View source for Order-5 hexagonal tiling honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Order-5 hexagonal tiling honeycomb
|-
|colspan=2 align=center|[[File:H3 635 FC boundary.png|320px]]<BR>[[Perspective projection]] view<BR>from center of [[Poincaré disk model]]
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb#.5B6.2C3.2C4.5D family|Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|6|node|3|node|5|node}}<BR>[[File:CDel_635_index120.png|80px]] ↔ {{CDD|node_1|6|node_g|3sg|node_g|5g|node_g}}
|-
|bgcolor=#e7dcc3|Cells||[[Hexagonal tiling|{6,3}]] [[File:Uniform tiling 63-t0.svg|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Edge figure]]||[[pentagon]] {5}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||<!--[[File:Order-5_hexagonal_tiling_honeycomb_verf.png|100px]]<BR>-->[[regular icosahedron|icosahedron]]
|-
|bgcolor=#e7dcc3|[[Dual polytope|Dual]]||[[Order-6 dodecahedral honeycomb]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{HV}_3</math>, [5,3,6]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the field of [[Hyperbolic space|hyperbolic geometry]], the '''order-5 hexagonal tiling honeycomb''' arises as one of 11 [[Paracompact uniform honeycombs#Regular paracompact honeycombs|regular paracompact honeycombs]] in 3-dimensional [[hyperbolic space]]. It is ''paracompact'' because it has [[Cell (geometry)|cells]] composed of an infinite number of faces.  Each cell consists of a [[hexagonal tiling]] whose vertices lie on a [[horosphere]], a flat plane in hyperbolic space that approaches a single [[ideal point]] at infinity.

The [[Schläfli symbol]] of the order-5 hexagonal tiling honeycomb is {6,3,5}. Since that of the [[hexagonal tiling]] is {6,3}, this honeycomb has five such hexagonal tilings meeting at each edge.  Since the Schläfli symbol of the [[regular icosahedron|icosahedron]] is {3,5}, the [[vertex figure]] of this honeycomb is an icosahedron.  Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.<ref>Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III</ref>

{{Honeycomb}}

== Symmetry==

A lower-symmetry construction of index 120, [6,(3,5)<sup>*</sup>], exists with regular [[regular dodecahedron|dodecahedral]] fundamental domains, and an [[regular icosahedron|icosahedral]] [[Coxeter-Dynkin diagram]] with 6 axial infinite-order (ultraparallel) branches.

== Images==
The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact [[order-5 apeirogonal tiling]], {&infin;,5}, with five [[apeirogon]]al faces meeting around every vertex.
: [[File:H2 tiling 25i-1.png|180px]]

== Related polytopes and honeycombs ==
The order-5 hexagonal tiling honeycomb is a [[List of regular polytopes#Tessellations of hyperbolic 3-space|regular hyperbolic honeycomb]] in 3-space, and one of 11 which are paracompact.
{{Regular_paracompact_H3_honeycombs}}

There are [[Convex uniform honeycombs in hyperbolic space#.5B5.2C3.2C6.5D family|15 uniform honeycombs]] in the [6,3,5] [[Coxeter group]] family, including this regular form, and its regular dual, the [[order-6 dodecahedral honeycomb]].
{{635 family}}

The order-5 hexagonal tiling honeycomb has a related [[Alternation (geometry)|alternation]] honeycomb, represented by {{CDD|node_h1|6|node|3|node|5|node}} ↔ {{CDD|branch_10ru|split2|node|5|node}}, with [[regular icosahedron|icosahedron]] and [[triangular tiling]] cells.

It is a part of sequence of regular hyperbolic honeycombs  of the form {6,3,p}, with [[hexagonal tiling]] facets:
{{Hexagonal tiling cell tessellations}}

It is also part of a sequence of [[regular polychora]] and honeycombs with [[regular icosahedron|icosahedral]] vertex figures:
{{Icosahedral vertex figure tessellations}}

=== Rectified order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Rectified order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||r{6,3,5} or t<sub>1</sub>{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node|6|node_1|3|node|5|node}}<BR>{{CDD|node_h0|6|node_1|3|node|5|node}} ↔ {{CDD|branch_11|split2|node|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[regular icosahedron|{3,5}]] [[File:Uniform polyhedron-53-t2.png|40px]]<BR>[[Trihexagonal tiling|r{6,3}]] or h<sub>2</sub>{6,3}<BR>[[File:Uniform tiling 63-t1.png|40px]][[File:Uniform polyhedron-63-t1-1.svg|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Rectified order-5 hexagonal tiling honeycomb verf.png|80px]]<BR>[[pentagonal prism]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HV}_3}</math>, [5,3,6]<BR><math>{\overline{HP}_3}</math>, [5,3<sup>[3]</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive
|}
The '''rectified order-5 hexagonal tiling honeycomb''', t<sub>1</sub>{6,3,5}, {{CDD|node|6|node_1|3|node|5|node}} has [[regular icosahedron|icosahedron]] and [[trihexagonal tiling]] facets, with a [[pentagonal prism]] [[vertex figure]].

