View source for Square tiling honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Square tiling honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[File:H3 443 FC boundary.png|320px]]
|-
|bgcolor=#e7dcc3|Type||Hyperbolic regular honeycomb<BR>Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||{4,4,3}<BR>r{4,4,4}<BR>{4<sup>1,1,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|node_1|4|node|4|node|3|node}}<BR>{{CDD|node|4|node_1|4|node|4|node}}<BR>{{CDD|node|4|node_1|split1-44|nodes}} ↔ {{CDD|node_1|4|node|4|node_g|3sg|node_g}}<BR>{{CDD|nodes_11|2a2b-cross|nodes|split2|node}} ↔ {{CDD|node_1|4|node_h0|4|node|3|node}}<BR>{{CDD|branchu_11|2|branchu_11|2|branchu_11|2|branchu_11}} ↔ {{CDD|node_1|4|node_g|4sg|node_g|3g|node_g}}
|-
|bgcolor=#e7dcc3|Cells||[[Square tiling|{4,4}]] [[File:Square_tiling_uniform_coloring_1.png|40px]] 40px [[File:Square_tiling_uniform_coloring_7.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Square|square]] {4}
|-
|bgcolor=#e7dcc3|Edge figure||[[Triangle|triangle]] {3}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Square tiling honeycomb verf.png|80px]]<BR>[[Cube|cube]], {4,3}
|-
|bgcolor=#e7dcc3|Dual||[[Order-4 octahedral honeycomb]]
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{R}_3</math>, [4,4,3]<BR><math>\overline{N}_3</math>, [4<sup>3</sup>]<BR><math>\overline{M}_3</math>, [4<sup>1,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[Geometry|geometry]] of [[Hyperbolic space|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 [[Horosphere|horosphere]]s and converge to a single [[Ideal point|ideal point]] at infinity. Given by [[Schläfli symbol]] {4,4,3}, it has three [[Square tiling|square tiling]]s, {4,4}, around each edge, and six square tilings around each vertex, in a [[Cube|cubic]] {4,3} [[Vertex figure|vertex figure]].<ref>Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III</ref>

{{Honeycomb}}

== Rectified order-4 square tiling ==
It is also seen as a rectified order-4 square tiling honeycomb, r{4,4,4}:
{| class=wikitable
![[Order-4 square tiling honeycomb|{4,4,4}]]
!r{4,4,4} = {4,4,3}
|-
!{{CDD|node_1|4|node|4|node|4|node}}||{{CDD|node|4|node_1|4|node|4|node}} = {{CDD|node_1|4|node|4|node|3|node}}
|-
|[[File:H3 444 FC boundary.png|240px]]||[[File:H3 444 boundary 0100.png|240px]]
|}

== Symmetry==
The square tiling honeycomb has three reflective symmetry constructions: {{CDD|node_1|4|node|4|node|3|node}} as a regular honeycomb, a half symmetry construction {{CDD|node_1|4|node_h0|4|node|3|node}} ↔ {{CDD|nodes_11|2a2b-cross|nodes|split2|node}}, and lastly a construction with three types (colors) of checkered square tilings {{CDD|node_1|4|node|4|node_g|3sg|node_g}} ↔ {{CDD|node|4|node_1|split1-44|nodes}}.

It also contains an index 6 subgroup [4,4,3<sup>*</sup>] ↔ [4<sup>1,1,1</sup>], and a radial subgroup [4,(4,3)<sup>*</sup>] of index 48, with a right [[Dihedral angle|dihedral-angled]] octahedral [[Fundamental domain|fundamental domain]], and four pairs of ultraparallel mirrors: {{CDD|branchu_11|2|branchu_11|2|branchu_11|2|branchu_11}}.

This honeycomb contains {{CDD|node|3|node|ultra|node_1}} that tile 2-[[Hypercycle (geometry)|hypercycle]] surfaces, which are similar to the paracompact [[Order-3 apeirogonal tiling|order-3 apeirogonal tiling]] {{CDD|node|3|node|infin|node_1}}:
: [[File:H2-I-3-dual.svg|120px]]

== Related polytopes and honeycombs ==
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|order-4 octahedral honeycomb]], {3,4,4}.

