In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.
It is also seen as a rectified order-4 square tiling honeycomb, r{4,4,4}:
The square tiling honeycomb has three reflective symmetry constructions: as a regular honeycomb, a half symmetry construction â , and lastly a construction with three types (colors) of checkered square tilings â .
It also contains an index 6 subgroup [4,4,3<sup>*</sup>] â [4<sup>1,1,1</sup>], and a radial subgroup [4,(4,3)<sup>*</sup>] of index 48, with a right dihedral-angled octahedral fundamental domain, and four pairs of ultraparallel mirrors: .
This honeycomb contains that tile 2-hypercycle surfaces, which are similar to the paracompact order-3 apeirogonal tiling :
The square tiling honeycomb is a regular hyperbolic honeycomb in 3-space. It is one of eleven regular paracompact honeycombs.
There are fifteen uniform honeycombs in the [4,4,3] Coxeter group family, including this regular form, and its dual, the order-4 octahedral honeycomb, {3,4,4}.
The square tiling honeycomb is part of the order-4 square tiling honeycomb family, as it can be seen as a rectified order-4 square tiling honeycomb.
It is related to the 24-cell, {3,4,3}, which also has a cubic vertex figure. It is also part of a sequence of honeycombs with square tiling cells:
The rectified square tiling honeycomb, t<sub>1</sub>{4,4,3}, has cube and square tiling facets, with a triangular prism vertex figure.
It is similar to the 2D hyperbolic uniform triapeirogonal tiling, r{âÂÂ,3}, with triangle and apeirogonal faces.
The truncated square tiling honeycomb, t{4,4,3}, has cube and truncated square tiling facets, with a triangular pyramid vertex figure. It is the same as the cantitruncated order-4 square tiling honeycomb, tr{4,4,4}, .
The bitruncated square tiling honeycomb, 2t{4,4,3}, has truncated cube and truncated square tiling facets, with a digonal disphenoid vertex figure.
The cantellated square tiling honeycomb, rr{4,4,3}, has cuboctahedron, square tiling, and triangular prism facets, with an isosceles triangular prism vertex figure.
The cantitruncated square tiling honeycomb, tr{4,4,3}, has truncated cube, truncated square tiling, and triangular prism facets, with an isosceles triangular pyramid vertex figure.
The runcinated square tiling honeycomb, t<sub>0,3</sub>{4,4,3}, has octahedron, triangular prism, cube, and square tiling facets, with an irregular triangular antiprism vertex figure.
The runcitruncated square tiling honeycomb, t<sub>0,1,3</sub>{4,4,3}, has rhombicuboctahedron, octagonal prism, triangular prism and truncated square tiling facets, with an isosceles-trapezoidal pyramid vertex figure.
The runcicantellated square tiling honeycomb is the same as the runcitruncated order-4 octahedral honeycomb.
The omnitruncated square tiling honeycomb, t<sub>0,1,2,3</sub>{4,4,3}, has truncated square tiling, truncated cuboctahedron, hexagonal prism, and octagonal prism facets, with an irregular tetrahedron vertex figure.
The alternated omnitruncated square tiling honeycomb (or omnisnub square tiling honeycomb), h(t<sub>0,1,2,3</sub>{4,4,3}), has snub square tiling, snub cube, triangular antiprism, square antiprism, and tetrahedron cells, with an irregular tetrahedron vertex figure.
The alternated square tiling honeycomb, h{4,4,3}, is a quasiregular paracompact uniform honeycomb in hyperbolic 3-space. It has cube and square tiling facets in a cuboctahedron vertex figure.
The cantic square tiling honeycomb, h<sub>2</sub>{4,4,3}, is a paracompact uniform honeycomb in hyperbolic 3-space. It has truncated square tiling, truncated cube, and cuboctahedron facets, with a rectangular pyramid vertex figure.
The runcic square tiling honeycomb, h<sub>3</sub>{4,4,3}, is a paracompact uniform honeycomb in hyperbolic 3-space. It has square tiling, rhombicuboctahedron, and octahedron facets in a square frustum vertex figure.
The runcicantic square tiling honeycomb, h<sub>2,3</sub>{4,4,3}, â , is a paracompact uniform honeycomb in hyperbolic 3-space. It has truncated square tiling, truncated cuboctahedron, and truncated octahedron facets in a mirrored sphenoid vertex figure.
The alternated rectified square tiling honeycomb is a paracompact uniform honeycomb in hyperbolic 3-space.