[[File:H3 635 boundary 0100.png|480px]]

It is similar to the 2D hyperbolic [[infinite-order square tiling]], r{&infin;,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.
: [[File:H2 tiling 25i-2.png|180px]]

{{Pentagonal prism vertex figure tessellations}}
{{Clear}}

=== Truncated order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Truncated order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t{6,3,5} or t<sub>0,1</sub>{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|6|node_1|3|node|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[regular icosahedron|{3,5}]] [[File:Uniform polyhedron-53-t2.png|40px]]<BR>[[truncated hexagonal tiling|t{6,3}]] [[File:Uniform tiling 63-t01.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[dodecagon]] {12}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Truncated order-5 hexagonal tiling honeycomb verf.png|80px]]<BR>[[pentagonal pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HV}}_3</math>, [5,3,6]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''truncated order-5 hexagonal tiling honeycomb''', t<sub>0,1</sub>{6,3,5}, {{CDD|node_1|6|node_1|3|node|5|node}} has [[regular icosahedron|icosahedron]] and [[truncated hexagonal tiling]] facets, with a [[pentagonal pyramid]] [[vertex figure]].

[[File:H3 635-1100.png|480px]]
{{Clear}}

=== Bitruncated order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Bitruncated order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||2t{6,3,5} or t<sub>1,2</sub>{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node|6|node_1|3|node_1|5|node}}<BR>{{CDD|node_h0|6|node_1|3|node_1|5|node}} ↔ {{CDD|branch_11|split2|node_1|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[hexagonal tiling|t{3,6}]] [[File:Uniform tiling 63-t12.svg|40px]]<BR>[[truncated icosahedron|t{3,5}]] [[File:Uniform polyhedron-53-t12.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[pentagon]] {5}<BR>[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Bitruncated order-5 hexagonal tiling honeycomb verf.png|80px]]<BR>[[digonal disphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HV}}_3</math>, [5,3,6]<BR><math>{\overline{HP}}_3</math>, [5,3<sup>[3]</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''bitruncated order-5 hexagonal tiling honeycomb''', t<sub>1,2</sub>{6,3,5}, {{CDD|node|6|node_1|3|node_1|5|node}} has [[hexagonal tiling]] and [[truncated icosahedron]] facets, with a [[digonal disphenoid]] [[vertex figure]].

[[File:H3 635-0110.png|480px]]
{{Clear}}

=== Cantellated order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantellated order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||rr{6,3,5} or t<sub>0,2</sub>{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|6|node|3|node_1|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Icosidodecahedron|r{3,5}]] [[File:Uniform polyhedron-53-t1.png|40px]]<BR>[[rhombitrihexagonal tiling|rr{6,3}]] [[File:Uniform tiling 63-t02.png|40px]]<BR>[[pentagonal prism|{}x{5}]] [[File:Pentagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[pentagon]] {5}<BR>[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Cantellated order-5 hexagonal tiling honeycomb verf.png|80px]]<BR>[[wedge (geometry)|wedge]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HV}}_3</math>, [5,3,6]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantellated order-5 hexagonal tiling honeycomb''', t<sub>0,2</sub>{6,3,5}, {{CDD|node_1|6|node|3|node_1|5|node}} has [[icosidodecahedron]], [[rhombitrihexagonal tiling]], and [[pentagonal prism]] facets, with a [[wedge (geometry)|wedge]] [[vertex figure]].

[[File:H3 635-1010.png|480px]]
{{Clear}}

=== Cantitruncated order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantitruncated order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||tr{6,3,5} or t<sub>0,1,2</sub>{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|6|node_1|3|node_1|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated icosahedron|t{3,5}]] [[File:Uniform polyhedron-53-t12.png|40px]]<BR>[[Truncated trihexagonal tiling|tr{6,3}]] [[File:Uniform tiling 63-t012.svg|40px]]<BR>[[pentagonal prism|{}x{5}]] [[File:Pentagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[pentagon]] {5}<BR>[[hexagon]] {6}<BR>[[dodecagon]] {12}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Cantitruncated order-5 hexagonal tiling honeycomb verf.png|80px]]<BR>[[mirrored sphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HV}}_3</math>, [5,3,6]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}

The '''cantitruncated order-5 hexagonal tiling honeycomb''', t<sub>0,1,2</sub>{6,3,5}, {{CDD|node_1|6|node_1|3|node_1|5|node}} has [[truncated icosahedron]], [[truncated trihexagonal tiling]], and [[pentagonal prism]] facets, with a [[mirrored sphenoid]] [[vertex figure]].