The square tiling honeycomb is part of the [[Order-4 square tiling honeycomb|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|24-cell]], {3,4,3}, which also has a cubic vertex figure.
It is also part of a sequence of honeycombs with [[Square tiling|square tiling]] cells:

=== Rectified square tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Rectified square tiling honeycomb 
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb<BR>Semiregular honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||r{4,4,3} or t<sub>1</sub>{4,4,3}<BR>2r{3,4<sup>1,1</sup>}<BR>r{4<sup>1,1,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|node|4|node_1|4|node|3|node}}<BR>{{CDD|nodes_11|split2-44|node|3|node}} ↔ {{CDD|node_h0|4|node_1|4|node|3|node}}<BR>{{CDD|nodes_11|split2-44|node|4|node_1}} ↔ {{CDD|node|4|node_1|4|node_g|3sg|node_g}}<BR>{{CDD|node_1|split1-uu|nodes_11|2a2b-cross|nodes_11|split2-uu|node_1}} ↔ {{CDD|node_h0|4|node_1|4|node_g|3sg|node_g}}
|-
|bgcolor=#e7dcc3|Cells||[[Cube|{4,3}]] [[File:Uniform_polyhedron-43-t0.png|40px]]<BR>[[Square tiling|r{4,4}]][[File:Uniform_tiling_44-t1.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Square|square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Rectified square tiling honeycomb verf.png|80px]]<BR>[[Triangular prism|triangular prism]]
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{R}_3</math>, [4,4,3]<BR><math>\overline{O}_3</math>, [3,4<sup>1,1</sup>]<BR><math>\overline{M}_3</math>, [4<sup>1,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive
|}
The '''rectified square tiling honeycomb''', t<sub>1</sub>{4,4,3}, {{CDD|node|4|node_1|4|node|3|node}} has [[Cube|cube]] and [[Square tiling|square tiling]] facets, with a [[Triangular prism|triangular prism]] [[Vertex figure|vertex figure]].

[[File:H3_443_boundary_0100.png|480px]]

It is similar to the 2D hyperbolic uniform triapeirogonal tiling, r{∞,3}, with [[Triangle|triangle]] and [[Apeirogon|apeirogon]]al faces.
: [[File:H2 tiling 23i-2.png|240px]]

{{Clear}}

=== Truncated square tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Truncated square tiling honeycomb 
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||t{4,4,3} or t<sub>0,1</sub>{4,4,3}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|node_1|4|node_1|4|node|3|node}}<BR>{{CDD|node_1|4|node_1|4|node_1|4|node}}<BR>{{CDD|node_1|4|node_1|split1-44|nodes_11}} ↔ {{CDD|node_1|4|node_1|4|node_1|4|node_h0}}<BR>{{CDD|nodes_11|split2-44|node_1|4|node_1}} ↔ {{CDD|node_1|4|node_1|4|node_g|3sg|node_g}}
|-
|bgcolor=#e7dcc3|Cells||[[Cube|{4,3}]] [[File:Uniform_polyhedron-43-t0.png|40px]]<BR>[[Square tiling|t{4,4}]][[File:Uniform_tiling_44-t01.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Square|square]] {4}<BR>[[Octagon|octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Truncated square tiling honeycomb verf.png|80px]]<BR>triangular pyramid
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{R}_3</math>, [4,4,3]<BR><math>\overline{N}_3</math>, [4<sup>3</sup>]<BR><math>\overline{M}_3</math>, [4<sup>1,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''truncated square tiling honeycomb''', t{4,4,3}, {{CDD|node_1|4|node_1|4|node|3|node}} has [[Cube|cube]] and [[Truncated square tiling|truncated square tiling]] facets, with a triangular pyramid [[Vertex figure|vertex figure]]. It is the same as the cantitruncated order-4 square tiling honeycomb, tr{4,4,4}, {{CDD|node_1|4|node_1|4|node_1|4|node}}.

