Order-4 octahedral honeycomb

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.

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:
 * H2chess 23ib.png

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.