[[File:H3 635-1110.png|480px]]

{{Clear}}

=== Runcinated order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcinated order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,3</sub>{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|6|node|3|node|5|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[Hexagonal tiling|{6,3}]] [[File:Uniform tiling 63-t0.svg|40px]]<BR>[[regular dodecahedron|{5,3}]] [[File:Uniform polyhedron-53-t0.png|40px]]<BR>[[Hexagonal prism|{}x{6}]] [[File:Hexagonal prism.png|40px]]<BR>[[Pentagonal prism|{}x{5}]] [[File:Pentagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[pentagon]] {5}<BR>[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcinated order-5 hexagonal tiling honeycomb verf.png|80px]]<BR>irregular [[triangular antiprism]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HV}}_3</math>, [5,3,6]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcinated order-5 hexagonal tiling honeycomb''', t<sub>0,3</sub>{6,3,5}, {{CDD|node_1|6|node|3|node|5|node_1}} has [[regular dodecahedron|dodecahedron]], [[hexagonal tiling]], [[pentagonal prism]], and [[hexagonal prism]] facets, with an irregular [[triangular antiprism]] [[vertex figure]].

[[File:H3 635-1001.png|480px]]
{{Clear}}

=== Runcitruncated order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcitruncated order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,3</sub>{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|6|node_1|3|node|5|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated hexagonal tiling|t{6,3}]] [[File:Uniform tiling 63-t01.png|40px]]<BR>[[rhombicosidodecahedron|rr{5,3}]] [[File:Uniform polyhedron-53-t02.png|40px]]<BR>[[pentagonal prism|{}x{5}]] [[File:Pentagonal prism.png|40px]]<BR>[[dodecagonal prism|{}x{12}]] [[File:Dodecagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[pentagon]] {5}<BR>[[dodecagon]] {12}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcitruncated order-5 hexagonal tiling honeycomb verf.png|80px]]<BR>[[isosceles trapezoid|isosceles-trapezoidal]] [[pyramid (geometry)|pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HV}}_3</math>, [5,3,6]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcitruncated order-5 hexagonal tiling honeycomb''', t<sub>0,1,3</sub>{6,3,5}, {{CDD|node_1|6|node_1|3|node|5|node_1}} has [[truncated hexagonal tiling]], [[rhombicosidodecahedron]], [[pentagonal prism]], and [[dodecagonal prism]] cells, with an [[isosceles trapezoid|isosceles-trapezoidal]] [[pyramid (geometry)|pyramid]] [[vertex figure]].

[[File:H3 635-1101.png|480px]]
{{Clear}}

=== Runcicantellated order-5 hexagonal tiling honeycomb ===

The '''runcicantellated order-5 hexagonal tiling honeycomb''' is the same as the [[runcitruncated order-6 dodecahedral honeycomb]].

=== Omnitruncated order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Omnitruncated order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,2,3</sub>{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|6|node_1|3|node_1|5|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated trihexagonal tiling|tr{6,3}]] [[File:Uniform tiling 63-t012.png|40px]]<BR>[[truncated icosidodecahedron|tr{5,3}]] [[File:Uniform polyhedron-53-t012.png|40px]]<BR>[[decagonal prism|{}x{10}]] [[File:Decagonal prism.png|40px]]<BR>[[dodecagonal prism|{}x{12}]] [[File:Dodecagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[hexagon]] {6}<BR>[[decagon]] {10}<BR>[[dodecagon]] {12}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Omnitruncated order-5 hexagonal tiling honeycomb verf.png|80px]]<BR>irregular [[tetrahedron]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HV}}_3</math>, [5,3,6]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''omnitruncated order-5 hexagonal tiling honeycomb''', t<sub>0,1,2,3</sub>{6,3,5}, {{CDD|node_1|6|node_1|3|node_1|5|node_1}} has [[truncated trihexagonal tiling]], [[truncated icosidodecahedron]], [[decagonal prism]], and [[dodecagonal prism]] facets, with an irregular [[tetrahedral]] [[vertex figure]].

[[File:H3 635-1111.png|480px]]
{{Clear}}

=== Alternated order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Alternated order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]<BR>[[Semiregular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h1|6|node|3|node|5|node}} ↔ {{CDD|branch_10ru|split2|node|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[triangular tiling|{3<sup>[3]</sup>}]] [[File:Uniform tiling 333-t1.svg|40px]]<BR>[[icosahedron|{3,5}]] [[File:Uniform polyhedron-53-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Uniform polyhedron-53-t12.png|40px]]<BR>[[truncated icosahedron]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HP}}_3</math>, [5,3<sup>[3]</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive, [[quasiregular honeycomb|quasiregular]]
|}
The '''alternated order-5 hexagonal tiling honeycomb''', h{6,3,5}, {{CDD|node_h1|6|node|3|node|5|node}} ↔ {{CDD|branch_10ru|split2|node|5|node}}, has [[triangular tiling]] and [[icosahedron]] facets, with a [[truncated icosahedron]] [[vertex figure]]. It is a [[quasiregular honeycomb]].

