View source for Rhombic dodecahedral honeycomb

{{Use American English|date = March 2019}}
{{Short description|Space-filling tesselation}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Rhombic dodecahedral honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[File:Rhombic dodecahedra.png|250px]]
|-
|bgcolor=#e7dcc3|Type||[[convex uniform honeycomb]] dual
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_fh|4|node|3|node|4|node}} = {{CDD|node_f1|3|node|split1-43|nodes}}<BR>{{CDD|node_f1|split1|nodes|split2|node}}
|-
|bgcolor=#e7dcc3|Cell type||[[File:Dodecahedrille cell.png|80px]]<BR>[[Rhombic dodecahedron]] ''V3.4.3.4''
|-
|bgcolor=#e7dcc3|Face types||[[Rhombus]]
|-
|bgcolor=#e7dcc3|[[Space group]]||Fm{{overline|3}}m (225)
|-
|bgcolor=#e7dcc3|[[Coxeter notation]]||½<math>{\tilde{C}}_3</math>, [1<sup>+</sup>,4,3,4]<BR><math>{\tilde{B}}_3</math>, [4,3<sup>1,1</sup>]<BR><math>{\tilde{A}}_3</math>×2, <[3<sup>[4]</sup>]>
|-
|bgcolor=#e7dcc3|Dual||[[tetrahedral-octahedral honeycomb]]
|-
|bgcolor=#e7dcc3|Properties||[[edge-transitive]], [[face-transitive]], [[cell-transitive]]
|}

The '''rhombic dodecahedral honeycomb''' (also '''dodecahedrille''') is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space.  It is the [[Voronoi diagram]] of the [[face-centered cubic]] sphere-packing, which has the densest possible packing of equal spheres in ordinary space (see [[Kepler conjecture]]).

== Geometry==
It consists of copies of a single [[Cell (mathematics)|cell]], the [[rhombic dodecahedron]].  All faces are [[rhombi]], with diagonals in the ratio 1:{{radic|2}}.  Three cells meet at each edge.  The honeycomb is thus [[cell-transitive]], [[face-transitive]], and [[edge-transitive]]; but it is not [[vertex-transitive]], as it has two kinds of vertex.  Each vertex with the obtuse rhombic face angles is shared by 4 cells; each vertex with the acute rhombic face angles is shared by 6 cells.

The rhombic dodecahedron can be twisted on one of its hexagonal cross-sections to form a [[trapezo-rhombic dodecahedron]], which is the cell of a somewhat similar tessellation, the [[Voronoi diagram]] of hexagonal [[close-packing]].

{| class=wikitable width=480
|- valign=top
|[[File:HC R1.png|240px]]
|[[File:Cubes-R1 ani.gif|240px]]<BR>The honeycomb can be derived from an alternate cube tessellation by augmenting each face of each cube with a pyramid.
|[[File:Rhombic_dodecahedral_honeycomb.png|400px]]<BR>The view from inside the rhombic dodecahedral honeycomb.
|}
=== Colorings ===
The tiling's cells can be [[graph coloring|4-colored]] in square layers of 2 colors each, such that two cells of the same color touch only at vertices; or they can be 6-colored in hexagonal layers of 3 colors each, such that same-colored cells have no contact at all.
{| class=wikitable width=640
!4-coloring||6-coloring
|-
|[[File:Rhombic_dodecahedral_honeycomb_4-color.gif|320px]]
|[[File:Rhombic_dodecahedral_honeycomb_6-color.gif|320px]]
|-
|Alternate square layers of yellow/blue and red/green
|Alternate hexagonal layers of red/green/blue and magenta/yellow/cyan
|}
{{-}}

==Related honeycombs==
The ''rhombic dodecahedral honeycomb'' can be dissected into a [[trigonal trapezohedral honeycomb]] with each rhombic dodecahedron dissected into 4 [[trigonal trapezohedron]]s. Each rhombic dodecahedra can also be dissected with a center point into 12 rhombic pyramids of the [[rhombic pyramidal honeycomb]].

