Order-4 octahedral honeycomb

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 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 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 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 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 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 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 ==