Order-6 hexagonal tiling honeycomb

Order-6 hexagonal tiling honeycomb
File:H3 636 FC boundary.png Perspective projection view from center of Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbol	{6,3,6} {6,3[3]}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	{6,3} File:Uniform tiling 63-t0.svg
Faces	hexagon {6}
Edge figure	hexagon {6}
Vertex figure	{3,6} or {3[3]} File:Uniform tiling 63-t2.svg Error creating thumbnail:
Dual	Self-dual
Coxeter group	${\bar{Z}}_3$, [6,3,6] ${\overline{VP}}_3$, [6,3[3]]
Properties	Regular, quasiregular

In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells with an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,6}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has six such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the triangular tiling is {3,6}, the vertex figure of this honeycomb is a triangular tiling. Thus, infinitely many hexagonal tilings meet at each vertex of this honeycomb.

Related tilings

The order-6 hexagonal 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 the paracompact tilings Template:CDD and Template:CDD (the truncated infinite-order triangular tiling and order-3 apeirogonal tiling, respectively):

File:H2-I-3-dual.svg

Symmetry

File:Hyperbolic subgroup tree 636.png

Subgroup relations:
Template:CDD ↔ Template:CDD

The order-6 hexagonal tiling honeycomb has a half-symmetry construction: Template:CDD.

It also has an index-6 subgroup, [6,3^*,6], with a non-simplex fundamental domain. This subgroup corresponds to a Coxeter diagram with six order-3 branches and three infinite-order branches in the shape of a triangular prism: Template:CDD.

Related polytopes and honeycombs

The order-6 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs in 3-space. Template:Regular paracompact H3 honeycombs

There are nine uniform honeycombs in the [6,3,6] Coxeter group family, including this regular form. Template:636 family

This honeycomb has a related alternated honeycomb, the triangular tiling honeycomb, but with a lower symmetry: Template:CDD ↔ Template:CDD.

The order-6 hexagonal tiling honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures: Template:Triangular tiling vertex figure tessellations small

It is also part of a sequence of regular polychora and honeycombs with hexagonal tiling cells: Template:Hexagonal tiling cell tessellations

It is also part of a sequence of regular polychora and honeycombs with regular deltahedral vertex figures: Template:Symmetric2 tessellations

Rectified order-6 hexagonal tiling honeycomb

Rectified order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	r{6,3,6} or t1{6,3,6}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD ↔ Template:CDD
Cells	{3,6} File:Uniform tiling 63-t2.svg r{6,3} File:Uniform tiling 63-t1.svg
Faces	triangle {3} hexagon {6}
Vertex figure	File:Rectified order-6 hexagonal tiling honeycomb verf.png hexagonal prism
Coxeter groups	${\bar{Z}}_3$, [6,3,6] ${\overline{VP}}_3$, [6,3[3]] ${\overline{PP}}_3$, [3[3,3]]
Properties	Vertex-transitive, edge-transitive

The rectified order-6 hexagonal tiling honeycomb, t₁{6,3,6}, Template:CDD has triangular tiling and trihexagonal tiling facets, with a hexagonal prism vertex figure.

it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q{6,3,6}, Template:CDD ↔ Template:CDD.

File:H3 636 boundary 0100.png

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

Error creating thumbnail:

Related honeycombs

The order-6 hexagonal tiling honeycomb is part of a series of honeycombs with hexagonal prism vertex figures: Template:Hexagonal tiling vertex figure tessellations

It is also part of a matrix of 3-dimensional quarter honeycombs: q{2p,4,2q} Template:Quarter hyperbolic honeycomb table Template:-

Truncated order-6 hexagonal tiling honeycomb

Truncated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t{6,3,6} or t0,1{6,3,6}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD
Cells	{3,6} File:Uniform tiling 63-t2.svg t{6,3} Error creating thumbnail:
Faces	triangle {3} dodecagon {12}
Vertex figure	File:Truncated order-6 hexagonal tiling honeycomb verf.png hexagonal pyramid
Coxeter groups	${\bar{Z}}_3$, [6,3,6] ${\overline{VP}}_3$, [6,3[3]]
Properties	Vertex-transitive

The truncated order-6 hexagonal tiling honeycomb, t_0,1{6,3,6}, Template:CDD has triangular tiling and truncated hexagonal tiling facets, with a hexagonal pyramid vertex figure.

