View source for Order-5 cubic honeycomb

{{Short description|Regular tiling of hyperbolic 3-space}}
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|'''Order-5 cubic honeycomb'''
|-
|bgcolor=#ffffff align=center colspan=2|[[File:H3_435_CC_center.png|320px]]<BR>[[Poincaré disk model]]s
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Uniform honeycombs in hyperbolic space|Uniform hyperbolic honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{{math|{4,3,5} }}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|4|node|3|node|5|node}}
|-
|bgcolor=#e7dcc3|[[Cell (geometry)|Cells]]||{{math|{4,3} }} ([[cube]])<br/>[[File:Uniform_polyhedron-43-t0.png|40px]]
|-
|bgcolor=#e7dcc3|[[Face (geometry)|Faces]]||{{math|{4} }} ([[square]])
|-
|bgcolor=#e7dcc3|[[Vertex figure#Edge figure|Edge figure]]||{{math|{5} }} ([[pentagon]])
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Order-5 cubic honeycomb verf.svg|80px]]<BR>[[regular icosahedron|icosahedron]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||{{math|{{overline|BH}}{{sub|3}}, [4,3,5]}}
|-
|bgcolor=#e7dcc3|Dual||[[Order-4 dodecahedral honeycomb]]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}

In [[hyperbolic geometry]], the '''order-5 cubic honeycomb''' is one of four compact [[regular polytope|regular]] space-filling [[tessellation]]s (or [[honeycomb (geometry)|honeycomb]]s) in [[Hyperbolic space|hyperbolic 3-space]]. With [[Schläfli symbol]] {{math|{4,3,5},}} it has five [[cube]]s {{math|{4,3} }} around each [[Edge (geometry)|edge]], and 20 cubes around each [[Vertex (geometry)|vertex]]. It is [[Dual polytope|dual]] with the [[order-4 dodecahedral honeycomb]].

{{Honeycomb}}

==Description ==
[[File:H2-5-4-primal.svg|200px|thumb|It is analogous to the 2D hyperbolic [[order-5 square tiling]], {4,5}]]
{| class=wikitable width=600
|[[File:Order-5 cubic honeycomb cell.png|200px]]<BR>One cell, centered in Poincare ball model
|[[Image:Hyperb_gcubic_hc_constr.png|200px]]<BR>Main cells
|[[Image:Hyperb_gcubic_hc.png|200px]]<BR>Cells with extended edges to ideal boundary
|}

== Symmetry ==
It has a radical subgroup symmetry construction with [[regular dodecahedron|dodecahedral]] fundamental domains: [[Coxeter notation#Extended symmetry|Coxeter notation]]: [4,(3,5)<sup>*</sup>], index 120.

== Related polytopes and honeycombs ==

The order-5 cubic honeycomb has a related [[Alternation (geometry)|alternated]] honeycomb, {{CDD|node_h1|4|node|3|node|5|node}} ↔ {{CDD|nodes_10ru|split2|node|5|node}}, with [[regular icosahedron|icosahedron]] and [[tetrahedron]] cells.

The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space:
{{Regular compact H3 honeycombs}}

There are [[Convex uniform honeycombs in hyperbolic space#.5B5.2C3.2C4.5D family|fifteen uniform honeycombs]] in the [5,3,4] [[Coxeter group]] family, including the order-5 cubic honeycomb as the regular form:
{{534 family}}

The order-5 cubic honeycomb is in a sequence of [[regular polychora]] and honeycombs with [[regular icosahedron|icosahedral]] [[vertex figure]]s.
{{Icosahedral vertex figure tessellations}}

It is also in a sequence of [[regular polychora]] and honeycombs with [[cube|cubic]] [[cell (geometry)|cells]]. The first polytope in the sequence is the [[tesseract]], and the second is the Euclidean [[cubic honeycomb]].
{{Cubic cell tessellations}}

