Triangular tiling honeycomb

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	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png ↔ File:CDel branch 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png File:CDel node h1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel branch.png ↔ File:CDel node 1.pngFile:CDel splitsplit1.pngFile:CDel branch4.pngFile:CDel splitsplit2.pngFile:CDel node.png ↔ File:CDel branch 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node h0.png
Cells	{3,6} File:Uniform tiling 63-t2.svg File:Uniform tiling 333-t1.svg
Faces	triangle {3}
Edge figure	triangle {3}
Vertex figure	File:Uniform tiling 63-t0.svg File:Uniform tiling 63-t12.svg File:Uniform tiling 333-t012.svg 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.

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 Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

Symmetry

File:Hyperbolic subgroup tree 363.png

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

It has two lower reflective symmetry constructions, as an alternated order-6 hexagonal tiling honeycomb, File:CDel node h1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png ↔ File:CDel branch 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png, and as File:CDel node 1.pngFile:CDel splitsplit1.pngFile:CDel branch4.pngFile:CDel splitsplit2.pngFile:CDel node.png from File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.png, 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]], File:CDel node.pngFile:CDel splitsplit1.pngFile:CDel branch4.pngFile:CDel splitsplit2.pngFile:CDel node.png, 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: File:CDel node c2.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png ↔ File:CDel node c2.pngFile:CDel splitsplit1.pngFile:CDel branch4 c1.pngFile:CDel splitsplit2.pngFile:CDel node c1.png.

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.

11 paracompact regular honeycombs
File:H3 633 FC boundary.png {6,3,3}	File:H3 634 FC boundary.png {6,3,4}	File:H3 635 FC boundary.png {6,3,5}	File:H3 636 FC boundary.png {6,3,6}	File:H3 443 FC boundary.png {4,4,3}	File:H3 444 FC boundary.png {4,4,4}
File:H3 336 CC center.png {3,3,6}	File:H3 436 CC center.png {4,3,6}	File:H3 536 CC center.png {5,3,6}	Error creating thumbnail: {3,6,3}	File:H3 344 CC center.png {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}, File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png with all truncated hexagonal tiling facets.

[3,6,3] family honeycombs
{3,6,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	r{3,6,3} File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	t{3,6,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	rr{3,6,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	t_0,3{3,6,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	2t{3,6,3} File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	tr{3,6,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	t_0,1,3{3,6,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	t_0,1,2,3{3,6,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png
Error creating thumbnail:	File:H3 363 boundary 0100.png	File:H3 363-1100.png	File:H3 363-1010.png	File:H3 363-1001.png	File:H3 363-0110.png	File:H3 363-1110.png	File:H3 363-1011.png	File:H3 363-1111.png

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	File:Stereographic polytope 5cell.png	File:Stereographic polytope 24cell.png	File:H3 353 CC center.png	Error creating thumbnail:	File:Hyperbolic honeycomb 3-7-3 poincare.png	File:Hyperbolic honeycomb 3-8-3 poincare.png	File:Hyperbolic honeycomb 3-i-3 poincare.png
Cells	File:Tetrahedron.png {3,3}	File:Octahedron.png {3,4}	File:Icosahedron.png {3,5}	File:Uniform tiling 63-t2.svg {3,6}	File:Order-7 triangular tiling.svg {3,7}	File:H2-8-3-primal.svg {3,8}	File:H2 tiling 23i-4.png {3,∞}
Vertex figure	File:5-cell verf.svg {3,3}	File:24 cell verf.svg {4,3}	File:Order-3 icosahedral honeycomb verf.svg {5,3}	File:Uniform tiling 63-t0.svg {6,3}	File:Heptagonal tiling.svg {7,3}	File:H2-8-3-dual.svg {8,3}	File:H2-I-3-dual.svg {∞,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	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.png ↔ File:CDel branch 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.png File:CDel node.pngFile:CDel splitsplit1.pngFile:CDel branch4 11.pngFile:CDel splitsplit2.pngFile:CDel node 1.png ↔ File:CDel branch 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node h0.png ↔ File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.png
Cells	r{3,6} File:Uniform polyhedron-63-t1.svg {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, File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, 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, File:CDel branch 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.png ↔ File:CDel node h1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.png. A second lower-index construction is File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.png ↔ File:CDel node.pngFile:CDel splitsplit1.pngFile:CDel branch4 11.pngFile:CDel splitsplit2.pngFile:CDel node 1.png.

File:H3 363 boundary 0100.png

Truncated triangular tiling honeycomb

Truncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t{3,6,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Cells	t{3,6} File:Uniform polyhedron-63-t12.png {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, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, is a lower-symmetry form of the hexagonal tiling honeycomb, File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png. It contains hexagonal tiling facets with a tetrahedral vertex figure.

