Hexagonal tiling honeycomb#Polytopes and honeycombs with tetrahedral vertex figures

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Hexagonal tiling honeycomb
bgcolor=#ffffff align=center colspan=2\|320px Perspective projection view within Poincaré disk model
bgcolor=#e7dcc3\|Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	{6,3,3} t{3,6,3} 2t{6,3,6} 2t{6,3^[3]} t{3^[3,3]}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|6\|node\|3\|node\|3\|node}} {{CDD\|node_1\|3\|node_1\|6\|node\|3\|node}} {{CDD\|node\|6\|node_1\|3\|node_1\|6\|node}} {{CDD\|branch_11\|split2\|node_1\|6\|node}} {{CDD\|branch_11\|splitcross\|branch_11}} ↔ {{CDD\|node_1\|6\|node_g\|3sg\|node_g\|3g\|node_g}} ↔ {{CDD\|node_1\|3\|node_1\|6\|node_g\|3sg\|node_g}} ↔ {{CDD\|node_h0\|6\|node_1\|3\|node_1\|6\|node_h0}} ↔ {{CDD\|branch_11\|split2\|node_1\|6\|node_h0}}
bgcolor=#e7dcc3\|Cells	{6,3} 40px
bgcolor=#e7dcc3\|Faces	hexagon {6}
bgcolor=#e7dcc3\|Edge figure	triangle {3}
bgcolor=#e7dcc3\|Vertex figure	80px tetrahedron {3,3}
bgcolor=#e7dcc3\|Dual	Order-6 tetrahedral honeycomb
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6] ${\overline{Y}}_3$ , [3,6,3] ${\overline{Z}}_3$ , [6,3,6] ${\overline{VP}}_3$ , [6,3^[3]] ${\overline{PP}}_3$ , [3^[3,3]]
bgcolor=#e7dcc3\|Properties	Regular

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

The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex.Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III

Images

480px

Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {∞,3} of H², with horocycles circumscribing vertices of apeirogonal faces.

class=wikitable width=480

!{6,3,3}

!{∞,3}

240px

|240px

align=center

|One hexagonal tiling cell of the hexagonal tiling honeycomb

|An order-3 apeirogonal tiling with a green apeirogon and its horocycle

Symmetry constructions

File:Hyperbolic subgroup tree 336-direct.png]]

It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: {{CDD|node_c1|6|node|3|node|3|node}} [6,3,3], {{CDD|node_c1|3|node_c1|6|node|3|node}} [3,6,3], {{CDD|node|6|node_c1|3|node_c1|6|node}} [6,3,6], {{CDD|branch_c1|split2|node_c1|6|node}} [6,3^[3]] and [3^[3,3]] {{CDD|branch_c1|splitcross|branch_c1}}, having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)^*] (remove 3 mirrors, index 24 subgroup); [3,6,3^*] or [3^*,6,3] (remove 2 mirrors, index 6 subgroup); [1⁺,6,3,6,1⁺] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3^[3,3]]. The ringed Coxeter diagrams are {{CDD|node_1|6|node|3|node|3|node}}, {{CDD|node_1|3|node_1|6|node|3|node}}, {{CDD|node|6|node_1|3|node_1|6|node}}, {{CDD|branch_11|split2|node_1|6|node}} and {{CDD|branch_11|splitcross|branch_11}}, representing different types (colors) of hexagonal tilings in the Wythoff construction.

Related polytopes and honeycombs

The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb.

It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures.

It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of hexagonal tiling cells:

= Rectified hexagonal tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Rectified hexagonal tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	r{6,3,3} or t₁{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node\|6\|node_1\|3\|node\|3\|node}} {{CDD\|node_h0\|6\|node_1\|3\|node\|3\|node}} ↔ {{CDD\|branch_11\|split2\|node\|3\|node}}
bgcolor=#e7dcc3\|Cells	{3,3} 40px r{6,3} 40px or 40px
bgcolor=#e7dcc3\|Faces	triangle {3} hexagon {6}
bgcolor=#e7dcc3\|Vertex figure	80px triangular prism
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6] ${\overline{P}}_3$ , [3,3^[3]]
bgcolor=#e7dcc3\|Properties	Vertex-transitive, edge-transitive

The rectified hexagonal tiling honeycomb, t₁{6,3,3}, {{CDD|node|6|node_1|3|node|3|node}} has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The {{CDD|branch_11|split2|node|3|node}} half-symmetry construction alternates two types of tetrahedra.

