Square tiling honeycomb

Square tiling honeycomb
File:H3 443 FC boundary.png
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{4,4,3} r{4,4,4} {41,1,1}
Coxeter diagrams	Template:CDD Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	{4,4} File:Square tiling uniform coloring 1.svg Error creating thumbnail: Error creating thumbnail:
Faces	square {4}
Edge figure	triangle {3}
Vertex figure	File:Square tiling honeycomb verf.png cube, {4,3}
Dual	Order-4 octahedral honeycomb
Coxeter groups	${\bar{R}}_3$, [4,4,3] ${\bar{N}}_3$, [43] ${\bar{M}}_3$, [41,1,1]
Properties	Regular

In the geometry of 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 horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.

Rectified order-4 square tiling

It is also seen as a rectified order-4 square tiling honeycomb, r{4,4,4}:

{4,4,4}	r{4,4,4} = {4,4,3}
Template:CDD	Template:CDD = Template:CDD

Symmetry

The square tiling honeycomb has three reflective symmetry constructions: Template:CDD as a regular honeycomb, a half symmetry construction Template:CDD ↔ Template:CDD, and lastly a construction with three types (colors) of checkered square tilings Template:CDD ↔ Template:CDD.

It also contains an index 6 subgroup [4,4,3^*] ↔ [4^1,1,1], and a radial subgroup [4,(4,3)^*] of index 48, with a right dihedral-angled octahedral fundamental domain, and four pairs of ultraparallel mirrors: Template:CDD.

This honeycomb contains Template:CDD that tile 2-hypercycle surfaces, which are similar to the paracompact order-3 apeirogonal tiling Template:CDD:

Error creating thumbnail:

Related polytopes and honeycombs

The square tiling honeycomb is a regular hyperbolic honeycomb in 3-space. It is one of eleven regular paracompact honeycombs. Template:Regular paracompact H3 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}. Template:443 family

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. Template:444 family

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: Template:Square tiling tessellations

Rectified 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	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	{4,3} Error creating thumbnail: r{4,4}
Faces	square {4}
Vertex figure	triangular prism
Coxeter groups	${\bar{R}}_3$, [4,4,3] ${\bar{O}}_3$, [3,41,1] ${\bar{M}}_3$, [41,1,1]
Properties	Vertex-transitive, edge-transitive

The rectified square tiling honeycomb, t₁{4,4,3}, Template:CDD has cube and square tiling facets, with a triangular prism vertex figure.

File:H3 443 boundary 0100.png

It is similar to the 2D hyperbolic uniform triapeirogonal tiling, r{∞,3}, with triangle and apeirogonal faces.

File:H2 tiling 23i-2.png

Truncated square tiling honeycomb

Truncated square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t{4,4,3} or t0,1{4,4,3}
Coxeter diagrams	Template:CDD Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	{4,3} Error creating thumbnail: t{4,4}File:Uniform tiling 44-t01.png
Faces	square {4} octagon {8}
Vertex figure	File:Truncated square tiling honeycomb verf.png triangular pyramid
Coxeter groups	${\bar{R}}_3$, [4,4,3] ${\bar{N}}_3$, [43] ${\bar{M}}_3$, [41,1,1]
Properties	Vertex-transitive

The truncated square tiling honeycomb, t{4,4,3}, Template:CDD 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}, Template:CDD.

File:H3 443-1100.png Template:Clear

Bitruncated square tiling honeycomb

Bitruncated square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	2t{4,4,3} or t1,2{4,4,3}
Coxeter diagram	Template:CDD
Cells	t{4,3} File:Uniform polyhedron-43-t01.png t{4,4}File:Uniform tiling 44-t01.png
Faces	triangle {3} square {4} octagon {8}
Vertex figure	File:Bitruncated square tiling honeycomb verf.png digonal disphenoid
Coxeter groups	${\bar{R}}_3$, [4,4,3]
Properties	Vertex-transitive

The bitruncated square tiling honeycomb, 2t{4,4,3}, Template:CDD has truncated cube and truncated square tiling facets, with a digonal disphenoid vertex figure.

Error creating thumbnail: Template:Clear

Cantellated square tiling honeycomb

Cantellated square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	rr{4,4,3} or t0,2{4,4,3}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	r{4,3} Error creating thumbnail: rr{4,4}File:Uniform tiling 44-t02.svg {}x{3}File:Triangular prism.png
Faces	triangle {3} square {4}
Vertex figure	File:Cantellated square tiling honeycomb verf.png isosceles triangular prism
Coxeter groups	${\bar{R}}_3$, [4,4,3]
Properties	Vertex-transitive

The cantellated square tiling honeycomb, rr{4,4,3}, Template:CDD has cuboctahedron, square tiling, and triangular prism facets, with an isosceles triangular prism vertex figure.

File:H3 443-1010.png Template:Clear

Cantitruncated square tiling honeycomb

Cantitruncated square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	tr{4,4,3} or t0,1,2{4,4,3}
Coxeter diagram	Template:CDD
Cells	t{4,3} File:Uniform polyhedron-43-t01.png tr{4,4}File:Uniform tiling 44-t012.png {}x{3} File:Triangular prism.png
Faces	triangle {3} square {4} octagon {8}
Vertex figure	File:Cantitruncated square tiling honeycomb verf.png isosceles triangular pyramid
Coxeter groups	${\bar{R}}_3$, [4,4,3]
Properties	Vertex-transitive

The cantitruncated square tiling honeycomb, tr{4,4,3}, Template:CDD has truncated cube, truncated square tiling, and triangular prism facets, with an isosceles triangular pyramid vertex figure.

