3 research outputs found
Polyhedra with hexagonal and triangular faces and three faces around each vertex
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 rotations. Every trihex
can also be described with either one or three "signatures": triples of numbers
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
in terms of the prime factorization of and to prove a conjecture
concerning trihexes that have no "belts" of hexagons.Comment: 26 pages, 19 figure