Rhombic dodecahedron

The rhombic dodecahedron, or rad, is one of the 13 Catalan solids. It has 12 rhombi as faces, with 6 order-4 and 8 order-3 vertices. It is the dual of the uniform cuboctahedron.

It can also be obtained as the convex hull of a cube and an octahedron scaled so that their edges are orthogonal. For this to happen, the octahedron's edge length must be $$\sqrt2 ≈ 1.41421$$ times that of the cube's edge length. Each edge of the cube or octahedron corresponds to one of the diagonals of the faces.

It can also be constructed by augmenting a cube with square pyramids of height 0.5.

Each face of this polyhedron is a rhombus with longer diagonal $$\sqrt2 ≈ 1.41421$$ times the shorter diagonal, with acute angle $$\arccos\left(\frac13\right) ≈ 70.52878°$$ and obtuse angle $$\arccos\left(-\frac13\right) ≈ 109.47122°$$.

The rhombic dodecahedron is the only Catalan solid that can tile 3D space by itself, forming the rhombic dodecahedral honeycomb.

Vertex coordinates
A rhombic dodecahedron of edge length 1 has vertex coordinates given by all permutations of
 * $$\left(±\frac{\sqrt3}{3},\,±\frac{\sqrt3}{3},\,±\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac{2\sqrt3}{3},\,0,\,0\right).$$

Variations
The rhombic dodecahedron can be varied to remain isohedral under tetrahedral symmetry. In the process the rhombic faces turn into kites, and the resulting polyhedron can be called a deltoidal dodecahedron.

A variant can be made with golden rhombi for faces. This variant can be dissected into 4 acute golden rhombohedra and 4 obtuse golden rhombohedra.