480px

class=wikitable !Hexagonal tiling honeycomb {{CDD\|node_1\|6\|node\|3\|node\|3\|node}} !Rectified hexagonal tiling honeycomb {{CDD\|node\|6\|node_1\|3\|node\|3\|node}} or {{CDD\|branch_11\|split2\|node\|3\|node}}
align=center \|200px \|200px
colspan=2\|Related H² tilings
Order-3 apeirogonal tiling {{CDD\|node_1\|infin\|node\|3\|node}} !Triapeirogonal tiling {{CDD\|node\|infin\|node_1\|3\|node}} or {{CDD\|labelinfin\|branch_11\|split2\|node}}
align=center \|100px \|100px 100px

class=wikitable

!Hexagonal tiling honeycomb
{{CDD|node_1|6|node|3|node|3|node}}

!Rectified hexagonal tiling honeycomb
{{CDD|node|6|node_1|3|node|3|node}} or {{CDD|branch_11|split2|node|3|node}}

align=center

|200px

colspan=2|Related H² tilings

Order-3 apeirogonal tiling
{{CDD|node_1|infin|node|3|node}}

!Triapeirogonal tiling
{{CDD|node|infin|node_1|3|node}} or {{CDD|labelinfin|branch_11|split2|node}}

align=center

|100px

|100px 100px

= Truncated hexagonal tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Truncated hexagonal tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbol	t{6,3,3} or t_0,1{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagram	{{CDD\|node_1\|6\|node_1\|3\|node\|3\|node}}
bgcolor=#e7dcc3\|Cells	{3,3} 40px t{6,3} 40px
bgcolor=#e7dcc3\|Faces	triangle {3} dodecagon {12}
bgcolor=#e7dcc3\|Vertex figure	80px triangular pyramid
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The truncated hexagonal tiling honeycomb, t_0,1{6,3,3}, {{CDD||node_1|6|node_1|3|node|3|node}} has tetrahedral and truncated hexagonal tiling facets, with a triangular pyramid vertex figure.

480px

It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t{∞,3} with apeirogonal and triangle faces:

: 240px

= Bitruncated hexagonal tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Bitruncated hexagonal tiling honeycomb Bitruncated order-6 tetrahedral honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbol	2t{6,3,3} or t_1,2{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagram	{{CDD\|node\|6\|node_1\|3\|node_1\|3\|node}} {{CDD\|branch_11\|split2\|node_1\|3\|node}} ↔ {{CDD\|node_h0\|6\|node_1\|3\|node_1\|3\|node}}
bgcolor=#e7dcc3\|Cells	t{3,3} 40px t{3,6} 40px
bgcolor=#e7dcc3\|Faces	triangle {3} hexagon {6}
bgcolor=#e7dcc3\|Vertex figure	80px digonal disphenoid
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6] ${\overline{P}}_3$ , [3,3^[3]]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t_1,2{6,3,3}, {{CDD|node|6|node_1|3|node_1|3|node}} has truncated tetrahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.

480px

= Cantellated hexagonal tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Cantellated hexagonal tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbol	rr{6,3,3} or t_0,2{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagram	{{CDD\|node_1\|6\|node\|3\|node_1\|3\|node}}
bgcolor=#e7dcc3\|Cells	r{3,3} 40px rr{6,3} 40px {}×{3} 40px
bgcolor=#e7dcc3\|Faces	triangle {3} square {4} hexagon {6}
bgcolor=#e7dcc3\|Vertex figure	80px wedge
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The cantellated hexagonal tiling honeycomb, t_0,2{6,3,3}, {{CDD||node_1|6|node|3|node_1|3|node}} has octahedron, rhombitrihexagonal tiling, and triangular prism cells, with a wedge vertex figure.

480px

= Cantitruncated hexagonal tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Cantitruncated hexagonal tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbol	tr{6,3,3} or t_0,1,2{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagram	{{CDD\|node_1\|6\|node_1\|3\|node_1\|3\|node}}
bgcolor=#e7dcc3\|Cells	t{3,3} 40px tr{6,3} 40px {}×{3} 40px
bgcolor=#e7dcc3\|Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
bgcolor=#e7dcc3\|Vertex figure	80px mirrored sphenoid
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The cantitruncated hexagonal tiling honeycomb, t_0,1,2{6,3,3}, {{CDD||node_1|6|node_1|3|node_1|3|node}} has truncated tetrahedron, truncated trihexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.