[[File:H3 443-1100.png|480px]]
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=== Bitruncated square tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Bitruncated square tiling honeycomb 
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||2t{4,4,3} or t<sub>1,2</sub>{4,4,3}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node|4|node_1|4|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated cube|t{4,3}]] [[File:Uniform_polyhedron-43-t01.png|40px]]<BR>[[Square tiling|t{4,4}]][[File:Uniform_tiling_44-t01.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}<BR>[[Octagon|octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Bitruncated square tiling honeycomb verf.png|80px]]<BR>digonal disphenoid
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{R}_3</math>, [4,4,3]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''bitruncated square tiling honeycomb''', 2t{4,4,3}, {{CDD|node|4|node_1|4|node_1|3|node}} has [[Truncated cube|truncated cube]] and [[Truncated square tiling|truncated square tiling]] facets, with a digonal disphenoid [[Vertex figure|vertex figure]].

[[File:H3 443-0110.png|480px]]
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=== Cantellated square tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantellated square tiling honeycomb 
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||rr{4,4,3} or t<sub>0,2</sub>{4,4,3}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|node_1|4|node|4|node_1|3|node}}<BR>{{CDD|nodes_11|2a2b-cross|nodes_11|split2|node}} ↔ {{CDD|node_1|4|node_h0|4|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Cuboctahedron|r{4,3}]] [[File:Uniform_polyhedron-43-t1.png|40px]]<BR>[[Square tiling|rr{4,4}]]40px<BR>[[Triangular prism|{}x{3}]][[File:triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Cantellated square tiling honeycomb verf.png|80px]]<BR>isosceles [[Triangular prism|triangular prism]]
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{R}_3</math>, [4,4,3]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantellated square tiling honeycomb''', rr{4,4,3}, {{CDD|node_1|4|node|4|node_1|3|node}} has [[Cuboctahedron|cuboctahedron]], [[Square tiling|square tiling]], and [[Triangular prism|triangular prism]] facets, with an isosceles [[Triangular prism|triangular prism]] [[Vertex figure|vertex figure]].

[[File:H3 443-1010.png|480px]]
{{Clear}}

=== Cantitruncated square tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantitruncated square tiling honeycomb 
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||tr{4,4,3} or t<sub>0,1,2</sub>{4,4,3}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|4|node_1|4|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated cube|t{4,3}]] [[File:Uniform_polyhedron-43-t01.png|40px]]<BR>[[Truncated square tiling|tr{4,4}]]40px<BR>[[Triangular prism|{}x{3}]] [[File:Triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}<BR>[[Octagon|octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Cantitruncated square tiling honeycomb verf.png|80px]]<BR>isosceles triangular pyramid
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{R}_3</math>, [4,4,3]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantitruncated square tiling honeycomb''', tr{4,4,3}, {{CDD|node_1|4|node_1|4|node_1|3|node}} has [[Truncated cube|truncated cube]], [[Truncated square tiling|truncated square tiling]], and [[Triangular prism|triangular prism]] facets, with an isosceles triangular pyramid [[Vertex figure|vertex figure]].

[[File:H3 443-1110.png|480px]]
{{Clear}}

=== Runcinated square tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcinated square tiling honeycomb 
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,3</sub>{4,4,3}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|node_1|4|node|4|node|3|node_1}}<BR>{{CDD|nodes_11|2a2b-cross|nodes|split2|node_1}} ↔ {{CDD|node_1|4|node_h0|4|node|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[Octahedron|{3,4}]] [[File:Uniform_polyhedron-43-t2.png|40px]]<BR>[[Square tiling|{4,4}]]40px<BR>[[Cube|{}x{4}]] 40px<BR>[[Triangular prism|{}x{3}]] [[File:Triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:runcinated square tiling honeycomb verf.png|80px]]<BR>irregular triangular antiprism
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{R}_3</math>, [4,4,3]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcinated square tiling honeycomb''', t<sub>0,3</sub>{4,4,3}, {{CDD|node_1|4|node|4|node|3|node_1}} has [[Octahedron|octahedron]], [[Triangular prism|triangular prism]], [[Cube|cube]], and [[Square tiling|square tiling]] facets, with an irregular triangular antiprism [[Vertex figure|vertex figure]].

