic honeycombRuncicantellated order-4 dodecahedral honeycombTypeUniform honeycombs in hyperbolic spaceSchläfli symbolt0,1,3{4,3,5}Coxeter diagramCellst{4,3} rr{5,3} {}x{5} {}x{8} Facestriangle {3}square {4}pentagon {5}octagon {8}Vertex figureisosceles-trapezoidal pyramidCoxeter group, [4,3,5]PropertiesVertex-transitive The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb, , has truncated cube, rhombicosidodecahedron, pentagonal prism, and octagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.
Related honeycombs It is similar to the Euclidean (order-4) runcitruncated cubic honeycomb, t0,1,3{4,3,4}:
Runcicantellated order-5 cubic honeycomb
The runcicantellated order-5 cubic honeycomb is the same as the runcitruncated order-4 dodecahedral honeycomb.
Omnitruncated order-5 cubic honeycomb Omnitruncated order-5 cubic honeycombTypeUniform honeycombs in hyperbolic spaceSemiregular honeycombSchläfli symbolt0,1,2,3{4,3,5}Coxeter diagramCellstr{5,3} tr{4,3} {10}x{} {8}x{} Facessquare {4}hexagon {6}octagon {8}decagon {10}Vertex figureirregular tetrahedronCoxeter group, [4,3,5]PropertiesVertex-transitive The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb, , has truncated icosidodecahedron, truncated cuboctahedron, decagonal prism, and octagonal prism cells, with an irregular tetrahedral vertex figure.
Related honeycombs It is similar to the Euclidean (order-4) omnitruncated cubic honeycomb, t0,1,2,3{4,3,4}:
Alternated order-5 cubic honeycomb
Alternated order-5 cubic honeycombTypeUniform honeycombs in hyperbolic spaceSchläfli symbolh{4,3,5}Coxeter diagram ↔ Cells{3,3} {3,5} Facestriangle {3}Vertex figureicosidodecahedronCoxeter group, [5,31,1]PropertiesVertex-transitive, edge-transitive, quasiregular In 3-dimensional hyperbolic geometry, the alternated order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). With Schläfli symbol h{4,3,5}, it can be considered a quasiregular honeycomb, alternating icosahedra and tetrahedra around each vertex in an icosidodecahedron vertex figure.
Related honeycombs It has 3 related forms: the cantic order-5 cubic honeycomb, , the runcic order-5 cubic honeycomb, , and the runcicantic order-5 cubic honeycomb, .
Cantic order-5 cubic honeycomb Cantic order-5 cubic honeycombTypeUniform honeycombs in hyperbolic spaceSchläfli symbolh2{4,3,5}Coxeter diagram ↔ Cellsr{5,3} t{3,5} t{3,3} Facestriangle {3}pentagon {5}hexagon {6}Vertex figurerectangular pyramidCoxeter group, [5,31,1]PropertiesVertex-transitive The cantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h2{4,3,5}. It has icosidodecahedron, truncated icosahedron, and truncated tetrahedron cells, with a rectangular pyramid vertex figure.
Runcic order-5 cubic honeycomb Runcic order-5 cubic honeycombTypeUniform honeycombs in hyperbolic spaceSchläfli symbolh3{4,3,5}Coxeter diagram ↔ Cells{5,3} rr{5,3} {3,3} Facestriangle {3}square {4}pentagon {5}Vertex figuretriangular frustumCoxeter group, [5,31,1]PropertiesVertex-transitive The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h3{4,3,5}. It has dodecahedron, rhombicosidodecahedron, and tetrahedron cells, with a triangular frustum vertex figure.
Runcicantic order-5 cubic honeycomb Runcicantic order-5 cubic honeycombTypeUniform honeycombs in hyperbolic spaceSchläfli symbolh2,3{4,3,5}Coxeter diagram ↔ Cellst{5,3} tr{5,3} t{3,3} Facestriangle {3}square {4}hexagon {6}decagon {10}Vertex figureirregular tetrahedronCoxeter group, [5,31,1]PropertiesVertex-transitive The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h2,3{4,3,5}. It has truncated dodecahedron, truncated icosidodecahedron, and truncated tetrahedron cells, with an irregular tetrahedron vertex figure.
See also Convex uniform honeycombs in hyperbolic space Regular tessellations of hyperbolic 3-space
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, pp. 212–213) 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, (2015) Chapter 13: Hyperbolic Coxeter groups