order-4 square tiling honeycomb

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Order-4 square tiling honeycomb
bgcolor=#ffffff align=center colspan=2\|320px
bgcolor=#e7dcc3\|Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	{4,4,4} h{4,4,4} ↔ {4,4^1,1} {4^[4]}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|4\|node\|4\|node\|4\|node}} {{CDD\|node_1\|4\|node\|4\|node\|4\|node_h0}} ↔ {{CDD\|node_1\|4\|node\|split1-44\|nodes}} ↔ {{CDD\|node_h1\|4\|node\|4\|node\|4\|node}} {{CDD\|nodes_10ru\|split2-44\|node\|4\|node_h0}} ↔ {{CDD\|node_h1\|4\|node\|4\|node\|4\|node_h0}} ↔ {{CDD\|node\|split1-44\|nodes_10luru\|split2-44\|node}} {{CDD\|node_1\|4\|node\|4\|node_h0\|4\|node}} ↔ {{CDD\|node_1\|split1-44\|nodes\|2a2b-cross\|nodes}} {{CDD\|node_1\|4\|node_h0\|4\|node\|4\|node}} ↔ {{CDD\|nodes_11\|2a2b-cross\|nodes\|split2-44\|node}} {{CDD\|node_1\|4\|node_h0\|4\|node\|4\|node_h0}} ↔ {{CDD\|node_1\|split1-uu\|nodes\|2a2b-cross\|nodes\|split2-uu\|node_1}} {{CDD\|node_1\|4\|node\|4\|node_g\|4sg\|node_g}} ↔ {{CDD\|branchu\|split2-44\|node_1\|split1-44\|branchu}} {{CDD\|node_1\|4\|node_g\|4sg\|node_g\|4\|node}} ↔ {{CDD\|branchu_10\|2\|branchu_10\|2\|branchu_10\|2\|branchu_10}}
bgcolor=#e7dcc3\|Cells	{4,4} 40px 40px 40px 40px
bgcolor=#e7dcc3\|Faces	square {4}
bgcolor=#e7dcc3\|Edge figure	square {4}
bgcolor=#e7dcc3\|Vertex figure	square tiling, {4,4} 40px 40px 40px 40px
bgcolor=#e7dcc3\|Dual	Self-dual
bgcolor=#e7dcc3\|Coxeter groups	$\overline{N}_3$ , [4,4,4] $\overline{M}_3$ , [4^1,1,1] $\widehat{RR}_3$ , [4^[4]]
bgcolor=#e7dcc3\|Properties	Regular, quasiregular

In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {4,4,4}, it has four square tilings around each edge, and infinite square tilings around each vertex in a square tiling vertex figure.Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III

Symmetry

The order-4 square tiling honeycomb has many reflective symmetry constructions: {{CDD|node_1|4|node|4|node|4|node}} as a regular honeycomb, {{CDD|node_1|4|node|4|node|4|node_h0}} ↔ {{CDD|node_1|4|node|split1-44|nodes}} with alternating types (colors) of square tilings, and {{CDD|node_1|split1-44|nodes|split2-44|node}} with 3 types (colors) of square tilings in a ratio of 2:1:1.

Two more half symmetry constructions with pyramidal domains have [4,4,1⁺,4] symmetry: {{CDD|node_1|4|node|4|node_h0|4|node}} ↔ {{CDD|node_1|split1-44|nodes|2a2b-cross|nodes}}, and {{CDD|node_1|4|node_h0|4|node|4|node}} ↔ {{CDD|nodes_11|2a2b-cross|nodes|split2-44|node}}.

There are two high-index subgroups, both index 8: [4,4,4^*] ↔ [(4,4,4,4,1⁺)], with a pyramidal fundamental domain: [((4,∞,4)),((4,∞,4))] or {{CDD|branchu|split2-44|node_1|split1-44|branchu}}; and [4,4^*,4], with 4 orthogonal sets of ultra-parallel mirrors in an octahedral fundamental domain: {{CDD|branchu_10|2|branchu_10|2|branchu_10|2|branchu_10}}.

