The Number of Boundary \(H\)-Points of \(H\)-Triangles

Abstract

An \(H\)-triangle is a triangle with corners in the set of vertices of a tiling of \(\mathbb{R}^2\) by regular hexagons of unit edge. Let \(b(\Delta)\) be the number of the boundary \(H\)-points of an \(H\)-triangle \(\Delta\). In [3] we made a conjecture that for any \(H\)-triangle with \(k\) interior \(H\)-points, we have \(b(\Delta) \in \{3, 4, \ldots, 3k+4, 3k+5, 3k+7\}\). In this note, we prove the conjecture is true for \(k = 4\), but not true for \(k = 5\) because \(b(\Delta)\) cannot equal \(15\).