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.
A half symmetry construction, [3,4,4,1<sup>+</sup>], exists as {3,4<sup>1,1</sup>}, with two alternating types (colors) of octahedral cells: â .
A second half symmetry is [3,4,1<sup>+</sup>,4]: â .
A higher index sub-symmetry, [3,4,4<sup>*</sup>], which is index 8, exists with a pyramidal fundamental domain, [((3,âÂÂ,3)),((3,âÂÂ,3))]: .
This honeycomb contains and that tile 2-hypercycle surfaces, which are similar to the paracompact infinite-order triangular tilings and , respectively:<BR>
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:
The rectified order-4 octahedral honeycomb, t<sub>1</sub>{3,4,4}, has cuboctahedron and square tiling facets, with a square prism vertex figure.
The truncated order-4 octahedral honeycomb, t<sub>0,1</sub>{3,4,4}, has truncated octahedron and square tiling facets, with a square pyramid vertex figure.
The bitruncated order-4 octahedral honeycomb is the same as the bitruncated square tiling honeycomb.
The cantellated order-4 octahedral honeycomb, t<sub>0,2</sub>{3,4,4}, has rhombicuboctahedron, cube, and square tiling facets, with a wedge vertex figure.
The cantitruncated order-4 octahedral honeycomb, t<sub>0,1,2</sub>{3,4,4}, has truncated cuboctahedron, cube, and truncated square tiling facets, with a mirrored sphenoid vertex figure.
The runcinated order-4 octahedral honeycomb is the same as the runcinated square tiling honeycomb.
The runcitruncated order-4 octahedral honeycomb, t<sub>0,1,3</sub>{3,4,4}, has truncated octahedron, hexagonal prism, and square tiling facets, with a square pyramid vertex figure.
The runcicantellated order-4 octahedral honeycomb is the same as the runcitruncated square tiling honeycomb.
The omnitruncated order-4 octahedral honeycomb is the same as the omnitruncated square tiling 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.