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{{Short description|Catalan solid with 12 faces}}
{{infobox polyhedron
| name = Rhombic dodecahedron
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{{multiple image
| image1 = Parallelohedron edges rhombic dodecahedron.png
| caption1 = The rhombic dodecahedron
| image2 = Rhombic dodecahedra.png
| caption2 = The rhombic dodecahedron can [[Rhombic dodecahedral honeycomb|tessellate space by translational copies of itself]]
| total_width = 400▼
}}
The rhombic dodecahedron is a [[space-filling polyhedron]], meaning it can be applied to [[Tessellation|tessellate]] three-dimensional space: it can be stacked to fill a space, much like [[hexagon]]s fill a plane. It is a [[parallelohedron]] because it can be space-filling a [[Honeycomb (geometry)|honeycomb]] in which all of its copies meet face-to-face.{{r|alexandrov}} More generally, every parellelohedron is [[zonohedron]], a [[centrally symmetric]] polyhedron with [[Zonogon|centrally symmetric faces]].{{r|eppstein}}
The [[rhombic dodecahedral honeycomb]] (or ''dodecahedrille'') is an example of a honeycomb constructed by filling all rhombic dodecahedrons. It is dual to the ''tetroctahedrille'' or [[half cubic honeycomb]], and it is described by two [[Coxeter diagram]]s: {{CDD|node_f1|3|node|split1-43|nodes}} and {{CDD|node_f1|split1|nodes|split2|node}}. With D<sub>3d</sub> symmetry, it can be seen as an [[Elongation (geometry)|elongated]] [[trigonal trapezohedron]]. It can be seen as the [[Voronoi tessellation]] of the [[Crystal structure|face-centered cubic lattice]]. It is the Brillouin zone of body-centered cubic (bcc) crystals. Some minerals such as [[garnet]] form a rhombic dodecahedral [[crystal habit]]. As [[Johannes Kepler]] noted in his 1611 book on snowflakes (''Strena seu de Nive Sexangula''), [[honey bee]]s use the geometry of rhombic dodecahedra to form [[honeycomb]]s from a tessellation of cells each of which is a [[hexagonal prism]] capped with half a rhombic dodecahedron. The rhombic dodecahedron also appears in the unit cells of [[diamond]] and [[diamondoid]]s. In these cases, four vertices (alternate threefold ones) are absent, but the chemical bonds lie on the remaining edges.<ref name=cryst1>[http://www.khulsey.com/jewelry/crystal_habit.html#h-5. Dodecahedral Crystal Habit] {{webarchive|url=https://web.archive.org/web/20090412012420/http://www.khulsey.com/jewelry/crystal_habit.html |date=2009-04-12 }}. khulsey.com</ref>
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=== Miscellaneous ===
The collections of the [[Louvre]] include a die in the shape of a rhombic dodecahedron dating from [[Ptolemaic Egypt]]. The faces are inscribed with Greek letters representing the numbers 1 through 12: Α Β Γ Δ Ε Ϛ Z Η Θ Ι ΙΑ ΙΒ. The function of the die is unknown.{{r|perdrizet}}
== Other related figures ==
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