[[File:H3 443-1001.png|480px]]
{{Clear}}

=== Runcitruncated square tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcitruncated square tiling honeycomb 
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||t<sub>0,1,3</sub>{4,4,3}<BR>s<sub>2,3</sub>{3,4,4}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|node_1|4|node_1|4|node|3|node_1}}<BR>{{CDD|node_1|4|node_1|4|node_h|3|node_h}}
|-
|bgcolor=#e7dcc3|Cells||[[Rhombicuboctahedron|rr{4,3}]] [[File:Uniform_polyhedron-43-t02.png|40px]]<BR>[[Truncated square tiling|t{4,4}]]40px<BR>[[Triangular prism|{}x{3}]] 40px<BR>[[Octagonal prism|{}x{8}]] [[File:Octagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}<BR>[[Octagon|octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:runcitruncated square tiling honeycomb verf.png|80px]]<BR>[[Isosceles trapezoid|isosceles-trapezoidal]] [[Pyramid (geometry)|pyramid]]
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{R}_3</math>, [4,4,3]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcitruncated square tiling honeycomb''', t<sub>0,1,3</sub>{4,4,3}, {{CDD|node_1|4|node_1|4|node|3|node_1}} has [[Rhombicuboctahedron|rhombicuboctahedron]], [[Octagonal prism|octagonal prism]], [[Triangular prism|triangular prism]] and [[Truncated square tiling|truncated square tiling]] facets, with an [[Isosceles trapezoid|isosceles-trapezoidal]] [[Pyramid (geometry)|pyramid]] [[Vertex figure|vertex figure]].

[[File:H3 443-1101.png|480px]]
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=== Runcicantellated square tiling honeycomb ===

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

=== Omnitruncated square tiling honeycomb ===

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Omnitruncated square tiling honeycomb 
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,2,3</sub>{4,4,3}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|4|node_1|4|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated square tiling|tr{4,4}]] [[File:Uniform_tiling 44-t012.png|40px]]<BR>[[Hexagonal prism|{}x{6}]] 40px<BR>[[Octagonal prism|{}x{8}]] 40px<BR>[[Truncated cuboctahedron|tr{4,3}]] [[File:Uniform_polyhedron-43-t012.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Square|square]] {4}<BR>[[Hexagon|hexagon]] {6}<BR>[[Octagon|octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Omnitruncated square tiling honeycomb verf.png|80px]]<BR>irregular [[Tetrahedron|tetrahedron]]
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{R}_3</math>, [4,4,3]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''omnitruncated square tiling honeycomb''', t<sub>0,1,2,3</sub>{4,4,3}, {{CDD|node_1|4|node_1|4|node_1|3|node_1}} has [[Truncated square tiling|truncated square tiling]], [[Truncated cuboctahedron|truncated cuboctahedron]], [[Hexagonal prism|hexagonal prism]], and [[Octagonal prism|octagonal prism]] facets, with an irregular [[Tetrahedron|tetrahedron]] [[Vertex figure|vertex figure]].

[[File:H3 443-1111.png|480px]]
{{Clear}}

=== Omnisnub square tiling honeycomb ===

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Omnisnub square tiling honeycomb 
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h(t<sub>0,1,2,3</sub>{4,4,3})
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_h|4|node_h|4|node_h|3|node_h}}
|-
|bgcolor=#e7dcc3|Cells||[[Snub square tiling|sr{4,4}]] [[File:Uniform_tiling 44-snub.png|40px]]<BR>[[Octahedron|sr{2,3}]] 40px<BR>[[Square antiprism|sr{2,4}]] 40px<BR>[[Snub cube|sr{4,3}]] [[File:Uniform_polyhedron-43-s012.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||irregular [[Tetrahedron|tetrahedron]]
|-
|bgcolor=#e7dcc3|Coxeter group||[4,4,3]<sup>+</sup>
|-
|bgcolor=#e7dcc3|Properties||Non-uniform, vertex-transitive
|}
The '''alternated omnitruncated square tiling honeycomb''' (or '''omnisnub square tiling honeycomb'''), h(t<sub>0,1,2,3</sub>{4,4,3}), {{CDD|node_h|4|node_h|4|node_h|3|node_h}} has [[Snub square tiling|snub square tiling]], [[Snub cube|snub cube]], triangular antiprism, [[Square antiprism|square antiprism]], and [[Tetrahedron|tetrahedron]] cells, with an irregular [[Tetrahedron|tetrahedron]] [[Vertex figure|vertex figure]].