{{Clear}}

=== Cantic order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantic order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>2</sub>{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h1|6|node|3|node_1|5|node}} ↔ {{CDD|branch_10ru|split2|node_1|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[trihexagonal tiling|h<sub>2</sub>{6,3}]] [[File:Uniform polyhedron-63-t1-1.svg|40px]]<BR>[[truncated icosahedron|t{3,5}]] [[File:Uniform polyhedron-53-t12.png|40px]]<BR>[[icosidodecahedron|r{5,3}]] [[File:Uniform polyhedron-53-t1.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[pentagon]] {5}<BR>[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Cantic order-5 hexagonal tiling honeycomb verf.png|80px]]<BR>[[triangular prism]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HP}}_3</math>, [5,3<sup>[3]</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantic order-5 hexagonal tiling honeycomb''', h<sub>2</sub>{6,3,5}, {{CDD|node_h1|6|node|3|node_1|5|node}} ↔ {{CDD|branch_10ru|split2|node_1|5|node}}, has [[trihexagonal tiling]], [[truncated icosahedron]], and [[icosidodecahedron]] facets, with a [[triangular prism]] [[vertex figure]].

{{Clear}}

=== Runcic order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcic order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>3</sub>{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h1|6|node|3|node|5|node_1}} ↔ {{CDD|branch_10ru|split2|node|5|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[triangular tiling|{3<sup>[3]</sup>}]] [[File:Uniform tiling 333-t1.svg|40px]]<BR>[[rhombicosidodecahedron|rr{5,3}]] [[File:Uniform polyhedron-53-t02.png|40px]]<BR>[[dodecahedron|{5,3}]] [[File:Uniform polyhedron-53-t0.png|40px]]<BR>[[triangular prism|{}x{3}]] [[File:Triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[pentagon]] {5}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcic order-5 hexagonal tiling honeycomb verf.png|80px]]<BR>[[triangular cupola]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HP}}_3</math>, [5,3<sup>[3]</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcic order-5 hexagonal tiling honeycomb''', h<sub>3</sub>{6,3,5}, {{CDD|node_h1|6|node|3|node|5|node_1}} ↔ {{CDD|branch_10ru|split2|node|5|node_1}}, has [[triangular tiling]], [[rhombicosidodecahedron]], [[dodecahedron]], and [[triangular prism]] facets, with a [[triangular cupola]] [[vertex figure]].

{{Clear}}

=== Runcicantic order-5 hexagonal tiling honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcicantic order-5 hexagonal tiling honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>2,3</sub>{6,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h1|6|node|3|node_1|5|node_1}} ↔ {{CDD|branch_10ru|split2|node_1|5|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[trihexagonal tiling|h<sub>2</sub>{6,3}]] [[File:Uniform polyhedron-63-t1-1.svg|40px]]<BR>[[truncated icosidodecahedron|tr{5,3}]] [[File:Uniform polyhedron-53-t012.png|40px]]<BR>[[truncated dodecahedron|t{5,3}]] [[File:Uniform polyhedron-53-t01.png|40px]]<BR>[[triangular prism|{}x{3}]] [[File:Triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[hexagon]] {6}<BR>[[decagon]] {10}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcicantic order-5 hexagonal tiling honeycomb verf.png|80px]]<BR>[[rectangle|rectangular]] [[pyramid (geometry)|pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>{\overline{HP}}_3</math>, [5,3<sup>[3]</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcicantic order-5 hexagonal tiling honeycomb''', h<sub>2,3</sub>{6,3,5}, {{CDD|node_h1|6|node|3|node_1|5|node_1}} ↔ {{CDD|branch_10ru|split2|node_1|5|node_1}}, has [[trihexagonal tiling]], [[truncated icosidodecahedron]], [[truncated dodecahedron]], and [[triangular prism]] facets, with a [[rectangle|rectangular]] [[pyramid (geometry)|pyramid]] [[vertex figure]].

{{Clear}}

== See also ==
* [[Convex uniform honeycombs in hyperbolic space]]
* [[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular tessellations of hyperbolic 3-space]]
* [[Paracompact uniform honeycomb]]s

== References ==

<references />
*[[H.S.M. Coxeter|Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;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]) Table III
* [[Jeffrey Weeks (mathematician)|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 (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript
** [[Norman Johnson (mathematician)|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

[[Category:Hexagonal tilings]]
[[Category:Regular 3-honeycombs]]