The rhombic dodecahedron can be constructed with 4 sets of 6 parallel edges. The rhombic dodecahedron can tessellate space by translational copies of itself, as can the stellated rhombic dodecahedron. With D 3d symmetry, it can be seen as an elongated trigonal trapezohedron. The rhombic dodecahedron is a parallelohedron, a space-filling polyhedron, dodecahedrille, being the dual to the tetroctahedrille or half cubic honeycomb, and described by two Coxeter diagrams: and. The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron, with permutation of coordinates (☑, ☑, ☑) and (0, 1 + h, 1 − h 2) with parameter h = 1.Ī topological rhombic dodecahedron can be seen inside of a hexagonal prism, with hexagons dissected into rhombi in complementary ways between top and bottom. The coordinates of the six vertices where four faces meet at their acute angles are: The eight vertices where three faces meet at their obtuse angles have Cartesian coordinates: Pyritohedron variations between a cube and rhombic dodecahedron The last two correspond to the B 2 and A 2 Coxeter planes. The rhombic dodecahedron has four special orthogonal projections along its axes of symmetry, centered on a face, an edge, and the two types of vertex, threefold and fourfold. R i = 6 3 a ≈ 0.8 a Orthogonal projections the radius of its inscribed sphere ( tangent to each of the rhombic dodecahedron's faces) is.This animation shows the construction of a rhombic dodecahedron from a cube, by inverting the center-face-pyramids of a cube.ĭenoting by a the edge length of a rhombic dodecahedron, 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 collections of the Louvre include a die in the shape of a rhombic dodecahedron dating from Ptolemaic Egypt.

These rhombi are the tiles of a rhombille. Analogy: a regular hexagon can be dissected into 3 rhombi around its center. These rhombohedra are the cells of a trigonal trapezohedral honeycomb. The graph of the rhombic dodecahedron is nonhamiltonian.Ī rhombic dodecahedron can be dissected into 4 obtuse trigonal trapezohedra around its center. In these cases, four vertices (alternate threefold ones) are absent, but the chemical bonds lie on the remaining edges. The rhombic dodecahedron also appears in the unit cells of diamond and diamondoids. As Johannes Kepler noted in his 1611 book on snowflakes ( Strena seu de Nive Sexangula), honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron.

Some minerals such as garnet form a rhombic dodecahedral crystal habit. It is the Brillouin zone of body centered cubic (bcc) crystals. This polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice. The rhombic dodecahedron can be used to tessellate three-dimensional space: it can be stacked to fill a space, much like hexagons fill a plane. The rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, and the rhombic triacontahedron. The 6 vertices where 4 rhombi meet correspond to the vertices of the octahedron, while the 8 vertices where 3 rhombi meet correspond to the vertices of the cube. The rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron. In elementary terms, this means that for any two faces A and B, there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B. The long face-diagonal length is exactly √ 2 times the short face-diagonal length thus, the acute angles on each face measure arccos( 1 / 3), or approximately 70.53°.īeing the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on its set of faces. Its polyhedral dual is the cuboctahedron. The rhombic dodecahedron is a zonohedron. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. It has 24 edges, and 14 vertices of 2 types. In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces.

Ipicture of a cuboctahedron flattened