Images

The order-4 square tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface.

: 160px

It contains {{CDD|node|4|node|ultra|node_1}} and {{CDD|node_1|4|node|ultra|node}} that tile 2-hypercycle surfaces, which are similar to these paracompact order-4 apeirogonal tilings {{CDD|node|4|node|infin|node_1}}:

: 120px 120px

Related polytopes and honeycombs

The order-4 square tiling honeycomb is a regular hyperbolic honeycomb in 3-space. It is one of eleven regular paracompact honeycombs.

There are nine uniform honeycombs in the [4,4,4] Coxeter group family, including this regular form.

It is part of a sequence of honeycombs with a square tiling vertex figure:

It is part of a sequence of honeycombs with square tiling cells:

It is part of a sequence of quasiregular polychora and honeycombs:

= Rectified order-4 square tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Rectified order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	r{4,4,4} or t₁{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node\|4\|node_1\|4\|node\|4\|node}} {{CDD\|node\|4\|node_1\|4\|node\|4\|node_h0}} ↔ {{CDD\|node\|4\|node_1\|split1-44\|nodes}}
bgcolor=#e7dcc3\|Cells	{4,4} 40px r{4,4} 40px
bgcolor=#e7dcc3\|Faces	square {4}
bgcolor=#e7dcc3\|Vertex figure	80px cube
bgcolor=#e7dcc3\|Coxeter groups	$\overline{N}_3$ , [4,4,4] $\overline{M}_3$ , [4^1,1,1]
bgcolor=#e7dcc3\|Properties	Quasiregular or regular, depending on symmetry

The rectified order-4 hexagonal tiling honeycomb, t₁{4,4,4}, {{CDD|node|4|node_1|4|node|4|node}} has square tiling facets, with a cubic vertex figure. It is the same as the regular square tiling honeycomb, {4,4,3}, {{CDD|node_1|4|node|4|node|3|node}}.

480px

= Truncated order-4 square tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Truncated order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	t{4,4,4} or t_0,1{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|4\|node_1\|4\|node\|4\|node}} {{CDD\|node_1\|4\|node_1\|4\|node\|4\|node_h0}} ↔ {{CDD\|node_1\|4\|node_1\|split1-44\|nodes}} {{CDD\|node_1\|4\|node_1\|4\|node_h0\|4\|node}} ↔ {{CDD\|node_1\|split1-44\|nodes_11\|2a2b-cross\|nodes}} {{CDD\|node_1\|4\|node_1\|4\|node_g\|4sg\|node_g}} ↔ {{CDD\|branchu_11\|split2-44\|node_1\|split1-44\|branchu_11}}
bgcolor=#e7dcc3\|Cells	{4,4} 40px t{4,4} 40px
bgcolor=#e7dcc3\|Faces	square {4} octagon {8}
bgcolor=#e7dcc3\|Vertex figure	80px square pyramid
bgcolor=#e7dcc3\|Coxeter groups	$\overline{N}_3$ , [4,4,4] $\overline{M}_3$ , [4^1,1,1]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The truncated order-4 square tiling honeycomb, t_0,1{4,4,4}, {{CDD|node_1|4|node_1|4|node|4|node}} has square tiling and truncated square tiling facets, with a square pyramid vertex figure.

