Rhombic dodecahedron

The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. 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°.

Related Topics:
Convex - Polyhedron - 12 - Rhombic - Dual - Cuboctahedron - Zonohedron - √2 - Acute

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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.

Related Topics:
Archimedean polyhedron - Symmetry group - Transitive - Rotation - Reflection - Platonic solid - Cuboctahedron - Icosidodecahedron - 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.

Related Topics:
Tessellate - Voronoi tessellation - Face-centred cubic lattice - Honeybee - Honeycomb

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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 parallelopipeds, giving 8 possible parallelopipeds. The 8 cells of the tesseract under this projection map precisely to these 8 parallelopipeds.

Related Topics:
Tesseract - Parallelopipeds

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~