480px

= Runcinated hexagonal tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Runcinated hexagonal tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbol	t_0,3{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagram	{{CDD\|node_1\|6\|node\|3\|node\|3\|node_1}}
bgcolor=#e7dcc3\|Cells	{3,3} 40px {6,3} 40px {}×{6}40px {}×{3} 40px
bgcolor=#e7dcc3\|Faces	triangle {3} square {4} hexagon {6}
bgcolor=#e7dcc3\|Vertex figure	80px irregular triangular antiprism
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The runcinated hexagonal tiling honeycomb, t_0,3{6,3,3}, {{CDD||node_1|6|node|3|node|3|node_1}} has tetrahedron, hexagonal tiling, hexagonal prism, and triangular prism cells, with an irregular triangular antiprism vertex figure.

480px

= Runcitruncated hexagonal tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Runcitruncated hexagonal tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbol	t_0,1,3{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagram	{{CDD\|node_1\|6\|node_1\|3\|node\|3\|node_1}}
bgcolor=#e7dcc3\|Cells	rr{3,3} 40px {}x{3} 40px {}x{12} 40px t{6,3} 40px
bgcolor=#e7dcc3\|Faces	triangle {3} square {4} dodecagon {12}
bgcolor=#e7dcc3\|Vertex figure	80px isosceles-trapezoidal pyramid
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The runcitruncated hexagonal tiling honeycomb, t_0,1,3{6,3,3}, {{CDD||node_1|6|node_1|3|node|3|node_1}} has cuboctahedron, triangular prism, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

480px

= Runcicantellated hexagonal tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Runcicantellated hexagonal tiling honeycomb runcitruncated order-6 tetrahedral honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbol	t_0,2,3{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagram	{{CDD\|node_1\|6\|node\|3\|node_1\|3\|node_1}}
bgcolor=#e7dcc3\|Cells	t{3,3} 40px {}x{6} 40px rr{6,3} 40px
bgcolor=#e7dcc3\|Faces	triangle {3} square {4} hexagon {6}
bgcolor=#e7dcc3\|Vertex figure	80px isosceles-trapezoidal pyramid
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t_0,2,3{6,3,3}, {{CDD||node_1|6|node|3|node_1|3|node_1}} has truncated tetrahedron, hexagonal prism, and rhombitrihexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

480px

= Omnitruncated hexagonal tiling honeycomb =

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Omnitruncated hexagonal tiling honeycomb Omnitruncated order-6 tetrahedral honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbol	t_0,1,2,3{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagram	{{CDD\|node_1\|6\|node_1\|3\|node_1\|3\|node_1}}
bgcolor=#e7dcc3\|Cells	tr{3,3} 40px {}x{6} 40px {}x{12} 40px tr{6,3} 40px
bgcolor=#e7dcc3\|Faces	square {4} hexagon {6} dodecagon {12}
bgcolor=#e7dcc3\|Vertex figure	80px irregular tetrahedron
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t_0,1,2,3{6,3,3}, {{CDD||node_1|6|node_1|3|node_1|3|node_1}} has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with an irregular tetrahedron vertex figure.

480px

References

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|99035678}}, {{ISBN|0-486-40919-8}} (Chapter 10, [http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf Regular Honeycombs in Hyperbolic Space] {{Webarchive|url=https://web.archive.org/web/20160610043106/http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf |date=2016-06-10 }}) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition {{ISBN|0-8247-0709-5}} (Chapters 16–17: Geometries on Three-manifolds I,II)
N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 [https://link.springer.com/article/10.1007%2FBF01238563] [https://web.archive.org/web/20140223225217/http://homeweb1.unifr.ch/kellerha/pub/TGarticle.pdf]
N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H³: p130. [http://www.sciencedirect.com/science/article/pii/S0024379501004773]

External links

John Baez, Visual Insight: [http://blogs.ams.org/visualinsight/2014/03/15/633-honeycomb/ {6,3,3} Honeycomb] (2014/03/15)
John Baez, Visual Insight: [http://blogs.ams.org/visualinsight/2013/09/15/633-honeycomb-in-upper-half-space/ {6,3,3} Honeycomb in Upper Half Space] (2013/09/15)
John Baez, Visual Insight: [http://blogs.ams.org/visualinsight/2016/12/01/truncated-633-honeycomb/ Truncated {6,3,3} Honeycomb] (2016/12/01)

