  We analyze polyhedra composed of hexagons and triangles with three faces
around each vertex, and their 3-regular planar graphs of edges and vertices,
which we call "trihexes". Trihexes are analogous to fullerenes, which are
3-regular planar graphs whose faces are all hexagons and pentagons. Every
trihex can be represented as the quotient of a hexagonal tiling of the plane
under a group of isometries generated by $180^\circ$ rotations. Every trihex
can also be described with either one or three "signatures": triples of numbers
$(s, b, f)$ that describe the arrangement of the rotocenters of these
rotations. Simple arithmetic rules relate the three signatures that describe
the same trihex. We obtain a bijection between trihexes and equivalence classes
of signatures as defined by these rules. Labeling trihexes with signatures
allows us to put bounds on the number of trihexes for a given number vertices
$v$ in terms of the prime factorization of $v$ and to prove a conjecture
concerning trihexes that have no "belts" of hexagons.
