Polyhedra with hexagonal and triangular faces and three faces around each vertex

TL;DR

This paper classifies and analyzes polyhedra made of hexagons and triangles with three faces meeting at each vertex, introducing a signature-based framework to enumerate and understand their structure and properties.

Contribution

It introduces a signature-based method to represent trihexes, establishing a bijection with equivalence classes and deriving bounds on their counts.

Findings

Established a bijection between trihexes and signature classes.

Derived bounds on the number of trihexes based on vertex count and prime factorization.

Proved a conjecture about trihexes lacking hexagon belts.

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 $180^\circ$ 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.