Waterman polyhedron

The Waterman polyhedra are an infinite family of convex polyhedra. They are based on an idea of Steve Waterman from 1990, generalized in full in November 2006 by Richard Klitzing. As such they require any lattice, some center point within any relative position and some radius. The center point and the radius then define an according ball and that very ball then encompasses a finite subset of lattice points. The hull of those points then defines the according Waterman polyhedron. For sure, this setup is not restricted to 3D alone.

In the sequel as an example we use Waterman's first application to the face-centered cubic (FCC) lattice (identical to the vertex locations of the tetrahedral-octahedral honeycomb), and take 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 square 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 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: W(FCC)1 O3*, W(FCC)2 O3*, W(FCC)1 O3, W(FCC)1 O4.
 * Regular octahedron: W(FCC)2 O1, W(FCC)1 O6.
 * Cube: W(FCC)2 O6.
 * Cuboctahedron: W(FCC)1 O1, W(FCC)4 O1.
 * Truncated octahedron: W(FCC)10 O1.
 * Truncated tetrahedron: W(FCC)4 O3, W(FCC)2 O4.
 * Truncated cuboctahedron with unequal edge lengths: W(FCC)7 O1.
 * Rhombicuboctahedron with unequal edge lengths: W(FCC)3 O1, W(FCC)12 O1.e