View source for Order-4 dodecahedral honeycomb

{{Short description|Regular tiling of hyperbolic 3-space}}
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|'''Order-4 dodecahedral honeycomb'''
|-
|bgcolor=#ffffff align=center colspan=2|[[File:H3_534_CC_center.png|320px]]
|-
|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|{5,3,4}<BR>{5,3{{sup|1,1}}} }}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node|3|node|4|node}}<BR>{{CDD|node_1|5|node|3|node|4|node_h0}} ↔ {{CDD|node_1|5|node|split1|nodes}}
|-
|bgcolor=#e7dcc3|Cells||{{math|{5,3} }}([[dodecahedron]])<br/>[[File:Uniform_polyhedron-53-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||{{math|{5} }} ([[pentagon]])
|-
|bgcolor=#e7dcc3|[[Edge figure]]||{{math|{4} }} ([[square (geometry)|square]])
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Order-4 dodecahedral honeycomb verf.png|80px]]<BR>[[octahedron]]
|-
|bgcolor=#e7dcc3|[[Dual polytope|Dual]]||[[Order-5 cubic honeycomb]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||{{math|{{overline|BH}}{{sub|3}}, [4,3,5]<BR>{{overline|DH}}{{sub|3}}, [5,3{{sup|1,1}}]}}
|-
|bgcolor=#e7dcc3|Properties||Regular, [[Quasiregular honeycomb]]
|}

In [[hyperbolic geometry]], the '''order-4 dodecahedral honeycomb''' is one of four compact [[regular polytope|regular]] space-filling [[tessellation]]s (or [[honeycomb (geometry)|honeycombs]]) of [[hyperbolic 3-space]]. With [[Schläfli symbol]] {{math|{5,3,4},}} it has four [[regular dodecahedron|dodecahedra]] around each [[Edge (geometry)|edge]], and 8 dodecahedra around each [[Vertex (geometry)|vertex]] in an [[octahedron|octahedral]] arrangement. Its vertices are constructed from 3 orthogonal axes. Its [[dual polytope|dual]] is the [[order-5 cubic honeycomb]].

{{Honeycomb}}

==Description==
The [[dihedral angle]] of a [[regular dodecahedron]] is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly-scaled regular dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge.

== Symmetry ==
It has a half symmetry construction, {5,3<sup>1,1</sup>}, with two types (colors) of dodecahedra in the [[Wythoff construction]]. {{CDD|node_1|5|node|3|node|4|node_h0}} ↔ {{CDD|node_1|5|node|split1|nodes}}.

== Images ==
[[File:H2-5-4-dual.svg|240px|thumb|It can be seen as analogous to the 2D hyperbolic [[order-4 pentagonal tiling]], {5,4}]]
[[Image:Hyperbolic orthogonal dodecahedral honeycomb.png|320px]]<BR>A view of the order-4 dodecahedral honeycomb under the [[Beltrami-Klein model]]

== Related polytopes and honeycombs ==
There are 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 this regular form. 
{{534 family}}

There are [[Convex uniform honeycombs in hyperbolic space#.5B5.2C31.2C1.5D family|eleven uniform honeycombs]] in the bifurcating [5,3<sup>1,1</sup>] Coxeter group family, including this honeycomb in its alternated form. 
This construction can be represented by alternation (checkerboard) with two colors of dodecahedral cells.

This honeycomb is also related to the [[16-cell]], [[cubic honeycomb]], and [[order-4 hexagonal tiling honeycomb]] all which have octahedral vertex figures:
{{Octahedral_vertex_figure_tessellations}}

This honeycomb is a part of a sequence of polychora and honeycombs with [[dodecahedral]] cells:
{{Dodecahedral_cell_tessellations}}

=== Rectified order-4 dodecahedral honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Rectified order-4 dodecahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||r{5,3,4}<BR>r{5,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node|5|node_1|3|node|4|node}}<BR>{{CDD|node|5|node_1|3|node|4|node_h0}} ↔ {{CDD|node|5|node_1|split1|nodes}}
|-
|bgcolor=#e7dcc3|Cells||[[icosidodecahedron|r{5,3}]] [[File:Uniform_polyhedron-53-t1.png|40px]]<BR>[[octahedron|{3,4}]] [[File:Uniform_polyhedron-43-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[pentagon]] {5}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Rectified order-4 dodecahedral honeycomb verf.png|80px]]<BR>[[square 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-4 dodecahedral honeycomb''', {{CDD|node|5|node_1|3|node|4|node}}, has alternating [[octahedron]] and [[icosidodecahedron]] cells, with a [[square prism]] [[vertex figure]].
: [[File:H3_534_CC_center_0100.png|320px]][[File:Rectified order 4 dodecahedral honeycomb.png|240px]]

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

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

{{Clear}}

=== Truncated order-4 dodecahedral honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Truncated order-4 dodecahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t{5,3,4}<BR>t{5,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node_1|3|node|4|node}}<BR>{{CDD|node_1|5|node_1|3|node|4|node_h0}} ↔ {{CDD|node_1|5|node_1|split1|nodes}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated dodecahedron|t{5,3}]] [[File:Uniform_polyhedron-53-t01.png|40px]]<BR>[[octahedron|{3,4}]] [[File:Uniform_polyhedron-43-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[decagon]] {10}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Truncated order-4 dodecahedral honeycomb verf.png|80px]]<BR>[[square pyramid]]
|-
|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
|}
The '''truncated order-4 dodecahedral honeycomb''', {{CDD|node_1|5|node_1|3|node|4|node}}, has [[octahedron]] and [[truncated dodecahedron]] cells, with a [[square pyramid]] [[vertex figure]].

