Rectified square tiling honeycomb

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

Runcicantellated square tiling honeycomb

The runcicantellated square tiling honeycomb is the same as the runcitruncated order-4 octahedral honeycomb.

Omnitruncated square tiling honeycomb

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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