Category:Hexagonal tilings

Category:Regular 3-honeycombs

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Hexagonal tiling honeycomb
bgcolor=#ffffff align=center colspan=2\|320px Perspective projection view within Poincaré disk model
bgcolor=#e7dcc3\|Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	{6,3,3} t{3,6,3} 2t{6,3,6} 2t{6,3^[3]} t{3^[3,3]}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node_1\|6\|node\|3\|node\|3\|node}} {{CDD\|node_1\|3\|node_1\|6\|node\|3\|node}} {{CDD\|node\|6\|node_1\|3\|node_1\|6\|node}} {{CDD\|branch_11\|split2\|node_1\|6\|node}} {{CDD\|branch_11\|splitcross\|branch_11}} ↔ {{CDD\|node_1\|6\|node_g\|3sg\|node_g\|3g\|node_g}} ↔ {{CDD\|node_1\|3\|node_1\|6\|node_g\|3sg\|node_g}} ↔ {{CDD\|node_h0\|6\|node_1\|3\|node_1\|6\|node_h0}} ↔ {{CDD\|branch_11\|split2\|node_1\|6\|node_h0}}
bgcolor=#e7dcc3\|Cells	{6,3} 40px
bgcolor=#e7dcc3\|Faces	hexagon {6}
bgcolor=#e7dcc3\|Edge figure	triangle {3}
bgcolor=#e7dcc3\|Vertex figure	80px tetrahedron {3,3}
bgcolor=#e7dcc3\|Dual	Order-6 tetrahedral honeycomb
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6] ${\overline{Y}}_3$ , [3,6,3] ${\overline{Z}}_3$ , [6,3,6] ${\overline{VP}}_3$ , [6,3^[3]] ${\overline{PP}}_3$ , [3^[3,3]]
bgcolor=#e7dcc3\|Properties	Regular

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Rectified hexagonal tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbols	r{6,3,3} or t₁{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagrams	{{CDD\|node\|6\|node_1\|3\|node\|3\|node}} {{CDD\|node_h0\|6\|node_1\|3\|node\|3\|node}} ↔ {{CDD\|branch_11\|split2\|node\|3\|node}}
bgcolor=#e7dcc3\|Cells	{3,3} 40px r{6,3} 40px or 40px
bgcolor=#e7dcc3\|Faces	triangle {3} hexagon {6}
bgcolor=#e7dcc3\|Vertex figure	80px triangular prism
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6] ${\overline{P}}_3$ , [3,3^[3]]
bgcolor=#e7dcc3\|Properties	Vertex-transitive, edge-transitive

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Truncated hexagonal tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbol	t{6,3,3} or t_0,1{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagram	{{CDD\|node_1\|6\|node_1\|3\|node\|3\|node}}
bgcolor=#e7dcc3\|Cells	{3,3} 40px t{6,3} 40px
bgcolor=#e7dcc3\|Faces	triangle {3} dodecagon {12}
bgcolor=#e7dcc3\|Vertex figure	80px triangular pyramid
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Bitruncated hexagonal tiling honeycomb Bitruncated order-6 tetrahedral honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbol	2t{6,3,3} or t_1,2{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagram	{{CDD\|node\|6\|node_1\|3\|node_1\|3\|node}} {{CDD\|branch_11\|split2\|node_1\|3\|node}} ↔ {{CDD\|node_h0\|6\|node_1\|3\|node_1\|3\|node}}
bgcolor=#e7dcc3\|Cells	t{3,3} 40px t{3,6} 40px
bgcolor=#e7dcc3\|Faces	triangle {3} hexagon {6}
bgcolor=#e7dcc3\|Vertex figure	80px digonal disphenoid
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6] ${\overline{P}}_3$ , [3,3^[3]]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2\|Cantellated hexagonal tiling honeycomb
bgcolor=#e7dcc3\|Type	Paracompact uniform honeycomb
bgcolor=#e7dcc3\|Schläfli symbol	rr{6,3,3} or t_0,2{6,3,3}
bgcolor=#e7dcc3\|Coxeter diagram	{{CDD\|node_1\|6\|node\|3\|node_1\|3\|node}}
bgcolor=#e7dcc3\|Cells	r{3,3} 40px rr{6,3} 40px {}×{3} 40px
bgcolor=#e7dcc3\|Faces	triangle {3} square {4} hexagon {6}
bgcolor=#e7dcc3\|Vertex figure	80px wedge
bgcolor=#e7dcc3\|Coxeter groups	${\overline{V}}_3$ , [3,3,6]
bgcolor=#e7dcc3\|Properties	Vertex-transitive

Hexagonal tiling honeycomb#Polytopes and honeycombs with tetrahedral vertex figures

Images

Symmetry constructions

Related polytopes and honeycombs

= Rectified hexagonal tiling honeycomb =

= Truncated hexagonal tiling honeycomb =

= Bitruncated hexagonal tiling honeycomb =

= Cantellated hexagonal tiling honeycomb =

= Cantitruncated hexagonal tiling honeycomb =

= Runcinated hexagonal tiling honeycomb =

= Runcitruncated hexagonal tiling honeycomb =

= Runcicantellated hexagonal tiling honeycomb =

= Omnitruncated hexagonal tiling honeycomb =

See also

References

External links