Rigidity of Infinite Hexagonal Triangulation of the Plane

TL;DR

This paper proves that under certain conformal and angle constraints, an infinite hexagonal triangulation of the plane must be regular, advancing understanding of geometric rigidity in triangulations.

Contribution

It establishes a rigidity result for infinite hexagonal triangulations under PL conformal changes with angle bounds, partially confirming Luo's conjecture.

Findings

Any PL conformal hexagonal triangulation with angles in [δ, π/2 - δ] is regular.

The proof introduces quasi-harmonic functions to analyze mesh properties.

The result confirms a conjecture of Luo regarding triangulation rigidity.

Abstract

In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the plane is PL conformal to the regular hexagonal triangulation and all inner angles are in $[\delta, \pi/2 -\delta]$ for any constant $\delta > 0$, then it is the regular hexagonal triangulation. This partially solves a conjecture of Luo [4]. The proof uses the concept of \emph{quasi-harmonic} functions to unfold the properties of the mesh.