File:H3 636-1100.png Template:-

Bitruncated order-6 hexagonal tiling honeycomb

Bitruncated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	bt{6,3,6} or t1,2{6,3,6}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD Template:CDD
Cells	t{3,6} Error creating thumbnail:
Faces	hexagon {6}
Vertex figure	File:Bitruncated order-6 hexagonal tiling honeycomb verf.png tetrahedron
Coxeter groups	$2\times {\bar{Z}}_3$, [[6,3,6]] ${\overline{VP}}_3$, [6,3[3]] ${\bar{V}}_3$, [3,3,6]
Properties	Regular

The bitruncated order-6 hexagonal tiling honeycomb is a lower symmetry construction of the regular hexagonal tiling honeycomb, Template:CDD ↔ Template:CDD. It contains hexagonal tiling facets, with a tetrahedron vertex figure.

File:H3 636-0110.png

Cantellated order-6 hexagonal tiling honeycomb

Cantellated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	rr{6,3,6} or t0,2{6,3,6}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD
Cells	r{3,6} File:Uniform tiling 63-t1.svg rr{6,3} File:Uniform tiling 63-t02.svg {}x{6} Error creating thumbnail:
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	File:Cantellated order-6 hexagonal tiling honeycomb verf.png wedge
Coxeter groups	${\bar{Z}}_3$, [6,3,6] ${\overline{VP}}_3$, [6,3[3]]
Properties	Vertex-transitive

The cantellated order-6 hexagonal tiling honeycomb, t_0,2{6,3,6}, Template:CDD has trihexagonal tiling, rhombitrihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure.

File:H3 636-1010.png Template:-

Cantitruncated order-6 hexagonal tiling honeycomb

Cantitruncated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	tr{6,3,6} or t0,1,2{6,3,6}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD
Cells	tr{3,6} File:Uniform tiling 63-t012.svg t{3,6} Error creating thumbnail: {}x{6} Error creating thumbnail:
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	File:Cantitruncated order-6 hexagonal tiling honeycomb verf.png mirrored sphenoid
Coxeter groups	${\bar{Z}}_3$, [6,3,6] ${\overline{VP}}_3$, [6,3[3]]
Properties	Vertex-transitive

The cantitruncated order-6 hexagonal tiling honeycomb, t_0,1,2{6,3,6}, Template:CDD has hexagonal tiling, truncated trihexagonal tiling, and hexagonal prism cells, with a mirrored sphenoid vertex figure.

Template:-

Runcinated order-6 hexagonal tiling honeycomb

Runcinated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,3{6,3,6}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD
Cells	{6,3} File:Uniform tiling 63-t0.svgFile:Uniform tiling 333-t012.svg {}×{6} Error creating thumbnail:
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	Error creating thumbnail: triangular antiprism
Coxeter groups	$2\times {\bar{Z}}_3$, [[6,3,6]]
Properties	Vertex-transitive, edge-transitive

The runcinated order-6 hexagonal tiling honeycomb, t_0,3{6,3,6}, Template:CDD has hexagonal tiling and hexagonal prism cells, with a triangular antiprism vertex figure.

File:H3 636-1001.png

It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr{6,6}, Template:CDD with square and hexagonal faces:

File:H2 tiling 266-5.png

Runcitruncated order-6 hexagonal tiling honeycomb

Runcitruncated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,3{6,3,6}
Coxeter diagram	Template:CDD
Cells	t{6,3} Error creating thumbnail: rr{6,3} File:Uniform tiling 63-t02.svg {}x{6}Error creating thumbnail: {}x{12} Error creating thumbnail:
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	File:Runcitruncated order-6 hexagonal tiling honeycomb verf.png isosceles-trapezoidal pyramid
Coxeter groups	${\bar{Z}}_3$, [6,3,6]
Properties	Vertex-transitive

