Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of two types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron. The long diagonal of each face is exactly Square root of two| times the length of the short diagonal, so that the acute angles on each face measure arccos, or approximately 70.53°.Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. 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 rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron. 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 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 rhombic dodecahedron can be used to tessellate three-dimensional space. It can be stacked to fill a space much like hexagons fill a plane.
This polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice. It is the Brillouin zone of body centered cubic crystals. Some minerals such as garnet form a rhombic dodecahedral crystal habit. As Johannes Kepler noted in his 1611 book on snowflakes, 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. The rhombic dodecahedron also appears in the unit cells of diamond and diamondoids. In these cases, four vertices are absent, but the chemical bonds lie on the remaining edges.
The graph of the rhombic dodecahedron is nonhamiltonian.
A rhombic dodecahedron can be dissected with its center into 4 trigonal trapezohedra. These rhombohedra are the cells of a trigonal trapezohedral honeycomb. This is analogous to the dissection of a regular hexagon dissected into rhombi, and tiled in the plane as a rhombille.
The collections of the Louvre include a die in the shape of a rhombic dodecahdron dating from Ptolemaic Egypt. The faces are inscribed with Greek letters representing the numbers 1 through 12: Α Β Γ Δ Ε Ζ Ϛ Η Θ Ι ΙΑ ΙΒ. The function of the die is unknown.
Dimensions
If the edge length of a rhombic dodecahedron is a, the radius of an inscribed sphere isand the radius of the midsphere is
Area and volume
The area A and the volume V of the rhombic dodecahedron of edge length a are:Orthogonal projections
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. The last two correspond to the B2 and A2 Coxeter planes.| Projective symmetry | ||||
| Rhombic dodecahedron | ||||
| Cuboctahedron |
Cartesian coordinates
The eight vertices where three faces meet at their obtuse angles have Cartesian coordinates:The coordinates of the six vertices where four faces meet at their acute angles are:
The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron, with permutation of coordinates and with parameter h = 1.
Topologically equivalent forms
Parallelohedron
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. With D3d symmetry, it can be seen as an elongated trigonal trapezohedron.The rhombic dodecahedron can tessellate space by translational copies of itself. So can the stellated rhombic dodecahedron. | The rhombic dodecahedron can be constructed with 4 sets of parallel edges. |
Dihedral rhombic dodecahedron
Other symmetry constructions of the rhombic dodecahedron are also space-filling, and as parallelotopes they are similar to variations of space-filling truncated octahedra.For example, with 4 square faces, and 60-degree rhombic faces, and D4h dihedral symmetry, order 16. It can be seen as a cuboctahedron with square pyramids augmented on the top and bottom.
Net | CoordinatesBilinski dodecahedronIn 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the Bilinski dodecahedron. It has the same topology but different geometry. The rhombic faces in this form have the golden ratio.
Deltoidal dodecahedronAnother topologically equivalent variation, sometimes called a deltoidal dodecahedron or trapezoidal dodecahedron, is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites. It has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by, where b is determined from a for planar faces. is the rhombic solution. As approaches, approaches infinity.Related polyhedraWhen projected onto a sphere, it can be seen that the edges make up the edges of two tetrahedra arranged in their dual positions. This trend continues on with the deltoidal icositetrahedron and deltoidal hexecontahedron for the dual pairings of the other regular polyhedra, giving this shape the alternative systematic name of deltoidal dodecahedron.This polyhedron is a part of a sequence of rhombic polyhedra and tilings with Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. Similarly it relates to the infinite series of tilings with the face configurations V3.2n.3.2n, the first in the Euclidean plane, and the rest in the hyperbolic plane.
StellationsLike many convex polyhedra, the rhombic dodecahedron can be stellated by extending the faces or edges until they meet to form a new polyhedron. Several such stellations have been described by Dorman Luke.by inverting the center-face pyramids of a rhombic dodecahedron. The first stellation, often simply called the stellated rhombic dodecahedron, is well known. It can be seen as a rhombic dodecahedron with each face augmented by attaching a rhombic-based pyramid to it, with a pyramid height such that the sides lie in the face planes of the neighbouring faces: Luke describes four more stellations: the second and third stellations, one formed by removing the second from the third, and another by adding the original rhombic dodecahedron back to the previous one.
Related polytopesThe rhombic dodecahedron forms the hull of the vertex-first projection of a tesseract to three dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into four congruent rhombohedra, giving eight possible rhombohedra as projections of the tesseracts 8 cubic cells. One set of projective vectors are: u=, v=, w=.The rhombic dodecahedron forms the maximal cross-section of a 24-cell, and also forms the hull of its vertex-first parallel projection into three dimensions. The rhombic dodecahedron can be decomposed into six congruent square dipyramids meeting at a single vertex in the center; these form the images of six 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 six 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 two tesseracts. Computer models
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