File:H3 443-1110.png Template:Clear

Runcinated square tiling honeycomb

Runcinated square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,3{4,4,3}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	{3,4} File:Uniform polyhedron-43-t2.png {4,4}File:Uniform tiling 44-t0.svg {}x{4} File:Tetragonal prism.png {}x{3} File:Triangular prism.png
Faces	triangle {3} square {4}
Vertex figure	File:Runcinated square tiling honeycomb verf.png irregular triangular antiprism
Coxeter groups	${\bar{R}}_3$, [4,4,3]
Properties	Vertex-transitive

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

File:H3 443-1001.png Template:Clear

Runcitruncated square tiling honeycomb

Runcitruncated square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t0,1,3{4,4,3} s2,3{3,4,4}
Coxeter diagrams	Template:CDD Template:CDD
Cells	rr{4,3} File:Uniform polyhedron-43-t02.png t{4,4}File:Uniform tiling 44-t01.png {}x{3} File:Triangular prism.png {}x{8} File:Octagonal prism.png
Faces	triangle {3} square {4} octagon {8}
Vertex figure	Error creating thumbnail: isosceles-trapezoidal pyramid
Coxeter groups	${\bar{R}}_3$, [4,4,3]
Properties	Vertex-transitive

The runcitruncated square tiling honeycomb, t_0,1,3{4,4,3}, Template:CDD has rhombicuboctahedron, octagonal prism, triangular prism and truncated square tiling facets, with an isosceles-trapezoidal pyramid vertex figure.

File:H3 443-1101.png Template:Clear

Runcicantellated square tiling honeycomb

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

Omnitruncated square tiling honeycomb

Omnitruncated square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,2,3{4,4,3}
Coxeter diagram	Template:CDD
Cells	tr{4,4} File:Uniform tiling 44-t012.png {}x{6} Error creating thumbnail: {}x{8} File:Octagonal prism.png tr{4,3} File:Uniform polyhedron-43-t012.png
Faces	square {4} hexagon {6} octagon {8}
Vertex figure	Error creating thumbnail: irregular tetrahedron
Coxeter groups	${\bar{R}}_3$, [4,4,3]
Properties	Vertex-transitive

The omnitruncated square tiling honeycomb, t_0,1,2,3{4,4,3}, Template:CDD has truncated square tiling, truncated cuboctahedron, hexagonal prism, and octagonal prism facets, with an irregular tetrahedron vertex figure.

Error creating thumbnail: Template:Clear

Omnisnub square tiling honeycomb

Omnisnub square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h(t0,1,2,3{4,4,3})
Coxeter diagram	Template:CDD
Cells	sr{4,4} Error creating thumbnail: sr{2,3} Error creating thumbnail: sr{2,4} File:Square antiprism.png sr{4,3} File:Uniform polyhedron-43-s012.png
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}), Template:CDD has snub square tiling, snub cube, triangular antiprism, square antiprism, and tetrahedron cells, with an irregular tetrahedron vertex figure.

Alternated square tiling honeycomb

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	Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD ↔ Template:CDD
Cells	{4,4} File:Uniform tiling 44-t0.svg {4,3}
Faces	square {4}
Vertex figure	Error creating thumbnail: cuboctahedron
Coxeter groups	${\bar{O}}_3$, [3,41,1] [4,1+,4,4] ↔ [∞,4,4,∞] ${\widehat{BR}}_3$, [(4,4,3,3)] [1+,41,1,1] ↔ [∞[6]]
Properties	Vertex-transitive, edge-transitive, quasiregular

The alternated square tiling honeycomb, h{4,4,3}, Template:CDD 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

Cantic square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h2{4,4,3}
Coxeter diagrams	Template:CDD ↔ Template:CDD
Cells	t{4,4} Error creating thumbnail: r{4,3} Error creating thumbnail: t{4,3} File:Uniform polyhedron-43-t01.png
Faces	triangle {3} square {4} octagon {8}
Vertex figure	rectangular pyramid
Coxeter groups	${\bar{O}}_3$, [3,41,1]
Properties	Vertex-transitive

The cantic square tiling honeycomb, h₂{4,4,3}, Template:CDD 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

Runcic square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h3{4,4,3}
Coxeter diagrams	Template:CDD ↔ Template:CDD
Cells	{4,4} File:Uniform tiling 44-t0.svg r{4,3} File:Uniform polyhedron-43-t02.png {3,4} File:Uniform polyhedron-43-t2.png
Faces	triangle {3} square {4}
Vertex figure	Error creating thumbnail: square frustum
Coxeter groups	${\bar{O}}_3$, [3,41,1]
Properties	Vertex-transitive

The runcic square tiling honeycomb, h₃{4,4,3}, Template:CDD 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

Runcicantic square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h2,3{4,4,3}
Coxeter diagrams	Template:CDD ↔ Template:CDD
Cells	t{4,4} tr{4,3} File:Uniform polyhedron-43-t012.png t{3,4} File:Uniform polyhedron-43-t12.png
Faces	square {4} hexagon {6} octagon {8}
Vertex figure	Error creating thumbnail: mirrored sphenoid
Coxeter groups	${\bar{O}}_3$, [3,41,1]
Properties	Vertex-transitive

The runcicantic square tiling honeycomb, h_2,3{4,4,3}, Template:CDD ↔ Template:CDD, 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

Alternated rectified square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	hr{4,4,3}
Coxeter diagrams	Template:CDD ↔ Template:CDD
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.

See also

• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• Paracompact uniform honeycombs

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. Template:ISBN. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, Template:LCCN, Template:ISBN (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition Template:ISBN (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 Quadratic Integers and Coxeter Groups PDF Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336