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:
Rectified square tiling honeycomb The
rectified square tiling honeycomb, t1{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{∞,3}, with
triangle and
apeirogonal faces. :
Truncated square tiling honeycomb 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}, .
Bitruncated square tiling honeycomb The
bitruncated square tiling honeycomb, 2t{4,4,3}, has
truncated cube and
truncated square tiling facets, with a
digonal disphenoid vertex figure.
Cantellated square tiling honeycomb The
cantellated square tiling honeycomb, rr{4,4,3}, has
cuboctahedron,
square tiling, and
triangular prism facets, with an isosceles
triangular prism vertex figure.
Cantitruncated square tiling honeycomb 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.
Runcinated square tiling honeycomb The
runcinated square tiling honeycomb, t0,3{4,4,3}, has
octahedron,
triangular prism,
cube, and
square tiling facets, with an irregular
triangular antiprism vertex figure.
Runcitruncated square tiling honeycomb The
runcitruncated square tiling honeycomb, t0,1,3{4,4,3}, has
rhombicuboctahedron,
octagonal prism,
triangular prism and
truncated square tiling facets, with an
isosceles-trapezoidal pyramid vertex figure.
Runcicantellated square tiling honeycomb The
runcicantellated square tiling honeycomb is the same as the
runcitruncated order-4 octahedral honeycomb.
Omnitruncated square tiling honeycomb The
omnitruncated square tiling honeycomb, t0,1,2,3{4,4,3}, has
truncated square tiling,
truncated cuboctahedron,
hexagonal prism, and
octagonal prism facets, with an irregular
tetrahedron vertex figure.
Omnisnub square tiling honeycomb The
alternated omnitruncated square tiling honeycomb (or
omnisnub square tiling honeycomb), h(t0,1,2,3{4,4,3}), has
snub square tiling,
snub cube,
triangular antiprism,
square antiprism, and
tetrahedron cells, with an irregular
tetrahedron vertex figure.
Alternated square tiling honeycomb 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.
Cantic square tiling honeycomb The
cantic square tiling honeycomb, h2{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.
Runcic square tiling honeycomb The
runcic square tiling honeycomb, h3{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.
Runcicantic square tiling honeycomb The
runcicantic square tiling honeycomb, h2,3{4,4,3}, ↔ , 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.
Alternated rectified square tiling honeycomb The
alternated rectified square tiling honeycomb is a paracompact uniform honeycomb in hyperbolic 3-space. == See also ==