Geometry of the rhombic dodecahedron

There are several different ways to describe a rhombic dodecahedron. It has 12 congruent rhombic faces. For each face, the ratio of the edge length to the short diagonal of the rhombus is √3/2. With this information you can easily make a model, but it is not clear why you can use rhombic dodecahedra to fill space!

Here is a 3D description of a rhombic dodecahedron. Start with a cube of edge length 1, and consider a pyramid whose base is a face of the cube, and whose apex is the center of the cube. The cube is divided into 6 such pyramids, one for each face. The faces of this pyramid are the square face of the cube and four isosceles triangles. These triangles have edge lengths 1, √3/2, √3/2. That's because the segment from a vertex of the cube to the center of the cube is half of a long diagonal, and the long diagonals have length √3. The dihedral angle between a triangular face and the square face of the pyramid is 45 degrees. (Proof?) The dihedral angle between two triangular faces of the pyramid is 120 degrees. (Proof?)

Now take 6 such pyramids and glue them onto the outside of six faces of another cube of edge length 1. The resulting solid is a rhombic dodecahedron with edge lengths √3/2. The triangular faces of the pyramids glued to two adjacent faces of the cube will lie in the same plane, and together these two triangles form a face of the rhombic dodecahedron.

Note that the volume of this rhombic dodecahedron is 2 (two times the volume of the cube), and its dihedral angles are 120 degrees.

Coxeter (Introduction to Geometry, section 22.4) gives the following way to visualize that rhombic dodecahedra fill space. Start with a tessellation of space by cubes. Color the cubes black and white in a 3D checkerboard pattern. Now cut each white cube into 6 congruent square pyramids. Glue each white square pyramid to the black cube that shares a square face with it. Then each black cube becomes a rhombic dodecahedron, and these rhombic dodecahedra fill space.