To tile shapes in the tetrahedron-cube-octahedron family of regular and semi-regular polyhedra all that one needs to know is how to arrange cubes, octahedrons or tetrahedrons in their lattice tilings and examine the spaces leftover. For example, arrange truncated cubes as if they were cubes: the spaces left over become octahedrons. The arrangement of both truncated octahedrons as if they were octahedrons and truncated tetrahedrons as if they tetrahedrons: a leftover space fit for that of a cuboctahedron. Arranging truncated shapes in these basic tiling patterns is analogous to truncating one of the regular lattices. Each of the vertices in the lattice will be cut out and each truncation of the lattice will yield a regular tiling. (Remember, each vertex in the lattice is both radially symmetrical and identical to all of the others). Since each of these polyhedra, except for the regular ones, has the symmetries of all three basic shapes, it becomes clear that there are scores of tilings possible. The only limitation is the shapes of the faces matching. You cannot get away with placing triangle faces on square faces.
