Triangular tiling honeycomb

Triangular tiling honeycomb
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Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbol	{3,6,3} h{6,3,6} h{6,3^[3]} ↔ {3^[3,3]}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD ↔ Template:CDD
Cells	{3,6} File:Uniform tiling 63-t2.svg
Faces	triangle {3}
Edge figure	triangle {3}
Vertex figure	Error creating thumbnail: File:Uniform tiling 333-t012.png hexagonal tiling
Dual	Self-dual
Coxeter groups	${\overline{Y}}_{3}$ , [3,6,3] ${\overline{V P}}_{3}$ , [6,3^[3]] ${\overline{P P}}_{3}$ , [3^[3,3]]
Properties	Regular

The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.

Template:Honeycomb

Symmetry

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Subgroups of [3,6,3] and [6,3,6]

It has two lower reflective symmetry constructions, as an alternated order-6 hexagonal tiling honeycomb, Template:CDD ↔ Template:CDD, and as Template:CDD from Template:CDD, which alternates 3 types (colors) of triangular tilings around every edge. In Coxeter notation, the removal of the 3rd and 4th mirrors, [3,6,3^*] creates a new Coxeter group [3^[3,3]], Template:CDD, subgroup index 6. The fundamental domain is 6 times larger. By Coxeter diagram there are 3 copies of the first original mirror in the new fundamental domain: Template:CDD ↔ Template:CDD.

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Related Tilings

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

File:H2 tiling 2ii-4.png

Related honeycombs

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

There are nine uniform honeycombs in the [3,6,3] Coxeter group family, including this regular form as well as the bitruncated form, t_1,2{3,6,3}, Template:CDD with all truncated hexagonal tiling facets. Template:363 family

The honeycomb is also part of a series of polychora and honeycombs with triangular edge figures. Template:Symmetric tessellations

Rectified triangular tiling honeycomb

Rectified triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	r{3,6,3} h₂{6,3,6}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD ↔ Template:CDD
Cells	r{3,6} Error creating thumbnail: {6,3} File:Uniform polyhedron-63-t0.png
Faces	triangle {3} hexagon {6}
Vertex figure	File:Rectified triangular tiling honeycomb verf.png triangular prism
Coxeter group	${\overline{Y}}_{3}$ , [3,6,3] ${\overline{V P}}_{3}$ , [6,3^[3]] ${\overline{P P}}_{3}$ , [3^[3,3]]
Properties	Vertex-transitive, edge-transitive

The rectified triangular tiling honeycomb, Template:CDD, has trihexagonal tiling and hexagonal tiling cells, with a triangular prism vertex figure.

Symmetry

A lower symmetry of this honeycomb can be constructed as a cantic order-6 hexagonal tiling honeycomb, Template:CDD ↔ Template:CDD. A second lower-index construction is Template:CDD ↔ Template:CDD.

Template:Clear

Truncated triangular tiling honeycomb

Truncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t{3,6,3}
Coxeter diagram	Template:CDD Template:CDD
Cells	t{3,6} {6,3} File:Uniform polyhedron-63-t0.png
Faces	hexagon {6}
Vertex figure	File:Truncated triangular tiling honeycomb verf.png tetrahedron
Coxeter group	${\overline{Y}}_{3}$ , [3,6,3] ${\overline{V}}_{3}$ , [3,3,6]
Properties	Regular

The truncated triangular tiling honeycomb, Template:CDD, is a lower-symmetry form of the hexagonal tiling honeycomb, Template:CDD. It contains hexagonal tiling facets with a tetrahedral vertex figure.

Template:Clear

Bitruncated triangular tiling honeycomb

Bitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	2t{3,6,3}
Coxeter diagram	Template:CDD
Cells	t{6,3} Error creating thumbnail:
Faces	triangle {3} dodecagon {12}
Vertex figure	Error creating thumbnail: tetragonal disphenoid
Coxeter group	$2 \times {\overline{Y}}_{3}$ , [[3,6,3]]
Properties	Vertex-transitive, edge-transitive, cell-transitive

The bitruncated triangular tiling honeycomb, Template:CDD, has truncated hexagonal tiling cells, with a tetragonal disphenoid vertex figure.