=== Rectified order-5 cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Rectified order-5 cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||r{4,3,5} or 2r{5,3,4}<BR>2r{5,3<sup>1,1</sup>}
|-
|width=120 bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node|5|node|3|node_1|4|node}}<BR>{{CDD|node|5|node|3|node_1|4|node_h0}} ↔ {{CDD|node|5|node|split1|nodes_11}}
|-
|bgcolor=#e7dcc3|Cells||[[Cuboctahedron|r{4,3}]] [[File:Uniform_polyhedron-43-t1.svg|40px]]<BR>[[regular icosahedron|{3,5}]] [[File:Uniform_polyhedron-53-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Rectified_order-5_cubic_honeycomb_verf.png|80px]]<BR>[[pentagonal prism]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{BH}_3</math>, [4,3,5]<BR><math>\overline{DH}_3</math>, [5,3<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive
|}
The '''rectified order-5 cubic honeycomb''', {{CDD|node|5|node|3|node_1|4|node}}, has alternating [[regular icosahedron|icosahedron]] and [[cuboctahedron]] cells, with a [[pentagonal prism]] [[vertex figure]].

[[File:H3_435_CC_center_0100.png|480px]]

==== Related honeycomb====
[[File:H2-5-4-rectified.svg|240px|thumb|It can be seen as analogous to the 2D hyperbolic [[tetrapentagonal tiling]], r{4,5} with square and pentagonal faces]]

There are four rectified compact regular honeycombs:
{{Rectified compact H3 honeycombs}}

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

=== Truncated order-5 cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Truncated order-5 cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t{4,3,5}
|-
|width=120 bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node|5|node|3|node_1|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated cube|t{4,3}]] [[File:Uniform_polyhedron-43-t01.png|40px]]<BR>[[regular icosahedron|{3,5}]] [[File:Uniform_polyhedron-53-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Truncated_order-5_cubic_honeycomb_verf.png|80px]]<BR>[[pentagonal pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{BH}_3</math>, [4,3,5]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''truncated order-5 cubic honeycomb''', {{CDD|node_1|4|node_1|3|node|5|node}}, has [[truncated cube]] and [[regular icosahedron|icosahedron]] cells, with a [[pentagonal pyramid]] [[vertex figure]].

[[File:H3 534-0011 center ultrawide.png|480px]]

It can be seen as analogous to the 2D hyperbolic [[truncated order-5 square tiling]], t{4,5}, with truncated square and pentagonal faces:
: [[File:H2-5-4-trunc-primal.svg|240px]]

It is similar to the Euclidean (order-4) [[truncated cubic honeycomb]], t{4,3,4}, which has octahedral cells at the truncated vertices.
:[[File:Truncated_cubic_honeycomb.png|240px]]

==== Related honeycombs ====
{{Truncated compact H3 honeycombs}}

{{Clear}}

=== Bitruncated order-5 cubic honeycomb ===

The '''bitruncated order-5 cubic honeycomb''' is the same as the [[bitruncated order-4 dodecahedral honeycomb]].

=== Cantellated order-5 cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantellated order-5 cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||rr{4,3,5}
|-
|width=120 bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node|5|node_1|3|node|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[rhombicuboctahedron|rr{4,3}]] [[File:Uniform_polyhedron-43-t02.png|40px]]<BR>[[Icosidodecahedron|r{3,5}]] [[File:Uniform_polyhedron-53-t1.png|40px]]<BR>[[Pentagonal prism|{}x{5}]] [[File:Pentagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[pentagon]] {5}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Cantellated_order-5_cubic_honeycomb_verf.png|80px]]<BR>[[wedge (geometry)|wedge]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{BH}_3</math>, [4,3,5]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantellated order-5 cubic honeycomb''', {{CDD|node_1|4|node|3|node_1|5|node}}, has [[rhombicuboctahedron]], [[icosidodecahedron]], and [[pentagonal prism]] cells, with a [[wedge (geometry)|wedge]] [[vertex figure]].

