Order-4 square tiling honeycomb

Order-4 square tiling honeycomb
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{4,4,4} h{4,4,4} ↔ {4,41,1} {4[4]}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD ↔ 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}
Faces	square {4}
Edge figure	square {4}
Vertex figure	square tiling, {4,4}
Dual	Self-dual
Coxeter groups	${\bar{N}}_3$, [4,4,4] ${\bar{M}}_3$, [41,1,1] ${\widehat{RR}}_3$, [4[4]]
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.

Symmetry

The order-4 square tiling honeycomb has many reflective symmetry constructions: Template:CDD as a regular honeycomb, Template:CDD ↔ Template:CDD with alternating types (colors) of square tilings, and Template:CDD 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: Template:CDD ↔ Template:CDD, and Template:CDD ↔ Template:CDD.

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 Template:CDD; and [4,4^*,4], with 4 orthogonal sets of ultra-parallel mirrors in an octahedral fundamental domain: Template:CDD.

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.

It contains Template:CDD and Template:CDD that tile 2-hypercycle surfaces, which are similar to these paracompact order-4 apeirogonal tilings Template:CDD:

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. Template:Regular paracompact H3 honeycombs

There are nine uniform honeycombs in the [4,4,4] Coxeter group family, including this regular form. Template:444 family

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

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

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

Quasiregular polychora and honeycombs: h{4,p,q}
Space	Finite	Affine	Compact	Paracompact
Schläfli symbol	h{4,3,3}	h{4,3,4}	h{4,3,5}	h{4,3,6}	h{4,4,3}	h{4,4,4}
	$\{3,\genfrac{}{}{0ex}{}{3}{3}\}$	$\{3,\genfrac{}{}{0ex}{}{4}{3}\}$	$\{3,\genfrac{}{}{0ex}{}{5}{3}\}$	$\{3,\genfrac{}{}{0ex}{}{6}{3}\}$	$\{4,\genfrac{}{}{0ex}{}{4}{3}\}$	$\{4,\genfrac{}{}{0ex}{}{4}{4}\}$
Coxeter diagram	Template:CDD ↔ Template:CDD	Template:CDD ↔ Template:CDD	Template:CDD ↔ Template:CDD	Template:CDD ↔ Template:CDD	Template:CDD ↔ Template:CDD	Template:CDD ↔ Template:CDD
	Template:CDD ↔ Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD ↔ Template:CDD
Image
Vertex figure r{p,3}	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD

Rectified order-4 square tiling honeycomb

Rectified order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	r{4,4,4} or t1{4,4,4}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	{4,4} r{4,4}
Faces	square {4}
Vertex figure	cube
Coxeter groups	${\bar{N}}_3$, [4,4,4] ${\bar{M}}_3$, [41,1,1]
Properties	Quasiregular or regular, depending on symmetry

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

Template:Clear

Truncated order-4 square tiling honeycomb

Truncated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t{4,4,4} or t0,1{4,4,4}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	{4,4} t{4,4}
Faces	square {4} octagon {8}
Vertex figure	square pyramid
Coxeter groups	${\bar{N}}_3$, [4,4,4] ${\bar{M}}_3$, [41,1,1]
Properties	Vertex-transitive

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

Template:Clear

Bitruncated order-4 square tiling honeycomb

Bitruncated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	2t{4,4,4} or t1,2{4,4,4}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD ↔ Template:CDD
Cells	t{4,4}
Faces	square {4} octagon {8}
Vertex figure	tetragonal disphenoid
Coxeter groups	$2\times {\bar{N}}_3$, [[4,4,4]] ${\bar{M}}_3$, [41,1,1] ${\widehat{RR}}_3$, [4[4]]
Properties	Vertex-transitive, edge-transitive, cell-transitive

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

Template:Clear

Cantellated order-4 square tiling honeycomb

Cantellated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	rr{4,4,4} or t0,2{4,4,4}
Coxeter diagrams	Template:CDD Template:CDD
Cells	{}x{4} r{4,4} rr{4,4}
Faces	square {4}
Vertex figure	triangular prism
Coxeter groups	${\bar{N}}_3$, [4,4,4] ${\bar{R}}_3$, [3,4,4]
Properties	Vertex-transitive, edge-transitive

The cantellated order-4 square tiling honeycomb, Template:CDD is the same thing as the rectified square tiling honeycomb, Template:CDD. It has cube and square tiling facets, with a triangular prism vertex figure.

Cantitruncated order-4 square tiling honeycomb

Cantitruncated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	tr{4,4,4} or t0,1,2{4,4,4}
Coxeter diagrams	Template:CDD Template:CDD Template:CDD ↔ Template:CDD
Cells	{}x{4} tr{4,4} t{4,4}
Faces	square {4} octagon {8}
Vertex figure	mirrored sphenoid
Coxeter groups	${\bar{N}}_3$, [4,4,4] ${\bar{R}}_3$, [3,4,4] ${\bar{M}}_3$, [41,1,1]
Properties	Vertex-transitive

The cantitruncated order-4 square tiling honeycomb, Template:CDD is the same as the truncated square tiling honeycomb, Template:CDD. It contains cube and truncated square tiling facets, with a mirrored sphenoid vertex figure.

It is the same as the truncated square tiling honeycomb, Template:CDD.

Template:Clear

Runcinated order-4 square tiling honeycomb

Runcinated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t0,3{4,4,4}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	{4,4} {}x{4}
Faces	square {4}
Vertex figure	square antiprism
Coxeter groups	$2\times {\bar{N}}_3$, [[4,4,4]]
Properties	Vertex-transitive, edge-transitive

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

Template:Clear

Runcitruncated order-4 square tiling honeycomb

Runcitruncated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t0,1,3{4,4,4}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	t{4,4} rr{4,4} {}x{4} {8}x{}
Faces	square {4} octagon {8}
Vertex figure	square pyramid
Coxeter groups	${\bar{N}}_3$, [4,4,4]
Properties	Vertex-transitive

The runcitruncated order-4 square tiling honeycomb, t_0,1,3{4,4,4}, Template:CDD 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.

Template:Clear

Omnitruncated order-4 square tiling honeycomb

Omnitruncated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t0,1,2,3{4,4,4}
Coxeter diagrams	Template:CDD
Cells	tr{4,4} {8}x{}
Faces	square {4} octagon {8}
Vertex figure	digonal disphenoid
Coxeter groups	$2\times {\bar{N}}_3$, [[4,4,4]]
Properties	Vertex-transitive

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

Template:Clear

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

Quarter order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	q{4,4,4}
Coxeter diagrams	Template:CDD Template:CDD
Cells	t{4,4} {4,4}
Faces	square {4} octagon {8}
Vertex figure	square antiprism
Coxeter groups	${\widehat{RR}}_3$, [4[4]]
Properties	Vertex-transitive, edge-transitive

The quarter order-4 square tiling honeycomb, q{4,4,4}, Template:CDD, or Template:CDD, has truncated square tiling and square tiling facets, with a square antiprism vertex figure.

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