Triangular tiling honeycomb
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	↔ ↔ ↔
Cells	{3,6}
Faces	triangle {3}
Edge figure	triangle {3}
Vertex figure	hexagonal tiling
Dual	Self-dual
Coxeter groups	${\displaystyle {\overline{Y}}_3}$, [3,6,3] ${\displaystyle {\overline{VP}}_3}$, [6,3[3]] ${\displaystyle {\overline{PP}}_3}$, [3[3,3]]
Properties	Regular

Thetriangular tiling honeycomb is one of 11 paracompact regular space-fillingtessellations (orhoneycombs) inhyperbolic 3-space. It is calledparacompact because it has infinitecells andvertex figures, with all vertices asideal points at infinity. It hasSchläfli symbol {3,6,3}, being composed oftriangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Itsvertex figure is ahexagonal tiling.

Ageometric honeycomb is aspace-filling ofpolyhedral or higher-dimensionalcells, so that there are no gaps. It is an example of the more general mathematicaltiling ortessellation in any number of dimensions.

Honeycombs are usually constructed in ordinaryEuclidean ("flat") space, like theconvex uniform honeycombs. They may also be constructed innon-Euclidean spaces, such ashyperbolic uniform honeycombs. Any finiteuniform polytope can be projected to itscircumsphere to form a uniform honeycomb in spherical space.

It has two lower reflective symmetry constructions, as analternatedorder-6 hexagonal tiling honeycomb, ↔, and as from, which alternates 3 types (colors) of triangular tilings around every edge. InCoxeter notation, the removal of the 3rd and 4th mirrors, [3,6,3^*] creates a newCoxeter group [3^[3,3]],, 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: ↔.

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

The triangular tiling honeycomb is aregular hyperbolic honeycomb in 3-space, and one of eleven 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 arenine uniform honeycombs in the [3,6,3]Coxeter group family, including this regular form as well as thebitruncated form, t_1,2{3,6,3}, with alltruncated hexagonal tiling facets.

[3,6,3] family honeycombs

{3,6,3}	r{3,6,3}	t{3,6,3}	rr{3,6,3}	t0,3{3,6,3}	2t{3,6,3}	tr{3,6,3}	t0,1,3{3,6,3}	t0,1,2,3{3,6,3}

The honeycomb is also part of a series ofpolychora and honeycombs with triangularedge figures.

{3,p,3} polytopes
Space	S3		H3
Form	Finite		Compact	Paracompact	Noncompact
{3,p,3}	{3,3,3}	{3,4,3}	{3,5,3}	{3,6,3}	{3,7,3}	{3,8,3}	...{3,∞,3}
Image
Cells	{3,3}	{3,4}	{3,5}	{3,6}	{3,7}	{3,8}	{3,∞}
Vertex figure	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

Rectified triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	r{3,6,3} h2{6,3,6}
Coxeter diagram	↔ ↔ ↔
Cells	r{3,6} {6,3}
Faces	triangle {3} hexagon {6}
Vertex figure	triangular prism
Coxeter group	${\displaystyle {\overline{Y}}_3}$, [3,6,3] ${\displaystyle {\overline{VP}}_3}$, [6,3[3]] ${\displaystyle {\overline{PP}}_3}$, [3[3,3]]
Properties	Vertex-transitive, edge-transitive

Therectified triangular tiling honeycomb,, hastrihexagonal tiling andhexagonal tiling cells, with atriangular prism vertex figure.

A lower symmetry of this honeycomb can be constructed as acantic order-6 hexagonal tiling honeycomb, ↔. A second lower-index construction is ↔.

Truncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t{3,6,3}
Coxeter diagram
Cells	t{3,6} {6,3}
Faces	hexagon {6}
Vertex figure	tetrahedron
Coxeter group	${\displaystyle {\overline{Y}}_3}$, [3,6,3] ${\displaystyle {\overline{V}}_3}$, [3,3,6]
Properties	Regular

Thetruncated triangular tiling honeycomb,, is a lower-symmetry form of thehexagonal tiling honeycomb,. It containshexagonal tiling facets with atetrahedral vertex figure.

Bitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	2t{3,6,3}
Coxeter diagram
Cells	t{6,3}
Faces	triangle {3} dodecagon {12}
Vertex figure	tetragonal disphenoid
Coxeter group	${\displaystyle 2\times {\overline{Y}}_3}$, [[3,6,3]]
Properties	Vertex-transitive, edge-transitive, cell-transitive

Thebitruncated triangular tiling honeycomb,, hastruncated hexagonal tiling cells, with atetragonal disphenoid vertex figure.

Cantellated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	rr{3,6,3} or t0,2{3,6,3} s2{3,6,3}
Coxeter diagram
Cells	rr{6,3} r{6,3} {}×{3}
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	wedge
Coxeter group	${\displaystyle {\overline{Y}}_3}$, [3,6,3]
Properties	Vertex-transitive

Thecantellated triangular tiling honeycomb,, hasrhombitrihexagonal tiling,trihexagonal tiling, andtriangular prism cells, with awedge vertex figure.

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

Cantitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	tr{3,6,3} or t0,1,2{3,6,3}
Coxeter diagram
Cells	tr{6,3} t{6,3} {}×{3}
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	mirrored sphenoid
Coxeter group	${\displaystyle {\overline{Y}}_3}$, [3,6,3]
Properties	Vertex-transitive

Thecantitruncated triangular tiling honeycomb,, hastruncated trihexagonal tiling,truncated hexagonal tiling, andtriangular prism cells, with amirrored sphenoid vertex figure.

Runcinated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,3{3,6,3}
Coxeter diagram
Cells	{3,6} {}×{3}
Faces	triangle {3} square {4}
Vertex figure	hexagonal antiprism
Coxeter group	${\displaystyle 2\times {\overline{Y}}_3}$, [[3,6,3]]
Properties	Vertex-transitive, edge-transitive

Theruncinated triangular tiling honeycomb,, hastriangular tiling andtriangular prism cells, with ahexagonal antiprism vertex figure.

Runcitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t0,1,3{3,6,3} s2,3{3,6,3}
Coxeter diagrams
Cells	t{3,6} rr{3,6} {}×{3} {}×{6}
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	isosceles-trapezoidalpyramid
Coxeter group	${\displaystyle {\overline{Y}}_3}$, [3,6,3]
Properties	Vertex-transitive

Theruncitruncated triangular tiling honeycomb,, hashexagonal tiling,rhombitrihexagonal tiling,triangular prism, andhexagonal prism cells, with anisosceles-trapezoidalpyramidvertex figure.

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

Omnitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,2,3{3,6,3}
Coxeter diagram
Cells	tr{3,6} {}×{6}
Faces	square {4} hexagon {6} dodecagon {12}
Vertex figure	phyllic disphenoid
Coxeter group	${\displaystyle 2\times {\overline{Y}}_3}$, [[3,6,3]]
Properties	Vertex-transitive, edge-transitive

Theomnitruncated triangular tiling honeycomb,, hastruncated trihexagonal tiling andhexagonal prism cells, with aphyllic disphenoid vertex figure.

Runcisnub triangular tiling honeycomb
Type	Paracompact scaliform honeycomb
Schläfli symbol	s3{3,6,3}
Coxeter diagram
Cells	r{6,3} {}x{3} {3,6} tricup
Faces	triangle {3} square {4} hexagon {6}
Vertex figure
Coxeter group	${\displaystyle {\overline{Y}}_3}$, [3+,6,3]
Properties	Vertex-transitive, non-uniform

Theruncisnub triangular tiling honeycomb,, hastrihexagonal tiling,triangular tiling,triangular prism, andtriangular cupola cells. It isvertex-transitive, but not uniform, since it containsJohnson solidtriangular cupola cells.

• 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. WeeksThe Shape of Space, 2nd editionISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I, II)
• Norman JohnsonUniform 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

• Regular 3-honeycombs
• Self-dual tilings
• Triangular tilings