[[File:H3_534-0101_center_ultrawide.png|480px]]

==== Related honeycombs====
It is similar to the Euclidean (order-4) [[cantellated cubic honeycomb]], rr{4,3,4}:
:[[File:Cantellated cubic honeycomb.png|240px]]

{{Cantellated compact H3 honeycombs}}
{{Clear}}

=== Cantitruncated order-5 cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantitruncated order-5 cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||tr{4,3,5}
|-
|width=120 bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node|5|node_1|3|node_1|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated cuboctahedron|tr{4,3}]] [[File:Uniform_polyhedron-43-t012.png|40px]]<BR>[[Truncated icosahedron|t{3,5}]] [[File:Uniform_polyhedron-53-t12.png|40px]]<BR>[[Pentagonal prism|{}x{5}]] [[File:Pentagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[pentagon]] {5}<BR>[[hexagon]] {6}<BR>[[octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Cantitruncated_order-5_cubic_honeycomb_verf.png|80px]]<BR>[[mirrored sphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{BH}_3</math>, [4,3,5]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantitruncated order-5 cubic honeycomb''', {{CDD|node_1|4|node_1|3|node_1|5|node}}, has [[truncated cuboctahedron]], [[truncated icosahedron]], and [[pentagonal prism]] cells, with a [[mirrored sphenoid]] [[vertex figure]].

[[File:H3_534-0111_center_ultrawide.png|480px]]

==== Related honeycombs====
It is similar to the Euclidean (order-4) [[cantitruncated cubic honeycomb]], tr{4,3,4}:
:[[File:2-Kuboktaederstumpf 1-Oktaederstumpf 1-Hexaeder.png|240px]]

{{Cantitruncated compact H3 honeycombs}}
{{Clear}}

=== Runcinated order-5 cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcinated order-5 cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]<BR>[[Semiregular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,3</sub>{4,3,5}
|-
|width=120 bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node|3|node|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[cube|{4,3}]] [[File:Uniform_polyhedron-43-t0.png|40px]]<BR>[[regular dodecahedron|{5,3}]] [[File:Uniform_polyhedron-53-t0.png|40px]]<BR>[[Pentagonal prism|{}x{5}]] [[File:Pentagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[pentagon]] {5}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcinated_order-5_cubic_honeycomb_verf.png|80px]]<BR>irregular [[triangular antiprism]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{BH}_3</math>, [4,3,5]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcinated order-5 cubic honeycomb''' or '''runcinated order-4 dodecahedral honeycomb''' {{CDD|node_1|5|node|3|node|4|node_1}}, has [[cube]], [[regular dodecahedron|dodecahedron]], and [[pentagonal prism]] cells, with an irregular [[triangular antiprism]] [[vertex figure]].

[[File:H3 534-1001 center ultrawide.png|480px]]

It is analogous to the 2D hyperbolic [[rhombitetrapentagonal tiling]], rr{4,5}, {{CDD|node_1|5|node|4|node_1}} with square and pentagonal faces:
: [[File:H2-5-4-cantellated.svg|240px]]

==== Related honeycombs====
It is similar to the Euclidean (order-4) [[runcinated cubic honeycomb]], t<sub>0,3</sub>{4,3,4}:
:[[File:Runcinated cubic honeycomb.png|240px]]

{{Runcinated compact H3 honeycombs}}
{{Clear}}

=== Runcitruncated order-5 cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runctruncated order-5 cubic honeycomb<BR>Runcicantellated order-4 dodecahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,3</sub>{4,3,5}
|-
|width=120 bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node|3|node_1|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated cube|t{4,3}]] [[File:Uniform_polyhedron-43-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>[[Octagonal prism|{}x{8}]] [[File:Octagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[pentagon]] {5}<BR>[[octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcitruncated_order-5_cubic_honeycomb_verf.png|80px]]<BR>[[isosceles trapezoid|isosceles-trapezoidal]] [[pyramid (geometry)|pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{BH}_3</math>, [4,3,5]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcitruncated order-5 cubic honeycomb''' or '''runcicantellated order-4 dodecahedral honeycomb''', {{CDD|node_1|4|node_1|3|node|5|node_1}}, has [[truncated cube]], [[rhombicosidodecahedron]], [[pentagonal prism]], and [[octagonal prism]] cells, with an [[isosceles trapezoid|isosceles-trapezoidal]] [[pyramid (geometry)|pyramid]] [[vertex figure]].