480px

= Bitruncated order-4 square tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Bitruncated order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	2t{4,4,4} or t_1,2{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node\|4\|node_1\|4\|node_1\|4\|node}} {{CDD\|node\|4\|node_1\|split1-44\|nodes_11}} ↔ {{CDD\|node\|4\|node_1\|4\|node_1\|4\|node_h0}} {{CDD\|node_1\|split1-44\|nodes_11\|split2-44\|node_1}} ↔ {{CDD\|node_h0\|4\|node_1\|split1-44\|nodes_11}} ↔ {{CDD\|node_h0\|4\|node_1\|4\|node_1\|4\|node_h0}}
bgcolor=#e7dcc3\|Cells	t{4,4} 40px
bgcolor=#e7dcc3\|Faces	square {4} octagon {8}
bgcolor=#e7dcc3\|Vertex figure	80px tetragonal disphenoid
bgcolor=#e7dcc3\|Coxeter groups	$2\times\overline{N}_3$ , 4,4,4 $\overline{M}_3$ , [4^1,1,1] $\widehat{RR}_3$ , [4^[4]]
bgcolor=#e7dcc3\|Properties	Vertex-transitive, edge-transitive, cell-transitive

The bitruncated order-4 square tiling honeycomb, t_1,2{4,4,4}, {{CDD|node|4|node_1|4|node_1|4|node}} has truncated square tiling facets, with a tetragonal disphenoid vertex figure.

480px

= Cantellated order-4 square tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Cantellated order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	rr{4,4,4} or t_0,2{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|4\|node\|4\|node_1\|4\|node}} {{CDD\|node\|4\|node_1\|4\|node\|3\|node}}
bgcolor=#e7dcc3\|Cells	{}x{4} 40px r{4,4} 40px rr{4,4} 40px
bgcolor=#e7dcc3\|Faces	square {4}
bgcolor=#e7dcc3\|Vertex figure	80px triangular prism
bgcolor=#e7dcc3\|Coxeter groups	$\overline{N}_3$ , [4,4,4] $\overline{R}_3$ , [3,4,4]
bgcolor=#e7dcc3\|Properties	Vertex-transitive, edge-transitive

The cantellated order-4 square tiling honeycomb, {{CDD|node_1|4|node|4|node_1|4|node}} is the same thing as the rectified square tiling honeycomb, {{CDD|node|4|node_1|4|node|3|node}}. It has cube and square tiling facets, with a triangular prism vertex figure.

480px

= Cantitruncated order-4 square tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Cantitruncated order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	tr{4,4,4} or t_0,1,2{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|4\|node_1\|4\|node_1\|4\|node}} {{CDD\|node_1\|4\|node_1\|4\|node\|3\|node}} {{CDD\|node_1\|4\|node_1\|split1-44\|nodes_11}} ↔ {{CDD\|node_1\|4\|node_1\|4\|node_1\|4\|node_h0}}
bgcolor=#e7dcc3\|Cells	{}x{4} 40px tr{4,4} 40px t{4,4} 40px
bgcolor=#e7dcc3\|Faces	square {4} octagon {8}
bgcolor=#e7dcc3\|Vertex figure	80px mirrored sphenoid
bgcolor=#e7dcc3\|Coxeter groups	$\overline{N}_3$ , [4,4,4] $\overline{R}_3$ , [3,4,4] $\overline{M}_3$ , [4^1,1,1]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The cantitruncated order-4 square tiling honeycomb, {{CDD|node_1|4|node_1|4|node_1|4|node}} is the same as the truncated square tiling honeycomb, {{CDD|node_1|4|node_1|4|node|3|node}}. It contains cube and truncated square tiling facets, with a mirrored sphenoid vertex figure.

It is the same as the truncated square tiling honeycomb, {{CDD|node_1|4|node_1|4|node|3|node}}.

