Order-4 octahedral honeycombPerspective projection viewwithin Poincaré disk modelTypeHyperbolic regular honeycombParacompact uniform honeycombSchläfli symbols{3,4,4}{3,41,1}Coxeter diagrams ↔ ↔ ↔ Cells{3,4} Facestriangle {3}Edge figuresquare {4}Vertex figuresquare tiling, {4,4} DualSquare tiling honeycomb, {4,4,3}Coxeter groups, [3,4,4], [3,41,1]PropertiesRegular The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.Coxeter, The Beauty of Geometry, 1999, Chapter 10, Table III
Symmetry A half symmetry construction, [3,4,4,1+], exists as {3,41,1}, with two alternating types (colors) of octahedral cells: ↔ .
A second half symmetry is [3,4,1+,4]: ↔ .
A higher index sub-symmetry, [3,4,4*], which is index 8, exists with a pyramidal fundamental domain, [((3,∞,3)),((3,∞,3))]: .
This honeycomb contains and that tile 2-hypercycle surfaces, which are similar to the paracompact infinite-order triangular tilings and , respectively:
Related polytopes and honeycombs The order-4 octahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and is one of eleven regular paracompact honeycombs.
There are fifteen uniform honeycombs in the [3,4,4] Coxeter group family, including this regular form.
It is a part of a sequence of honeycombs with a square tiling vertex figure:
It a part of a sequence of regular polychora and honeycombs with octahedral cells:
Rectified order-4 octahedral honeycomb Rectified order-4 octahedral honeycomb TypeParacompact uniform honeycombSchläfli symbolsr{3,4,4} or t1{3,4,4}Coxeter diagrams ↔ ↔ ↔ Cellsr{4,3} {4,4}Facestriangle {3}square {4}Vertex figuresquare prismCoxeter groups, [3,4,4], [3,41,1]PropertiesVertex-transitive, edge-transitive The rectified order-4 octahedral honeycomb, t1{3,4,4}, has cuboctahedron and square tiling facets, with a square prism vertex figure.
Truncated order-4 octahedral honeycomb Truncated order-4 octahedral honeycomb TypeParacompact uniform honeycombSchläfli symbolst{3,4,4} or t0,1{3,4,4}Coxeter diagrams ↔ ↔ ↔ Cellst{3,4} {4,4}Facessquare {4}hexagon {6}Vertex figuresquare pyramidCoxeter groups, [3,4,4], [3,41,1]PropertiesVertex-transitive The truncated order-4 octahedral honeycomb, t0,1{3,4,4}, has truncated octahedron and square tiling facets, with a square pyramid vertex figure.
Bitruncated order-4 octahedral honeycomb The bitruncated order-4 octahedral honeycomb is the same as the bitruncated square tiling honeycomb.
Cantellated order-4 octahedral honeycomb Cantellated order-4 octahedral honeycomb TypeParacompact uniform honeycombSchläfli symbolsrr{3,4,4} or t0,2{3,4,4}s2{3,4,4}Coxeter diagrams ↔ Cellsrr{3,4} {}x4 r{4,4} Facestriangle {3}square {4}Vertex figurewedgeCoxeter groups, [3,4,4], [3,41,1]PropertiesVertex-transitive The cantellated order-4 octahedral honeycomb, t0,2{3,4,4}, has rhombicuboctahedron, cube, and square tiling facets, with a wedge vertex figure.
Cantitruncated order-4 octahedral honeycomb Cantitruncated order-4 octahedral honeycomb TypeParacompact uniform honeycombSchläfli symbolstr{3,4,4} or t0,1,2{3,4,4}Coxeter diagrams ↔ Cellstr{3,4} {}x{4} t{4,4} Facessquare {4}hexagon {6}octagon {8}Vertex figuremirrored sphenoidCoxeter groups, [3,4,4], [3,41,1]PropertiesVertex-transitive The cantitruncated order-4 octahedral honeycomb, t0,1,2{3,4,4}, has truncated cuboctahedron, cube, and truncated square tiling facets, with a mirrored sphenoid vertex figure.
Runcinated order-4 octahedral honeycomb The runcinated order-4 octahedral honeycomb is the same as the runcinated square tiling honeycomb.
Runcitruncated order-4 octahedral honeycomb Runcitruncated order-4 octahedral honeycomb TypeParacompact uniform honeycombSchläfli symbolst0,1,3{3,4,4}Coxeter diagrams ↔ Cellst{3,4} {6}x{} rr{4,4} Facessquare {4}hexagon {6}octagon {8}Vertex figuresquare pyramidCoxeter groups, [3,4,4]PropertiesVertex-transitive The runcitruncated order-4 octahedral honeycomb, t0,1,3{3,4,4}, has truncated octahedron, hexagonal prism, and square tiling facets, with a square pyramid vertex figure.
Runcicantellated order-4 octahedral honeycomb The runcicantellated order-4 octahedral honeycomb is the same as the runcitruncated square tiling honeycomb.
Omnitruncated order-4 octahedral honeycomb The omnitruncated order-4 octahedral honeycomb is the same as the omnitruncated square tiling honeycomb.
Snub order-4 octahedral honeycomb Snub order-4 octahedral honeycomb TypeParacompact scaliform honeycombSchläfli symbolss{3,4,4}Coxeter diagrams ↔ ↔ ↔ Cellssquare tilingicosahedronsquare pyramidFacestriangle {3}square {4}Vertex figureCoxeter groups[4,4,3+][41,1,3+][(4,4,(3,3)+)]PropertiesVertex-transitive The snub order-4 octahedral honeycomb, s{3,4,4}, has Coxeter diagram . It is a scaliform honeycomb, with square pyramid, square tiling, and icosahedron facets.
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) 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, (2015), Chapter 13: Hyperbolic Coxeter groups Norman W. Johnson and Asia Ivić Weiss, Quadratic Integers and Coxeter Groups, Can. J. Math. Vol. 51 (6), 1999, pp. 1307–1336