[[File:H3_534-1011_center_ultrawide.png|480px]]

==== Related honeycombs====
It is similar to the Euclidean (order-4) [[runcitruncated cubic honeycomb]], t<sub>0,1,3</sub>{4,3,4}:
:[[File:Runcitruncated cubic honeycomb.jpg|240px]]

{{Runcitruncated compact H3 honeycombs}}
{{Clear}}

=== Runcicantellated order-5 cubic honeycomb ===

The '''runcicantellated order-5 cubic honeycomb''' is the same as the [[runcitruncated order-4 dodecahedral honeycomb]].

=== Omnitruncated order-5 cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Omnitruncated order-5 cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]<BR>[[Semiregular honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,2,3</sub>{4,3,5}
|-
|width=120 bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node_1|3|node_1|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated icosidodecahedron|tr{5,3}]] [[File:Uniform_polyhedron-53-t012.png|40px]]<BR>[[truncated cuboctahedron|tr{4,3}]] [[File:Uniform_polyhedron-43-t012.png|40px]]<BR>[[decagonal prism|{10}x{}]] [[File:decagonal prism.png|40px]]<BR>[[octagonal prism|{8}x{}]] [[File:octagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[hexagon]] {6}<BR>[[octagon]] {8}<BR>[[decagon]] {10}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Omnitruncated_order-4 dodecahedral honeycomb_verf.png|80px]]<BR>irregular [[tetrahedron]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{BH}_3</math>, [4,3,5]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''omnitruncated order-5 cubic honeycomb''' or '''omnitruncated order-4 dodecahedral honeycomb''', {{CDD|node_1|5|node_1|3|node_1|4|node_1}}, has [[truncated icosidodecahedron]], [[truncated cuboctahedron]], [[decagonal prism]], and [[octagonal prism]] cells, with an irregular [[tetrahedron|tetrahedral]] vertex figure.

[[File:H3 534-1111 center ultrawide.png|480px]]

==== Related honeycombs====
It is similar to the Euclidean (order-4) [[omnitruncated cubic honeycomb]], t<sub>0,1,2,3</sub>{4,3,4}:
:[[File:Omnitruncated cubic honeycomb1.png|240px]]

{{Omnitruncated compact H3 honeycombs}}
{{Clear}}

=== Alternated order-5 cubic honeycomb ===

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Alternated order-5 cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|[[File:Alternated order 5 cubic honeycomb.png|320px]]-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h{4,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h1|4|node|3|node|5|node}} ↔ {{CDD|nodes_10ru|split2|node|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Tetrahedron|{3,3}]] [[File:Uniform polyhedron-33-t0.png|40px]]<BR>[[regular icosahedron|{3,5}]] [[Image:Uniform polyhedron-53-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Alternated order-5 cubic honeycomb verf.png|80px]]<BR>[[icosidodecahedron]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{DH}_3</math>, [5,3<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive, [[Quasiregular honeycomb|quasiregular]]
|}
In 3-dimensional hyperbolic geometry, the '''alternated order-5 cubic honeycomb''' is a uniform compact space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). With [[Schläfli symbol]] h{4,3,5}, it can be considered a [[quasiregular honeycomb]], alternating [[regular icosahedron|icosahedra]] and [[tetrahedron|tetrahedra]] around each vertex in an [[icosidodecahedron]] vertex figure.

[[File:Alternated order 5 cubic honeycomb.png|480px]]
{{Clear}}

==== Related honeycombs====
It has 3 related forms: the [[cantic order-5 cubic honeycomb]], {{CDD|node_h1|4|node|3|node_1|5|node}}, the [[runcic order-5 cubic honeycomb]], {{CDD|node_h1|4|node|3|node|5|node_1}}, and the [[runcicantic order-5 cubic honeycomb]], {{CDD|node_h1|4|node|3|node_1|5|node_1}}.