480px

= Runcinated order-4 square tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Runcinated order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	t_0,3{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|4\|node\|4\|node\|4\|node_1}} {{CDD\|node_1\|4\|node\|4\|node_h0\|4\|node_1}} ↔ {{CDD\|node_1\|split1-44\|nodes\|2a2b-cross\|nodes_11}} {{CDD\|node_1\|4\|node_g\|4sg\|node_g\|4\|node_1}} ↔ {{CDD\|branchu_11\|2\|branchu_11\|2\|branchu_11\|2\|branchu_11}}
bgcolor=#e7dcc3\|Cells	{4,4} 40px {}x{4} 40px
bgcolor=#e7dcc3\|Faces	square {4}
bgcolor=#e7dcc3\|Vertex figure	80px square antiprism
bgcolor=#e7dcc3\|Coxeter groups	$2\times\overline{N}_3$ , 4,4,4
bgcolor=#e7dcc3\|Properties	Vertex-transitive, edge-transitive

The runcinated order-4 square tiling honeycomb, t_0,3{4,4,4}, {{CDD|node_1|4|node|4|node|4|node_1}} has square tiling and cube facets, with a square antiprism vertex figure.

480px

= Runcitruncated order-4 square tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Runcitruncated order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	t_0,1,3{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|4\|node_1\|4\|node\|4\|node_1}} {{CDD\|node_1\|4\|node_1\|4\|node_h0\|4\|node_1}} ↔ {{CDD\|node_1\|split1-44\|nodes_11\|2a2b-cross\|nodes_11}}
bgcolor=#e7dcc3\|Cells	t{4,4} 40px rr{4,4} 40px {}x{4} 40px {8}x{} 40px
bgcolor=#e7dcc3\|Faces	square {4} octagon {8}
bgcolor=#e7dcc3\|Vertex figure	80px square pyramid
bgcolor=#e7dcc3\|Coxeter groups	$\overline{N}_3$ , [4,4,4]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The runcitruncated order-4 square tiling honeycomb, t_0,1,3{4,4,4}, {{CDD|node_1|4|node_1|4|node|4|node_1}} has square tiling, truncated square tiling, cube, and octagonal prism facets, with a square pyramid vertex figure.

The runcicantellated order-4 square tiling honeycomb is equivalent to the runcitruncated order-4 square tiling honeycomb.

480px

= Omnitruncated order-4 square tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Omnitruncated order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	t_0,1,2,3{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|4\|node_1\|4\|node_1\|4\|node_1}}
bgcolor=#e7dcc3\|Cells	tr{4,4} 40px {8}x{} 40px
bgcolor=#e7dcc3\|Faces	square {4} octagon {8}
bgcolor=#e7dcc3\|Vertex figure	80px digonal disphenoid
bgcolor=#e7dcc3\|Coxeter groups	$2\times\overline{N}_3$ , 4,4,4
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The omnitruncated order-4 square tiling honeycomb, t_0,1,2,3{4,4,4}, {{CDD|node_1|4|node_1|4|node_1|4|node_1}} has truncated square tiling and octagonal prism facets, with a digonal disphenoid vertex figure.

480px

= Alternated order-4 square tiling honeycomb =

The alternated order-4 square tiling honeycomb is a lower-symmetry construction of the order-4 square tiling honeycomb itself.

= Cantic order-4 square tiling honeycomb =

The cantic order-4 square tiling honeycomb is a lower-symmetry construction of the truncated order-4 square tiling honeycomb.

= Runcic order-4 square tiling honeycomb =

The runcic order-4 square tiling honeycomb is a lower-symmetry construction of the order-3 square tiling honeycomb.

= Runcicantic order-4 square tiling honeycomb =

The runcicantic order-4 square tiling honeycomb is a lower-symmetry construction of the bitruncated order-4 square tiling honeycomb.

= Quarter order-4 square tiling honeycomb=

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Quarter order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	q{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_h1\|4\|node\|4\|node\|4\|node_h1}} {{CDD\|label4\|branch_10r\|4a4b\|branch_10l\|label4}}
bgcolor=#e7dcc3\|Cells	t{4,4} 40px {4,4} 40px
bgcolor=#e7dcc3\|Faces	square {4} octagon {8}
bgcolor=#e7dcc3\|Vertex figure	80px square antiprism
bgcolor=#e7dcc3\|Coxeter groups	$\widehat{RR}_3$ , [4^[4]]
bgcolor=#e7dcc3\|Properties	Vertex-transitive, edge-transitive

The quarter order-4 square tiling honeycomb, q{4,4,4}, {{CDD|label4|branch_10r|4a4b|branch_10l|label4}}, or {{CDD|node_h1|4|node|4|node|4|node_h1}}, has truncated square tiling and square tiling facets, with a square antiprism vertex figure.

