Order-4 square tiling honeycomb

Order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	{4,4,4} h{4,4,4} ↔ {4,41,1} {4[4]}
Coxeter diagrams	↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔
Cells	{4,4}
Faces	square {4}
Edge figure	square {4}
Vertex figure	square tiling, {4,4}
Dual	Self-dual
Coxeter groups	${\displaystyle {\overline {N}}_{3}}$, [4,4,4] ${\displaystyle {\overline {M}}_{3}}$, [41,1,1] ${\displaystyle {\widehat {RR}}_{3}}$, [4[4]]
Properties	Regular, quasiregular

In the

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary

Symmetry

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: .

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

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.

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

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 v t e
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 v t e
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

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

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

It is the same as the

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

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

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

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.

See also

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

References

• 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,
  ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space
  ) Table III
• 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