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| Rhombic dodecahedron | |
|---|---|
| Type | Catalan |
| Face polygon | rhombus |
| Faces | 12 |
| Edges | 24 |
| Vertices | 14 = 6 + 8 |
| Face configuration | V3.4.3.4 |
| Symmetry group | octahedral (Oh) |
| Dual polyhedron | cuboctahedron |
| Properties | convex, face-uniform, edge-uniform, zonohedron |
It is the polyhedral dual of the cuboctahedron and a zonohedron. 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 2 tan−1(1/√2), or approximately 70.53°.
Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-uniform, 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 is also somewhat special in being one of the nine edge-uniform 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 3-dimensional space. This tessellation can be seen as the Voronoi tessellation of the face-centred cubic lattice. Honeybees use the geometry of rhombic dodecahedra to form honeycomb from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron.
The rhombic dodecahedron forms the hull of the vertex-first projection of a tesseract to 3 dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelepipeds, giving 8 possible parallelepipeds. The 8 cells of the tesseract under this projection map precisely to these 8 parallelepipeds.
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)
See also
External links
- Rhombic Dodecahedron – from MathWorld
- Virtual Reality Polyhedra – The Encyclopedia of Polyhedra
Catalan solids | Quasiregular polyhedra | Zonohedra
Rhombendodekaeder | Dodecaedro rómbico | Romboëdrisch dodecaëder | 菱形十二面体 | Dodecaedro rômbico
"Rhombic dodecahedron"