{{Clear}}

=== Alternated square tiling honeycomb===

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2| Alternated square tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb<BR>Semiregular honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h{4,4,3}<BR>hr{4,4,4}<BR>{(4,3,3,4)}<BR>h{4<sup>1,1,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter diagrams|| {{CDD|nodes_10ru|split2-44|node|3|node}} ↔ {{CDD|node_h1|4|node|4|node|3|node}}<BR>{{CDD|node|4|node_h1|4|node|4|node}} ↔ {{CDD|nodes_10|2a2b-cross|nodes_10ru|split2-44|node}}<BR>{{CDD|node_1|split1-44|nodes|split2|node}} ↔ {{CDD|node_h0|4|node|split1-43|nodes_10lu}}<BR>{{CDD|node_h|split1-44|nodes|split2-44|node_h}} ↔ {{CDD|node_h0|4|node_h1|4|node|4|node_h0}}<BR>{{CDD|nodes|split2-44|node_h1|4|node}} ↔ {{CDD|node|4|node_h1|4|node|4|node_h0}} ↔ {{CDD|node_1|split1-uu|nodes|2a2b-cross|nodes_11|split2-uu|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Square tiling|{4,4}]] [[File:Uniform tiling 44-t0.svg|40px]]<BR>[[Cube|{4,3}]] [[File:Uniform polyhedron-43-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Square|square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]|||[[File:Uniform polyhedron-43-t1.png|40px]]<BR>[[Cuboctahedron|cuboctahedron]]
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{O}_3</math>, [3,4<sup>1,1</sup>]<BR>[4,1<sup>+</sup>,4,4] ↔ [&infin;,4,4,&infin;]<BR><math>\widehat{BR}_3</math>, [(4,4,3,3)]<BR>[1<sup>+</sup>,4<sup>1,1,1</sup>] ↔ [&infin;<sup>[6]</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive, quasiregular
|}
The '''alternated square tiling honeycomb''', h{4,4,3}, {{CDD|nodes_10ru|split2-44|node|3|node}} is a quasiregular paracompact uniform honeycomb in hyperbolic 3-space. It has [[Cube|cube]] and [[Square tiling|square tiling]] facets in a [[Cuboctahedron|cuboctahedron]] vertex figure.

{{Clear}}

=== Cantic square tiling honeycomb===

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2| Cantic square tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>2</sub>{4,4,3}
|-
|bgcolor=#e7dcc3|Coxeter diagrams|| {{CDD|nodes_10ru|split2-44|node_1|3|node}} ↔ {{CDD|node_h1|4|node|4|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated square tiling|t{4,4}]] [[File:Uniform tiling 44-t01.svg|40px]]<BR>[[Cuboctahedron|r{4,3}]] 40px<BR>[[Truncated cube|t{4,3}]] [[File:Uniform polyhedron-43-t01.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}<BR>[[Octagon|octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]|||[[File:Cantic square tiling honeycomb verf.png|80px]]<BR>[[Rectangle|rectangular]] [[Pyramid (geometry)|pyramid]]
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{O}_3</math>, [3,4<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantic square tiling honeycomb''', h<sub>2</sub>{4,4,3}, {{CDD|nodes_10ru|split2-44|node_1|3|node}} is a paracompact uniform honeycomb in hyperbolic 3-space. It has [[Truncated square tiling|truncated square tiling]], [[Truncated cube|truncated cube]], and [[Cuboctahedron|cuboctahedron]] facets, with a [[Rectangle|rectangular]] [[Pyramid (geometry)|pyramid]] [[Vertex figure|vertex figure]].

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=== Runcic square tiling honeycomb===

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcic square tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>3</sub>{4,4,3}
|-
|bgcolor=#e7dcc3|Coxeter diagrams|| {{CDD|nodes_10ru|split2-44|node|3|node_1}} ↔ {{CDD|node_h1|4|node|4|node|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[Square tiling|{4,4}]] [[File:Uniform tiling 44-t0.svg|40px]]<BR>[[Rhombicuboctahedron|r{4,3}]] 40px<BR>[[Octahedron|{3,4}]] [[File:Uniform polyhedron-43-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]|||[[File:Runcic square tiling honeycomb verf.png|80px]]<BR>[[Square|square]] [[Frustum|frustum]]
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{O}_3</math>, [3,4<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcic square tiling honeycomb''', h<sub>3</sub>{4,4,3}, {{CDD|nodes_10ru|split2-44|node|3|node_1}} is a paracompact uniform honeycomb in hyperbolic 3-space. It has [[Square tiling|square tiling]], [[Rhombicuboctahedron|rhombicuboctahedron]], and [[Octahedron|octahedron]] facets in a [[Square|square]] [[Frustum|frustum]] vertex figure.