The runcitruncated order-6 hexagonal tiling honeycomb, t_0,1,3{6,3,6}, Template:CDD has truncated hexagonal tiling, rhombitrihexagonal tiling, hexagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

File:H3 636-1011.png Template:-

Omnitruncated order-6 hexagonal tiling honeycomb

Omnitruncated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,2,3{6,3,6}
Coxeter diagram	Template:CDD
Cells	tr{6,3} File:Uniform tiling 63-t012.svg {}x{12} Error creating thumbnail:
Faces	square {4} hexagon {6} dodecagon {12}
Vertex figure	File:Omnitruncated order-6 hexagonal tiling honeycomb verf.png phyllic disphenoid
Coxeter groups	$2\times {\bar{Z}}_3$, [[6,3,6]]
Properties	Vertex-transitive

The omnitruncated order-6 hexagonal tiling honeycomb, t_0,1,2,3{6,3,6}, Template:CDD has truncated trihexagonal tiling and dodecagonal prism cells, with a phyllic disphenoid vertex figure.

Error creating thumbnail: Template:-

Alternated order-6 hexagonal tiling honeycomb

Alternated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h{6,3,6}
Coxeter diagrams	Template:CDD ↔ Template:CDD
Cells	{3,6} File:Uniform tiling 63-t2.svg {3[3]} File:Uniform tiling 333-t0.svg
Faces	triangle {3}
Vertex figure	File:Uniform tiling 63-t0.svg hexagonal tiling
Coxeter groups	${\overline{VP}}_3$, [6,3[3]]
Properties	Regular, quasiregular

The alternated order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the regular triangular tiling honeycomb, Template:CDD ↔ Template:CDD. It contains triangular tiling facets in a hexagonal tiling vertex figure.

Cantic order-6 hexagonal tiling honeycomb

Cantic order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h2{6,3,6}
Coxeter diagrams	Template:CDD ↔ Template:CDD
Cells	t{3,6} Error creating thumbnail: r{6,3} File:Uniform tiling 63-t1.svg h2{6,3}
Faces	triangle {3} hexagon {6}
Vertex figure	File:Rectified triangular tiling honeycomb verf.png triangular prism
Coxeter groups	${\overline{VP}}_3$, [6,3[3]]
Properties	Vertex-transitive, edge-transitive

The cantic order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the rectified triangular tiling honeycomb, Template:CDD ↔ Template:CDD, with trihexagonal tiling and hexagonal tiling facets in a triangular prism vertex figure.

Runcic order-6 hexagonal tiling honeycomb

Runcic order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h3{6,3,6}
Coxeter diagrams	Template:CDD ↔ Template:CDD
Cells	rr{3,6} File:Uniform tiling 63-t02.svg {6,3} File:Uniform tiling 63-t0.svg {3[3]} File:Uniform tiling 333-t0.svg {3}x{} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	File:Runcic order-6 hexagonal tiling honeycomb verf.png triangular cupola
Coxeter groups	${\overline{VP}}_3$, [6,3[3]]
Properties	Vertex-transitive

The runcic hexagonal tiling honeycomb, h₃{6,3,6}, Template:CDD, or Template:CDD, has hexagonal tiling, rhombitrihexagonal tiling, triangular tiling, and triangular prism facets, with a triangular cupola vertex figure. Template:-

Runicantic order-6 hexagonal tiling honeycomb

Runcicantic order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h2,3{6,3,6}
Coxeter diagrams	Template:CDD ↔ Template:CDD
Cells	tr{6,3} File:Uniform tiling 63-t012.png t{6,3} h2{6,3} Error creating thumbnail: {}x{3} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	File:Runcicantic order-6 hexagonal tiling honeycomb verf.png rectangular pyramid
Coxeter groups	${\overline{VP}}_3$, [6,3[3]]
Properties	Vertex-transitive

The runcicantic order-6 hexagonal tiling honeycomb, h_2,3{6,3,6}, Template:CDD, or Template:CDD, contains truncated trihexagonal tiling, truncated hexagonal tiling, trihexagonal tiling, and triangular prism facets, with a rectangular pyramid vertex figure. Template:-

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