Triangular tiling honeycomb

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

It has two lower reflective symmetry constructions, as an alternated order-6 hexagonal tiling honeycomb, ↔ , and as from , 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]], , 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 hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface.

The triangular tiling honeycomb is a regular 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 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}, with all truncated 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 of polychora and honeycombs with triangular edge 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

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

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

Truncated triangular tiling honeycomb

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

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

Bitruncated triangular tiling honeycomb

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

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

Cantellated triangular tiling honeycomb

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

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

Cantitruncated triangular tiling honeycomb

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

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

Runcinated triangular tiling honeycomb

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

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

Runcitruncated triangular tiling honeycomb

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-trapezoidal pyramid
Coxeter group	${\displaystyle {\overline {Y}}_{3}}$, [3,6,3]
Properties	Vertex-transitive

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

Omnitruncated triangular tiling honeycomb

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

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

Runcisnub triangular tiling honeycomb

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

The runcisnub triangular tiling honeycomb, , 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.

• 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