Template:Clear

Cantellated triangular tiling honeycomb

Cantellated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	rr{3,6,3} or t_0,2{3,6,3} s₂{3,6,3}
Coxeter diagram	Template:CDD Template:CDD
Cells	rr{6,3} File:Uniform polyhedron-63-t02.png r{6,3} {}×{3} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	wedge
Coxeter group	${\overline{Y}}_{3}$ , [3,6,3]
Properties	Vertex-transitive

The cantellated triangular tiling honeycomb, Template:CDD, has rhombitrihexagonal tiling, trihexagonal tiling, and triangular prism cells, with a wedge vertex figure.

Symmetry

It can also be constructed as a cantic snub triangular tiling honeycomb, Template:CDD, a half-symmetry form with symmetry [3⁺,6,3].

Error creating thumbnail: Template:Clear

Cantitruncated triangular tiling honeycomb

Cantitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	tr{3,6,3} or t_0,1,2{3,6,3}
Coxeter diagram	Template:CDD
Cells	tr{6,3} t{6,3} {}×{3} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	File:Cantitruncated triangular tiling honeycomb verf.png mirrored sphenoid
Coxeter group	${\overline{Y}}_{3}$ , [3,6,3]
Properties	Vertex-transitive

The cantitruncated triangular tiling honeycomb, Template:CDD, has truncated trihexagonal tiling, truncated hexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.

File:H3 363-1110.png Template:Clear

Runcinated triangular tiling honeycomb

Runcinated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t_0,3{3,6,3}
Coxeter diagram	Template:CDD
Cells	{3,6} {}×{3} File:Triangular prism.png
Faces	triangle {3} square {4}
Vertex figure	hexagonal antiprism
Coxeter group	$2 \times {\overline{Y}}_{3}$ , [[3,6,3]]
Properties	Vertex-transitive, edge-transitive

The runcinated triangular tiling honeycomb, Template:CDD, has triangular tiling and triangular prism cells, with a hexagonal antiprism vertex figure.

Template:Clear

Runcitruncated triangular tiling honeycomb

Runcitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t_0,1,3{3,6,3} s_2,3{3,6,3}
Coxeter diagrams	Template:CDD Template:CDD
Cells	t{3,6} rr{3,6} File:Uniform polyhedron-63-t02.png {}×{3} File:Triangular prism.png {}×{6}
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	isosceles-trapezoidal pyramid
Coxeter group	${\overline{Y}}_{3}$ , [3,6,3]
Properties	Vertex-transitive

The runcitruncated triangular tiling honeycomb, Template:CDD, has hexagonal tiling, rhombitrihexagonal tiling, triangular prism, and hexagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

Symmetry

It can also be constructed as a runcicantic snub triangular tiling honeycomb, Template:CDD, a half-symmetry form with symmetry [3⁺,6,3].

Template:Clear

Omnitruncated triangular tiling honeycomb

Omnitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t_0,1,2,3{3,6,3}
Coxeter diagram	Template:CDD
Cells	tr{3,6} {}×{6}
Faces	square {4} hexagon {6} dodecagon {12}
Vertex figure	phyllic disphenoid
Coxeter group	$2 \times {\overline{Y}}_{3}$ , [[3,6,3]]
Properties	Vertex-transitive, edge-transitive

The omnitruncated triangular tiling honeycomb, Template:CDD, has truncated trihexagonal tiling and hexagonal prism cells, with a phyllic disphenoid vertex figure.

Template:Clear

Runcisnub triangular tiling honeycomb

Runcisnub triangular tiling honeycomb
Type	Paracompact scaliform honeycomb
Schläfli symbol	s₃{3,6,3}
Coxeter diagram	Template:CDD
Cells	r{6,3} {}x{3} File:Triangular prism.png {3,6} tricup
Faces	triangle {3} square {4} hexagon {6}
Vertex figure
Coxeter group	${\overline{Y}}_{3}$ , [3⁺,6,3]
Properties	Vertex-transitive, non-uniform

The runcisnub triangular tiling honeycomb, Template:CDD, has trihexagonal tiling, triangular tiling, triangular prism, and triangular cupola cells. It is vertex-transitive, but not uniform, since it contains Johnson solid triangular cupola cells.

Template:Clear

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

Triangular tiling honeycomb

Contents

Symmetry

Related Tilings