===Cantic order-5 cubic honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantic order-5 cubic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>2</sub>{4,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h1|4|node|3|node_1|5|node}} ↔ {{CDD|nodes_10ru|split2|node_1|5|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Icosidodecahedron|r{5,3}]] [[File:Uniform polyhedron-53-t1.png|40px]]<br>[[truncated icosahedron|t{3,5}]] [[Image:Uniform polyhedron-53-t12.png|40px]]<br>[[Truncated tetrahedron|t{3,3}]] [[Image:Uniform polyhedron-33-t01.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[pentagon]] {5}<BR>[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Truncated_alternated_order-5_cubic_honeycomb_verf.png|80px]]<BR>rectangular [[pyramid (geometry)|pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{DH}_3</math>, [5,3<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantic order-5 cubic honeycomb''' is a uniform compact space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]), with [[Schläfli symbol]] h<sub>2</sub>{4,3,5}. It has [[icosidodecahedron]], [[truncated icosahedron]], and [[truncated tetrahedron]] cells, with a rectangular [[pyramid (geometry)|pyramid]] [[vertex figure]].

[[File:H3 5311-0110 center ultrawide.png|480px]]
{{Clear}}

===Runcic order-5 cubic honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcic order-5 cubic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>3</sub>{4,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h1|4|node|3|node|5|node_1}} ↔ {{CDD|nodes_10ru|split2|node|5|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[regular dodecahedron|{5,3}]] [[Image:Uniform polyhedron-53-t0.png|40px]]<br>[[rhombicosidodecahedron|rr{5,3}]] [[Image:Uniform polyhedron-53-t02.png|40px]]<br>[[tetrahedron|{3,3}]] [[Image:Uniform polyhedron-33-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[pentagon]] {5}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcinated_alternated_order-5_cubic_honeycomb_verf.png|80px]]<BR>[[triangular frustum]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{DH}_3</math>, [5,3<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcic order-5 cubic honeycomb''' is a uniform compact space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]), with [[Schläfli symbol]] h<sub>3</sub>{4,3,5}. It has [[regular dodecahedron|dodecahedron]], [[rhombicosidodecahedron]], and [[tetrahedron]] cells, with a [[triangular frustum]] [[vertex figure]].

[[File:H3_5311-1010_center_ultrawide.png|480px]]
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===Runcicantic order-5 cubic honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcicantic order-5 cubic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>2,3</sub>{4,3,5}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h1|4|node|3|node_1|5|node_1}} ↔ {{CDD|nodes_10ru|split2|node_1|5|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated dodecahedron|t{5,3}]] [[Image:Uniform polyhedron-53-t01.png|40px]]<br>[[truncated icosidodecahedron|tr{5,3}]] [[Image:Uniform polyhedron-53-t012.png|40px]]<br>[[truncated tetrahedron|t{3,3}]] [[Image:Uniform polyhedron-33-t01.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[hexagon]] {6}<BR>[[decagon]] {10}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Runcitruncated alternated order-5 cubic honeycomb verf.png|80px]]<BR>irregular [[tetrahedron]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{DH}_3</math>, [5,3<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcicantic order-5 cubic honeycomb''' is a uniform compact space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]), with [[Schläfli symbol]] h<sub>2,3</sub>{4,3,5}. It has [[truncated dodecahedron]], [[truncated icosidodecahedron]], and [[truncated tetrahedron]] cells, with an irregular [[tetrahedron]] [[vertex figure]].

[[File:H3_5311-1110_center_ultrawide.png|480px]]
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==See also==
* [[Convex uniform honeycombs in hyperbolic space]]
* [[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular tessellations of hyperbolic 3-space]]

== 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)
*[[H.S.M. Coxeter|Coxeter]], ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999 {{isbn|0-486-40919-8}} (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
* [[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'', (2015) Chapter 13: Hyperbolic Coxeter groups

[[Category:Regular 3-honeycombs]]