References

Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp. 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
Jeffrey R. Weeks The Shape of Space, 2nd edition {{isbn|0-8247-0709-5}} (Chapter 16-17: Geometries on Three-manifolds I, II)
Norman Johnson Uniform Polytopes, Manuscript
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. 1307–1336

Category:Regular 3-honeycombs

Category:Self-dual tilings

Category:Square tilings

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Order-4 square tiling honeycomb
bgcolor=#ffffff align=center colspan=2\|320px
bgcolor=#e7dcc3\|Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	{4,4,4} h{4,4,4} ↔ {4,4^1,1} {4^[4]}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|4\|node\|4\|node\|4\|node}} {{CDD\|node_1\|4\|node\|4\|node\|4\|node_h0}} ↔ {{CDD\|node_1\|4\|node\|split1-44\|nodes}} ↔ {{CDD\|node_h1\|4\|node\|4\|node\|4\|node}} {{CDD\|nodes_10ru\|split2-44\|node\|4\|node_h0}} ↔ {{CDD\|node_h1\|4\|node\|4\|node\|4\|node_h0}} ↔ {{CDD\|node\|split1-44\|nodes_10luru\|split2-44\|node}} {{CDD\|node_1\|4\|node\|4\|node_h0\|4\|node}} ↔ {{CDD\|node_1\|split1-44\|nodes\|2a2b-cross\|nodes}} {{CDD\|node_1\|4\|node_h0\|4\|node\|4\|node}} ↔ {{CDD\|nodes_11\|2a2b-cross\|nodes\|split2-44\|node}} {{CDD\|node_1\|4\|node_h0\|4\|node\|4\|node_h0}} ↔ {{CDD\|node_1\|split1-uu\|nodes\|2a2b-cross\|nodes\|split2-uu\|node_1}} {{CDD\|node_1\|4\|node\|4\|node_g\|4sg\|node_g}} ↔ {{CDD\|branchu\|split2-44\|node_1\|split1-44\|branchu}} {{CDD\|node_1\|4\|node_g\|4sg\|node_g\|4\|node}} ↔ {{CDD\|branchu_10\|2\|branchu_10\|2\|branchu_10\|2\|branchu_10}}
bgcolor=#e7dcc3\|Cells	{4,4} 40px 40px 40px 40px
bgcolor=#e7dcc3\|Faces	square {4}
bgcolor=#e7dcc3\|Edge figure	square {4}
bgcolor=#e7dcc3\|Vertex figure	square tiling, {4,4} 40px 40px 40px 40px
bgcolor=#e7dcc3\|Dual	Self-dual
bgcolor=#e7dcc3\|Coxeter groups	$\overline{N}_3$ , [4,4,4] $\overline{M}_3$ , [4^1,1,1] $\widehat{RR}_3$ , [4^[4]]
bgcolor=#e7dcc3\|Properties	Regular, quasiregular

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Rectified order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	r{4,4,4} or t₁{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node\|4\|node_1\|4\|node\|4\|node}} {{CDD\|node\|4\|node_1\|4\|node\|4\|node_h0}} ↔ {{CDD\|node\|4\|node_1\|split1-44\|nodes}}
bgcolor=#e7dcc3\|Cells	{4,4} 40px r{4,4} 40px
bgcolor=#e7dcc3\|Faces	square {4}
bgcolor=#e7dcc3\|Vertex figure	80px cube
bgcolor=#e7dcc3\|Coxeter groups	$\overline{N}_3$ , [4,4,4] $\overline{M}_3$ , [4^1,1,1]
bgcolor=#e7dcc3\|Properties	Quasiregular or regular, depending on symmetry

