Waterman polyhedron

The Waterman polyhedra are an infinite family of convex polyhedra. A Waterman polyhedron is formed by starting with the face-centered cubic (FCC) packing of unit spheres, finding all centers of the spheres (identical to the vertex locations of the tetrahedral-octahedral honeycomb), and taking the convex hull of all points enclosed in a particular sphere of any radius or location, provided that the resulting polyhedron is not degenerate.

Most Waterman polyhedra are asymmetrical, but constraining the center of the enclosing sphere (called the "origin") to certain locations will guarantee certain symmetries. There are seven notable origins distinguished by the FCC's symmetry group, here described in terms of the corresponding tetrahedral-octahedral honeycomb:


 * O1: vertex. Produces cubic symmetry.
 * O2: midpoint of an edge. Produces cuboidal symmetry.
 * O3: center of a triangular face. Produces triangular pyramidal symmetry.
 * O3*: center of an equilateral triangle joining the centers of three mutually adjacent octahedra. Also produces triangular pyramidal symmetry.
 * O4: center of a tetrahedron. Produces tetrahedral symmetry.
 * O5: halfway between a vertex and the center of an octahedron. Produces rectangular pyramidal symmetry.
 * O6: center of an octahedron. Produces octahedral symmetry.

Other origins that always generate symmetrical polyhedra exist, but they have "degrees of freedom."

A notation for these particular symmetric Waterman polyhedra goes as follows. Fix one of the seven origins On. Then start with a sphere of radius zero at On, and expand its radius until a nondegenerate polyhedron is enclosed. This is W1 On. Then continue expanding the sphere until more points are enclosed. This is W2 On. The next distinct polyhedron is W3 On, and so forth.

Some Archimedean solids can be represented with this notation, some in multiple ways:


 * Regular tetrahedron: W1 O3*, W2 O3*, W1 O3, W1 O4.
 * Regular octahedron: W2 O1, W1 O6.
 * Cube: W2 O6.
 * Cuboctahedron: W1 O1, W4 O1.
 * Truncated octahedron: W10 O1.
 * Truncated tetrahedron: W4 O3, W2 O4.
 * Truncated cuboctahedron with unequal edge lengths: W7 O1.
 * Rhombicuboctahedron with unequal edge lengths: W3 O1, W12 O1.