{{Clear}}

=== Runcicantic square tiling honeycomb===

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2| Runcicantic square tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>2,3</sub>{4,4,3}
|-
|bgcolor=#e7dcc3|Coxeter diagrams|| {{CDD|nodes_10ru|split2-44|node_1|3|node_1}} ↔ {{CDD|node_h1|4|node|4|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated square tiling|t{4,4}]] [[File:Uniform tiling 44-t01.svg|40px]]<BR>[[Truncated cuboctahedron|tr{4,3}]] 40px<BR>[[Truncated octahedron|t{3,4}]] [[File:Uniform polyhedron-43-t12.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Square|square]] {4}<BR>[[Hexagon|hexagon]] {6}<BR>[[Octagon|octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]|||[[File:Runcicantic square tiling honeycomb verf.png|80px]]<BR>mirrored sphenoid
|-
|bgcolor=#e7dcc3|Coxeter groups||<math>\overline{O}_3</math>, [3,4<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcicantic square tiling honeycomb''', h<sub>2,3</sub>{4,4,3}, {{CDD|nodes_10ru|split2-44|node_1|3|node_1}} ↔ {{CDD|node_h1|4|node|4|node_1|3|node_1}}, is a paracompact uniform honeycomb in hyperbolic 3-space. It has [[Truncated square tiling|truncated square tiling]], [[Truncated cuboctahedron|truncated cuboctahedron]], and [[Truncated octahedron|truncated octahedron]] facets in a mirrored sphenoid [[Vertex figure|vertex figure]].

{{Clear}}

=== Alternated rectified square tiling honeycomb===

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2| Alternated rectified square tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||hr{4,4,3}
|-
|bgcolor=#e7dcc3|Coxeter diagrams|| {{CDD|node|4|node_h1|4|node|3|node}} ↔ {{CDD|nodes_10|2a2b-cross|nodes_10ru|split2|node}}
|-
|bgcolor=#e7dcc3|Cells|||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|[[Vertex figure]]|||[[Triangular prism|triangular prism]]
|-
|bgcolor=#e7dcc3|Coxeter groups||[4,1<sup>+</sup>,4,3] = [∞,3,3,∞]
|-
|bgcolor=#e7dcc3|Properties||Nonsimplectic, vertex-transitive
|}
The '''alternated rectified square tiling honeycomb''' is a paracompact uniform honeycomb in hyperbolic 3-space.

{{Clear}}

== See also ==
* Convex uniform honeycombs in hyperbolic space
* Regular tessellations of hyperbolic 3-space
* Paracompact uniform honeycombs

== References ==
{{reflist}}
*Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. {{ISBN|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;294–296)
* ''The Beauty of Geometry: Twelve Essays'' (1999), Dover Publications, {{LCCN|99035678}}, {{ISBN|0-486-40919-8}} (Chapter 10, [http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf Regular Honeycombs in Hyperbolic Space]) Table III
* [[Biography:Jeffrey Weeks (mathematician)|Jeffrey R. Weeks]] ''The Shape of Space, 2nd edition'' {{ISBN|0-8247-0709-5}} (Chapter 16-17: Geometries on Three-manifolds I, II)
* [[Biography:Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript
** [[Biography:Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 
** N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 13: Hyperbolic Coxeter groups
** Norman W. Johnson and Asia Ivic Weiss [https://cms.math.ca/cjm/v51/weisscox8.pdf Quadratic Integers and Coxeter Groups] [[PDF]] Can. J. Math. Vol. 51 (6), 1999 pp.&nbsp;1307–1336

[[Category:Honeycombs (geometry)]]
[[Category:Square tilings]]

{{Sourceattribution|Square tiling honeycomb}}