File:H3 363-1100.png

Bitruncated triangular tiling honeycomb

Bitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	2t{3,6,3}
Coxeter diagram	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
Cells	t{6,3} File:Uniform polyhedron-63-t01.png
Faces	triangle {3} dodecagon {12}
Vertex figure	File:Bitruncated triangular tiling honeycomb verf.png tetragonal disphenoid
Coxeter group	$2 \times {\overline{Y}}_{3}$ , [[3,6,3]]
Properties	Vertex-transitive, edge-transitive, cell-transitive

The bitruncated triangular tiling honeycomb, File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png, has truncated hexagonal tiling cells, with a tetragonal disphenoid vertex figure.

File:H3 363-0110.png

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	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
Cells	rr{6,3} File:Uniform polyhedron-63-t02.png r{6,3} File:Uniform polyhedron-63-t1.svg {}×{3} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	File:Cantellated triangular tiling honeycomb verf.png wedge
Coxeter group	${\overline{Y}}_{3}$ , [3,6,3]
Properties	Vertex-transitive

The cantellated triangular tiling honeycomb, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png, 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, File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png, a half-symmetry form with symmetry [3⁺,6,3].

File:H3 363-1010.png

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	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
Cells	tr{6,3} File:Uniform polyhedron-63-t012.png t{6,3} File:Uniform polyhedron-63-t01.png {}×{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, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png, has truncated trihexagonal tiling, truncated hexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.

File:H3 363-1110.png

Runcinated triangular tiling honeycomb

Runcinated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t_0,3{3,6,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
Cells	{3,6} File:Uniform polyhedron-63-t2.svg {}×{3} File:Triangular prism.png
Faces	triangle {3} square {4}
Vertex figure	File:Runcinated triangular tiling honeycomb verf.png hexagonal antiprism
Coxeter group	$2 \times {\overline{Y}}_{3}$ , [[3,6,3]]
Properties	Vertex-transitive, edge-transitive

The runcinated triangular tiling honeycomb, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png, has triangular tiling and triangular prism cells, with a hexagonal antiprism vertex figure.

File:H3 363-1001.png

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	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png
Cells	t{3,6} File:Uniform polyhedron-63-t12.png rr{3,6} File:Uniform polyhedron-63-t02.png {}×{3} File:Triangular prism.png {}×{6} File:Hexagonal prism.png
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	File:Runcitruncated triangular tiling honeycomb verf.png isosceles-trapezoidal pyramid
Coxeter group	${\overline{Y}}_{3}$ , [3,6,3]
Properties	Vertex-transitive

The runcitruncated triangular tiling honeycomb, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png, 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, File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png, a half-symmetry form with symmetry [3⁺,6,3].

File:H3 363-1101.png

Omnitruncated triangular tiling honeycomb

Omnitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t_0,1,2,3{3,6,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png
Cells	tr{3,6} File:Uniform polyhedron-63-t012.png {}×{6} File:Hexagonal prism.png
Faces	square {4} hexagon {6} dodecagon {12}
Vertex figure	File:Omnitruncated triangular tiling honeycomb verf.png phyllic disphenoid
Coxeter group	$2 \times {\overline{Y}}_{3}$ , [[3,6,3]]
Properties	Vertex-transitive, edge-transitive

The omnitruncated triangular tiling honeycomb, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png, has truncated trihexagonal tiling and hexagonal prism cells, with a phyllic disphenoid vertex figure.

File:H3 363-1111.png

Runcisnub triangular tiling honeycomb

Runcisnub triangular tiling honeycomb
Type	Paracompact scaliform honeycomb
Schläfli symbol	s₃{3,6,3}
Coxeter diagram	File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
Cells	r{6,3} File:Uniform tiling 333-t02.svg {}x{3} File:Triangular prism.png {3,6} File:Uniform tiling 333-t1.svg tricup File:Triangular cupola.png
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, File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png, 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.

References

Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (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

Related honeycombs

Rectified triangular tiling honeycomb

Symmetry

Truncated triangular tiling honeycomb

Bitruncated triangular tiling honeycomb

Cantellated triangular tiling honeycomb

Symmetry

Cantitruncated triangular tiling honeycomb

Runcinated triangular tiling honeycomb

Runcitruncated triangular tiling honeycomb

Symmetry

Omnitruncated triangular tiling honeycomb

Runcisnub triangular tiling honeycomb

See also

References

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Triangular tiling honeycomb

Symmetry

Related Tilings

Related honeycombs

Rectified triangular tiling honeycomb

Symmetry

Truncated triangular tiling honeycomb

Bitruncated triangular tiling honeycomb

Cantellated triangular tiling honeycomb

Symmetry

Cantitruncated triangular tiling honeycomb

Runcinated triangular tiling honeycomb

Runcitruncated triangular tiling honeycomb

Symmetry

Omnitruncated triangular tiling honeycomb

Runcisnub triangular tiling honeycomb

See also

References

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