[[File:H3 435-0011 center ultrawide.png|480px]]

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

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

{{Clear}}

=== Bitruncated order-4 dodecahedral honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Bitruncated order-4 dodecahedral honeycomb<BR>Bitruncated order-5 cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||2t{5,3,4}<BR>2t{5,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node|5|node_1|3|node_1|4|node}}<BR>{{CDD|node|5|node_1|3|node_1|4|node_h0}} ↔ {{CDD|node|5|node_1|split1|nodes_11}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated icosahedron|t{3,5}]] [[File:Uniform_polyhedron-53-t12.png|40px]]<BR>[[Truncated octahedron|t{3,4}]] [[File:Uniform_polyhedron-43-t12.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[pentagon]] {5}<BR>[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Bitruncated order-4 dodecahedral honeycomb verf.png|80px]]<BR>[[digonal disphenoid]]
|-
|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
|}
The '''bitruncated order-4 dodecahedral honeycomb''', or '''bitruncated order-5 cubic honeycomb''', {{CDD|node|5|node_1|3|node_1|4|node}}, has [[truncated octahedron]] and [[truncated icosahedron]] cells, with a [[digonal disphenoid]] [[vertex figure]].

[[File:H3 534-0110 center ultrawide.png|480px]]

==== Related honeycombs====
{{Bitruncated compact H3 honeycombs}}
{{Clear}}

=== Cantellated order-4 dodecahedral honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantellated order-4 dodecahedral honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||rr{5,3,4}<BR>rr{5,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node|3|node_1|4|node}}<BR>{{CDD|node_1|5|node|3|node_1|4|node_h0}} ↔ {{CDD|node_1|5|node|split1|nodes_11}}
|-
|bgcolor=#e7dcc3|Cells||[[rhombicosidodecahedron|rr{3,5}]] [[File:Uniform_polyhedron-53-t02.png|40px]]<BR>[[cuboctahedron|r{3,4}]] [[File:Uniform_polyhedron-43-t1.svg|40px]]<BR>[[cube|{}x{4}]] [[File:Tetragonal_prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[pentagon]] {5}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Cantellated order-4 dodecahedral honeycomb verf.png|80px]]<BR>[[wedge (geometry)|wedge]]
|-
|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
|}
The '''cantellated order-4 dodecahedral honeycomb''', {{CDD|node_1|5|node|3|node_1|4|node}}, has [[rhombicosidodecahedron]], [[cuboctahedron]], and [[cube]] cells, with a [[wedge (geometry)|wedge]] [[vertex figure]].

[[File:H3 534-1010 center ultrawide.png|480px]]

==== Related honeycombs====
{{Cantellated compact H3 honeycombs}}
{{Clear}}

=== Cantitruncated order-4 dodecahedral honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantitruncated order-4 dodecahedral honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||tr{5,3,4}<BR>tr{5,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node_1|3|node_1|4|node}}<BR>{{CDD|node_1|5|node_1|3|node_1|4|node_h0}} ↔ {{CDD|node_1|5|node_1|split1|nodes_11}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated icosidodecahedron|tr{3,5}]] [[File:Uniform_polyhedron-53-t012.png|40px]]<BR>[[truncated octahedron|t{3,4}]] [[File:Uniform_polyhedron-43-t12.png|40px]]<BR>[[cube|{}x{4}]] [[File:Tetragonal_prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[hexagon]] {6}<BR>[[decagon]] {10}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Cantitruncated order-4 dodecahedral honeycomb verf.png|80px]]<BR>[[mirrored sphenoid]]
|-
|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
|}
The '''cantitruncated order-4 dodecahedral honeycomb''', {{CDD|node_1|5|node_1|3|node_1|4|node}}, has [[truncated icosidodecahedron]], [[truncated octahedron]], and [[cube]] cells, with a [[mirrored sphenoid]] [[vertex figure]].

[[File:H3 534-1110 center ultrawide.png|480px]]

==== Related honeycombs====
{{Cantitruncated compact H3 honeycombs}}
{{Clear}}

=== Runcinated order-4 dodecahedral honeycomb ===

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

=== Runcitruncated order-4 dodecahedral honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcitruncated order-4 dodecahedral honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Uniform honeycombs in hyperbolic space]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,3</sub>{5,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node_1|3|node|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated dodecahedron|t{5,3}]] [[File:Uniform_polyhedron-53-t01.png|40px]]<BR>[[rhombicuboctahedron|rr{3,4}]] [[File:Uniform_polyhedron-43-t02.png|40px]]<BR>[[decagonal prism|{}x{10}]] [[File:decagonal_prism.png|40px]]<BR>[[cube|{}x{4}]] [[File:Tetragonal_prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[decagon]] {10}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcitruncated order-4 dodecahedral 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-4 dodecahedral honeycomb''', {{CDD|node_1|5|node_1|3|node|4|node_1}}, has [[truncated dodecahedron]], [[rhombicuboctahedron]], [[decagonal prism]], and [[cube]] cells, with an [[isosceles trapezoid|isosceles-trapezoidal]] [[pyramid (geometry)|pyramid]] [[vertex figure]].

[[File:H3 534-1101 center ultrawide.png|480px]]

==== Related honeycombs====
{{Runcitruncated compact H3 honeycombs}}

{{Clear}}

=== Runcicantellated order-4 dodecahedral honeycomb ===

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

=== Omnitruncated order-4 dodecahedral honeycomb ===

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

== See also ==
* [[Convex uniform honeycombs in hyperbolic space]]
* [[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular tessellations of hyperbolic 3-space]]
* [[Poincaré homology sphere]] Poincaré dodecahedral space
* [[Seifert–Weber space]] Seifert–Weber dodecahedral 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)
* [[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

{{DEFAULTSORT:Order-4 Dodecahedral Honeycomb}}
[[Category:Regular 3-honeycombs]]