Polyhedra with Hexagonal and Triangular Faces and Three Faces Around Each Vertex

Abstract 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 $$180^\circ $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>180</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> 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 of vertices v in terms of the prime factorization of v and to prove a conjecture concerning trihexes that have no “belts” of hexagons.