- Rhombic dodecahedron
The

**rhombic dodecahedron**is a convexpolyhedron with 12 rhombic faces. It is an Archimedean dual solid, or aCatalan solid . Its dual is thecuboctahedron .**Properties**It is the polyhedral dual of the

cuboctahedron , and azonohedron . The long diagonal of each face is exactly √2 times the length of the short diagonal, so that the acute angles on each face measure cos^{−1}(1/3), or approximately 70.53°.Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is

face-transitive , meaning thesymmetry group of the solid actstransitive ly on the set of faces. In elementary terms, this means that for any two faces A and B there is arotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B.The rhombic dodecahedron is one of the nine

edge-transitive convex polyhedra, the others being the fivePlatonic solid s, thecuboctahedron , theicosidodecahedron and therhombic triacontahedron .The rhombic dodecahedron can be used to tessellate 3-dimensional space. It can be stacked to fill a space much like

hexagon s fill a plane.This tessellation can be seen as the

Voronoi tessellation of the face-centred cubic lattice. Some minerals such asgarnet form a rhombic dodecahedralcrystal habit .Honeybee s use the geometry of rhombic dodecahedra to formhoneycomb from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron.**Area and volume**The area "A" and the volume "V" of the rhombic dodecahedron of edge length "a" are::$A\; =\; 8sqrt\{2\}a^2\; approx\; 11.3137085a^2$

:$V\; =\; frac\{16\}\{9\}\; sqrt\{3\}a^3\; approx\; 3.07920144a^3$

**Cartesian coordinates**The eight vertices where three faces meet at their obtuse angles have

Cartesian coordinates : (±1, ±1, ±1)The six vertices where four faces meet at their acute angles are given by the permutations of: (0, 0, ±2)

**Related polyhedra**This polyhedron is related to an infinite series of tilings with the

face configuration s V3.2n.3.2n, the first in the Euclidean plane, and the rest in the hyperbolic plane.**Related polytopes**s, giving 8 possible parallelepipeds. The 8 cells of the tesseract under this projection map precisely to these 8 parallelepipeds.

The rhombic dodecahedron forms the maximal cross-section of a

24-cell , and also forms the hull of its vertex-first parallel projection into 3 dimensions. The rhombic dodecahedron can be decomposed into 6 congruent (but non-regular)square dipyramid s meeting at a single vertex in the center; these form the images of 6 pairs of the 24-cell's octahedral cells. The remaining 12 octahedral cells project onto the faces of the rhombic dodecahedron. The non-regularity of these images are due to projective distortion; the facets of the 24-cell are regular octahedra in 4-space.This decomposition gives an interesting method for constructing the rhombic dodecahedron: cut a

cube into 6 congruent square pyramids, and attach them to the faces of a second cube. The triangular faces of each pair of adjacent pyramids lie on the same plane, and so merge into rhombuses. The 24-cell may also be constructed in an analogous way using twotesseract s.**ee also***

Dodecahedron

*Rhombic triacontahedron

*Quasiregular rhombic tiling

*Truncated rhombic dodecahedron

*24-cell - 4D analog of rhombic dodecahedron

*Rhombic dodecahedral honeycomb **References*** (Section 3-9)

**External links***

* [*http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra*] – The Encyclopedia of Polyhedra**Computer models*** [

*http://polyhedra.org/poly/show/33/rhombic_dodecahedron Rhombic Dodecahedron*] -- interactive 3-d model

* [*http://demonstrations.wolfram.com/RelatingARhombicTriacontahedronAndARhombicDodecahedron/ Relating a Rhombic Triacontahedron and a Rhombic Dodecahedron*] , [*http://demonstrations.wolfram.com/RhombicDodecahedron5Compound/ Rhombic Dodecahedron 5-Compound*] and [*http://demonstrations.wolfram.com/RhombicDodecahedron5Compound/ Rhombic Dodecahedron 5-Compound*] by Sándor Kabai,The Wolfram Demonstrations Project .**Paper projects*** [

*http://www.ii.uib.no/~arntzen/kalender/ Rhombic Dodecahedron Calendar*] – make a rhombic dodecahedron calendar without glue

* [*http://www.southernct.edu/~fields/2008-Calendars/ Another Rhombic Dodecahedron Calendar*] – made by plaiting paper strips**Practical applications*** [

*http://archinstitute.blogspot.com Archimede Institute*] Examples of actual housing construction projects using this geometry

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**rhombic dodecahedron**— rombinis dodekaedras statusas T sritis fizika atitikmenys: angl. rhombic dodecahedron; rhombododecahedron vok. Rhombendodekaeder, n rus. ромбический додекаэдр, m; ромбододекаэдр, m pranc. dodécaèdre rhomboïdal, m; rhombododécaèdre, m … Fizikos terminų žodynas**Truncated rhombic dodecahedron**— The truncated rhombic dodecahedron is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 (order 4) vertices. The 6 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become… … Wikipedia**Trapezo-rhombic dodecahedron**— The Trapezo rhombic dodecahedron is a convex polyhedron with 6 rhombic and 6 trapezoidal faces.This shape could be constructed by taking a tall uniform hexagonal prism, and making 3 angled cuts on the top and bottom. The trapezoids represent what … Wikipedia**Dodecahedron**— Do*dec a*he dron, n. [Gr. ?; ? twelve + ? seat, bottom, base: cf. F. dod[ e]ca[ e]dre.] (Geom. & Crystallog.) A solid having twelve faces. [1913 Webster] Note: The regular dodecahedron is bounded by twelve equal and regular pentagons; the… … The Collaborative International Dictionary of English**Dodecahedron**— Regular Dodecahedron (Click here for rotating model) Type Platonic solid Elements F = 12, E = 30 V = 20 (χ = 2) Faces by sides 12{5} … Wikipedia**Rhombic dodecahedral honeycomb**— The rhombic dodecahedra honeycomb is a space filling tessellation (or honeycomb) in Euclidean 3 space. It is the Voronoi diagram of the face centered cubic sphere packing, which is believed to be the densest possible packing of equal spheres in… … Wikipedia**Rhombic triacontahedron**— In geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. It is the polyhedral dual of the icosidodecahedron, and it is a zonohedron.The ratio of the long diagonal… … Wikipedia**Rhombic enneacontahedron**— A rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 rhombus shaped faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a… … Wikipedia**Rhombo-hexagonal dodecahedron**— Infobox Polyhedron Polyhedron Type=Dodecahedron Face List=8 rhombi 4 hexagons Edge Count=28 Vertex Count=18 Vertex List=(8) 4.6.6 (8) 4.4.6 (2) 4.4.4.4 Symmetry Group=D4h Dual= Property List=convex, ZonohedronThe rhombo hexagonal dodecahedron is… … Wikipedia**Disdyakis dodecahedron**— Click on picture for large version Spinning version Type Catalan Face polygon scalene triangle Faces 48 Edges … Wikipedia