Order-4 square tiling honeycomb

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

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

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

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

11 paracompact regular honeycombs
{6,3,3}	{6,3,4}	{6,3,5}	{6,3,6}	{4,4,3}	{4,4,4}
{3,3,6}	{4,3,6}	{5,3,6}	{3,6,3}	{3,4,4}

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

[4,4,4] family honeycombs
{4,4,4}	r{4,4,4}	t{4,4,4}	rr{4,4,4}	t0,3{4,4,4}	2t{4,4,4}	tr{4,4,4}	t0,1,3{4,4,4}	t0,1,2,3{4,4,4}

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

{p,4,4} honeycombs
Space	E3	H3
Form	Affine	Paracompact		Noncompact
Name	{2,4,4}	{3,4,4}	{4,4,4}	{5,4,4}	{6,4,4}	..{∞,4,4}
Coxeter
Image
Cells	{2,4}	{3,4}	{4,4}	{5,4}	{6,4}	{∞,4}

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

{4,4,p} honeycombs
Space	E3	H3
Form	Affine	Paracompact		Noncompact
Name	{4,4,2}	{4,4,3}	{4,4,4}	{4,4,5}	{4,4,6}	...{4,4,∞}
Coxeter
Image
Vertex figure	{4,2}	{4,3}	{4,4}	{4,5}	{4,6}	{4,∞}

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}
	${\displaystyle \left\{3,{3 \atop 3}\right\}}$	${\displaystyle \left\{3,{4 \atop 3}\right\}}$	${\displaystyle \left\{3,{5 \atop 3}\right\}}$	${\displaystyle \left\{3,{6 \atop 3}\right\}}$	${\displaystyle \left\{4,{4 \atop 3}\right\}}$	${\displaystyle \left\{4,{4 \atop 4}\right\}}$
Coxeter diagram	↔	↔	↔	↔	↔	↔
	↔					↔
Image
Vertex figure r{p,3}

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	↔
Cells	{4,4} r{4,4}
Faces	square {4}
Vertex figure	cube
Coxeter groups	${\displaystyle {\overline {N}}_{3}}$, [4,4,4] ${\displaystyle {\overline {M}}_{3}}$, [41,1,1]
Properties	Quasiregular or regular, depending on symmetry

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

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	↔ ↔ ↔
Cells	{4,4} t{4,4}
Faces	square {4} octagon {8}
Vertex figure	square pyramid
Coxeter groups	${\displaystyle {\overline {N}}_{3}}$, [4,4,4] ${\displaystyle {\overline {M}}_{3}}$, [41,1,1]
Properties	Vertex-transitive

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

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	↔ ↔ ↔
Cells	t{4,4}
Faces	square {4} octagon {8}
Vertex figure	tetragonal disphenoid
Coxeter groups	${\displaystyle 2\times {\overline {N}}_{3}}$, [[4,4,4]] ${\displaystyle {\overline {M}}_{3}}$, [41,1,1] ${\displaystyle {\widehat {RR}}_{3}}$, [4[4]]
Properties	Vertex-transitive, edge-transitive, cell-transitive

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

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
Cells	{}x{4} r{4,4} rr{4,4}
Faces	square {4}
Vertex figure	triangular prism
Coxeter groups	${\displaystyle {\overline {N}}_{3}}$, [4,4,4] ${\displaystyle {\overline {R}}_{3}}$, [3,4,4]
Properties	Vertex-transitive, edge-transitive

The cantellated order-4 square tiling honeycomb, is the same thing as the rectified square tiling honeycomb, . 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	↔
Cells	{}x{4} tr{4,4} t{4,4}
Faces	square {4} octagon {8}
Vertex figure	mirrored sphenoid
Coxeter groups	${\displaystyle {\overline {N}}_{3}}$, [4,4,4] ${\displaystyle {\overline {R}}_{3}}$, [3,4,4] ${\displaystyle {\overline {M}}_{3}}$, [41,1,1]
Properties	Vertex-transitive

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

It is the same as the truncated square tiling honeycomb, .

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	↔ ↔
Cells	{4,4} {}x{4}
Faces	square {4}
Vertex figure	square antiprism
Coxeter groups	${\displaystyle 2\times {\overline {N}}_{3}}$, [[4,4,4]]
Properties	Vertex-transitive, edge-transitive

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

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	↔
Cells	t{4,4} rr{4,4} {}x{4} {8}x{}
Faces	square {4} octagon {8}
Vertex figure	square pyramid
Coxeter groups	${\displaystyle {\overline {N}}_{3}}$, [4,4,4]
Properties	Vertex-transitive

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

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
Cells	tr{4,4} {8}x{}
Faces	square {4} octagon {8}
Vertex figure	digonal disphenoid
Coxeter groups	${\displaystyle 2\times {\overline {N}}_{3}}$, [[4,4,4]]
Properties	Vertex-transitive

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

Quarter order-4 square tiling honeycomb

Quarter order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	q{4,4,4}
Coxeter diagrams
Cells	t{4,4} {4,4}
Faces	square {4} octagon {8}
Vertex figure	square antiprism
Coxeter groups	${\displaystyle {\widehat {RR}}_{3}}$, [4[4]]
Properties	Vertex-transitive, edge-transitive

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

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

• 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 99-35678, ISBN 0-486-40919-8 (Chapter 10, 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 Quadratic Integers and Coxeter Groups PDF Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336