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Truncated order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	t{4,4,4} or t_0,1{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|4\|node_1\|4\|node\|4\|node}} {{CDD\|node_1\|4\|node_1\|4\|node\|4\|node_h0}} ↔ {{CDD\|node_1\|4\|node_1\|split1-44\|nodes}} {{CDD\|node_1\|4\|node_1\|4\|node_h0\|4\|node}} ↔ {{CDD\|node_1\|split1-44\|nodes_11\|2a2b-cross\|nodes}} {{CDD\|node_1\|4\|node_1\|4\|node_g\|4sg\|node_g}} ↔ {{CDD\|branchu_11\|split2-44\|node_1\|split1-44\|branchu_11}}
bgcolor=#e7dcc3\|Cells	{4,4} 40px t{4,4} 40px
bgcolor=#e7dcc3\|Faces	square {4} octagon {8}
bgcolor=#e7dcc3\|Vertex figure	80px square pyramid
bgcolor=#e7dcc3\|Coxeter groups	$\overline{N}_3$ , [4,4,4] $\overline{M}_3$ , [4^1,1,1]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Bitruncated order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	2t{4,4,4} or t_1,2{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node\|4\|node_1\|4\|node_1\|4\|node}} {{CDD\|node\|4\|node_1\|split1-44\|nodes_11}} ↔ {{CDD\|node\|4\|node_1\|4\|node_1\|4\|node_h0}} {{CDD\|node_1\|split1-44\|nodes_11\|split2-44\|node_1}} ↔ {{CDD\|node_h0\|4\|node_1\|split1-44\|nodes_11}} ↔ {{CDD\|node_h0\|4\|node_1\|4\|node_1\|4\|node_h0}}
bgcolor=#e7dcc3\|Cells	t{4,4} 40px
bgcolor=#e7dcc3\|Faces	square {4} octagon {8}
bgcolor=#e7dcc3\|Vertex figure	80px tetragonal disphenoid
bgcolor=#e7dcc3\|Coxeter groups	$2\times\overline{N}_3$ , 4,4,4 $\overline{M}_3$ , [4^1,1,1] $\widehat{RR}_3$ , [4^[4]]
bgcolor=#e7dcc3\|Properties	Vertex-transitive, edge-transitive, cell-transitive

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Cantellated order-4 square tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	rr{4,4,4} or t_0,2{4,4,4}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|4\|node\|4\|node_1\|4\|node}} {{CDD\|node\|4\|node_1\|4\|node\|3\|node}}
bgcolor=#e7dcc3\|Cells	{}x{4} 40px r{4,4} 40px rr{4,4} 40px
bgcolor=#e7dcc3\|Faces	square {4}
bgcolor=#e7dcc3\|Vertex figure	80px triangular prism
bgcolor=#e7dcc3\|Coxeter groups	$\overline{N}_3$ , [4,4,4] $\overline{R}_3$ , [3,4,4]
bgcolor=#e7dcc3\|Properties	Vertex-transitive, edge-transitive

order-4 square tiling honeycomb

Symmetry

Images

Related polytopes and honeycombs

= Rectified order-4 square tiling honeycomb =

= Truncated order-4 square tiling honeycomb =

= Bitruncated order-4 square tiling honeycomb =

= Cantellated order-4 square tiling honeycomb =

= Cantitruncated order-4 square tiling honeycomb =

= Runcinated order-4 square tiling honeycomb =

= Runcitruncated order-4 square tiling honeycomb =

= Omnitruncated order-4 square tiling honeycomb =

= Alternated order-4 square tiling honeycomb =

= Cantic order-4 square tiling honeycomb =

= Runcic order-4 square tiling honeycomb =

= Runcicantic order-4 square tiling honeycomb =

= Quarter order-4 square tiling honeycomb=

See also

References