Square tiling honeycomb

Rectified square tiling honeycomb
Type	Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbols	r{4,4,3} or t1{4,4,3} 2r{3,41,1} r{41,1,1}
Coxeter diagrams	↔ ↔ ↔
Cells	{4,3} r{4,4}
Faces	square {4}
Vertex figure	triangular prism
Coxeter groups	${\displaystyle {\overline {R}}_{3}}$, [4,4,3] ${\displaystyle {\overline {O}}_{3}}$, [3,41,1] ${\displaystyle {\overline {M}}_{3}}$, [41,1,1]
Properties	Vertex-transitive, edge-transitive

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
Type	Paracompact uniform honeycomb
Schläfli symbols	t{4,4,3} or t0,1{4,4,3}
Coxeter diagrams	↔ ↔
Cells	{4,3} t{4,4}
Faces	square {4} octagon {8}
Vertex figure	triangular pyramid
Coxeter groups	${\displaystyle {\overline {R}}_{3}}$, [4,4,3] ${\displaystyle {\overline {N}}_{3}}$, [43] ${\displaystyle {\overline {M}}_{3}}$, [41,1,1]
Properties	Vertex-transitive

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
Type	Paracompact uniform honeycomb
Schläfli symbols	2t{4,4,3} or t1,2{4,4,3}
Coxeter diagram
Cells	t{4,3} t{4,4}
Faces	triangle {3} square {4} octagon {8}
Vertex figure	digonal disphenoid
Coxeter groups	${\displaystyle {\overline {R}}_{3}}$, [4,4,3]
Properties	Vertex-transitive

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
Type	Paracompact uniform honeycomb
Schläfli symbols	rr{4,4,3} or t0,2{4,4,3}
Coxeter diagrams	↔
Cells	r{4,3} rr{4,4} {}x{3}
Faces	triangle {3} square {4}
Vertex figure	isosceles triangular prism
Coxeter groups	${\displaystyle {\overline {R}}_{3}}$, [4,4,3]
Properties	Vertex-transitive

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
Type	Paracompact uniform honeycomb
Schläfli symbols	tr{4,4,3} or t0,1,2{4,4,3}
Coxeter diagram
Cells	t{4,3} tr{4,4} {}x{3}
Faces	triangle {3} square {4} octagon {8}
Vertex figure	isosceles triangular pyramid
Coxeter groups	${\displaystyle {\overline {R}}_{3}}$, [4,4,3]
Properties	Vertex-transitive

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
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,3{4,4,3}
Coxeter diagrams	↔
Cells	{3,4} {4,4} {}x{4} {}x{3}
Faces	triangle {3} square {4}
Vertex figure	irregular triangular antiprism
Coxeter groups	${\displaystyle {\overline {R}}_{3}}$, [4,4,3]
Properties	Vertex-transitive

The runcinated square tiling honeycomb, t_0,3{4,4,3}, has octahedron, triangular prism, cube, and square tiling facets, with an irregular triangular antiprism vertex figure.

Runcitruncated square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t0,1,3{4,4,3} s2,3{3,4,4}
Coxeter diagrams
Cells	rr{4,3} t{4,4} {}x{3} {}x{8}
Faces	triangle {3} square {4} octagon {8}
Vertex figure	isosceles-trapezoidal pyramid
Coxeter groups	${\displaystyle {\overline {R}}_{3}}$, [4,4,3]
Properties	Vertex-transitive

The runcitruncated square tiling honeycomb, t_0,1,3{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.

Omnitruncated square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,2,3{4,4,3}
Coxeter diagram
Cells	tr{4,4} {}x{6} {}x{8} tr{4,3}
Faces	square {4} hexagon {6} octagon {8}
Vertex figure	irregular tetrahedron
Coxeter groups	${\displaystyle {\overline {R}}_{3}}$, [4,4,3]
Properties	Vertex-transitive

The omnitruncated square tiling honeycomb, t_0,1,2,3{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
Type	Paracompact uniform honeycomb
Schläfli symbol	h(t0,1,2,3{4,4,3})
Coxeter diagram
Cells	sr{4,4} sr{2,3} sr{2,4} sr{4,3}
Faces	triangle {3} square {4}
Vertex figure	irregular tetrahedron
Coxeter group	[4,4,3]+
Properties	Non-uniform, vertex-transitive

The alternated omnitruncated square tiling honeycomb (or omnisnub square tiling honeycomb), h(t_0,1,2,3{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
Type	Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbol	h{4,4,3} hr{4,4,4} {(4,3,3,4)} h{41,1,1}
Coxeter diagrams	↔ ↔ ↔ ↔ ↔ ↔
Cells	{4,4} {4,3}
Faces	square {4}
Vertex figure	cuboctahedron
Coxeter groups	${\displaystyle {\overline {O}}_{3}}$, [3,41,1] [4,1+,4,4] ↔ [∞,4,4,∞] ${\displaystyle {\widehat {BR}}_{3}}$, [(4,4,3,3)] [1+,41,1,1] ↔ [∞[6]]
Properties	Vertex-transitive, edge-transitive, quasiregular

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
Type	Paracompact uniform honeycomb
Schläfli symbol	h2{4,4,3}
Coxeter diagrams	↔
Cells	t{4,4} r{4,3} t{4,3}
Faces	triangle {3} square {4} octagon {8}
Vertex figure	rectangular pyramid
Coxeter groups	${\displaystyle {\overline {O}}_{3}}$, [3,41,1]
Properties	Vertex-transitive

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
Type	Paracompact uniform honeycomb
Schläfli symbol	h3{4,4,3}
Coxeter diagrams	↔
Cells	{4,4} r{4,3} {3,4}
Faces	triangle {3} square {4}
Vertex figure	square frustum
Coxeter groups	${\displaystyle {\overline {O}}_{3}}$, [3,41,1]
Properties	Vertex-transitive

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
Type	Paracompact uniform honeycomb
Schläfli symbol	h2,3{4,4,3}
Coxeter diagrams	↔
Cells	t{4,4} tr{4,3} t{3,4}
Faces	square {4} hexagon {6} octagon {8}
Vertex figure	mirrored sphenoid
Coxeter groups	${\displaystyle {\overline {O}}_{3}}$, [3,41,1]
Properties	Vertex-transitive

The runcicantic square tiling honeycomb, h_2,3{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
Type	Paracompact uniform honeycomb
Schläfli symbol	hr{4,4,3}
Coxeter diagrams	↔
Cells
Faces
Vertex figure	triangular prism
Coxeter groups	[4,1+,4,3] = [∞,3,3,∞]
Properties	Nonsimplectic, vertex-transitive

The alternated rectified square tiling honeycomb is a paracompact uniform honeycomb in hyperbolic 3-space.