## Tilings

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.