=== Trapezo-rhombic dodecahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Trapezo-rhombic dodecahedral honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Trapez rhombic dodeca hb.png|250px]]
|-
|bgcolor=#e7dcc3|Type||[[convex uniform honeycomb]] dual
|-
|bgcolor=#e7dcc3|Cell type||[[trapezo-rhombic dodecahedron]] ''VG3.4.3.4''<BR>[[File:Trapezo-rhombic dodecahedron.png|60px]]
|-
|bgcolor=#e7dcc3|Face types||[[rhombus]],<BR>[[trapezoid]]
|-
|bgcolor=#e7dcc3|[[Symmetry group]]||P6<sub>3</sub>/mmc
|-
|bgcolor=#e7dcc3|Dual||[[gyrated tetrahedral-octahedral honeycomb]]
|-
|bgcolor=#e7dcc3|Properties||edge-uniform, face-uniform, cell-uniform
|}
The '''trapezo-rhombic dodecahedral honeycomb''' is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space. It consists of copies of a single cell, the [[trapezo-rhombic dodecahedron]].  It is similar to the higher symmetric rhombic dodecahedral honeycomb which has all 12 faces as rhombi.

:[[File:Trapezo-rhombic dodecahedron honeycomb.png|320px]]

==== Related honeycombs====
It is a dual to the [[vertex-transitive]] [[gyrated tetrahedral-octahedral honeycomb]].
: [[File:Gyrated alternated cubic honeycomb.png|160px]]

{{-}}

=== Rhombic pyramidal honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Rhombic pyramidal honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||Dual uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_f1|3|node_f1|split1-43|nodes}}<BR>{{CDD|node_fh|4|node|3|node_f1|4|node}}
|-
|bgcolor=#e7dcc3|[[Cell (geometry)|Cell]]
|[[File:Half oblate octahedrille cell.png|30px]]<BR>rhombic [[pyramid]]
|-
|bgcolor=#e7dcc3|[[Face (geometry)|Faces]]
|[[Rhombus]]<BR>[[Triangle]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||[4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math><BR>[3<sup>[4]</sup>], <math>{\tilde{A}}_3</math>
|-
|bgcolor=#e7dcc3|[[Space group|Symmetry group]]||Fm{{overline|3}}m (225)
|-
|bgcolor=#e7dcc3|[[vertex figure]]s
|[[File:Tetrakis cube.png|30px]][[File:Rhombic dodecahedron.jpg|30px]][[File:Triakis tetrahedron.png|30px]]<BR>{{CDD|node_f1|3|node_f1|4|node}}, {{CDD|node|3|node_f1|4|node}}, {{CDD|node_f1|3|node_f1|3|node}}
|-
|bgcolor=#e7dcc3|Dual||[[Cantic cubic honeycomb]]
|-
|bgcolor=#e7dcc3|Properties||[[Cell-transitive]]
|}
The '''rhombic pyramidal honeycomb''' or '''half oblate octahedrille''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space. 

This honeycomb can be seen as a rhombic dodecahedral honeycomb, with the [[rhombic dodecahedron|rhombic dodecahedra]] [[dissection (geometry)|dissected]] with its center into 12 rhombic pyramids.
{| class=wikitable
|- valign=bottom align=center
|[[File:HC R1.png|160px]]<BR>rhombic dodecahedral honeycomb
|[[File:Half oblate octahedrille cell.png|160px]]<BR>Rhombohedral dissection
|[[File:Half oblate octahedrille cell-cube.png|240px]]<BR>Within a cube
|}
==== Related honeycombs====
It is dual to the [[cantic cubic honeycomb]]:
:[[File:Truncated Alternated Cubic Honeycomb.svg|160px]]

== See also==
*[[Architectonic and catoptric tessellation]]

== References ==
* {{The Geometrical Foundation of Natural Structure (book)|page=168}}

== External links ==
{{Commons category|Rhombic dodecahedral honeycomb}}
* {{Mathworld | urlname = Space-FillingPolyhedron  | title = Space-filling polyhedron}}
* [http://www.archinstitute.blogspot.com Examples of Housing Construction using this geometry]

[[Category:3-honeycombs]]