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 $b(\mathrm{\Delta })$ be the number of the boundary $H$-points of an $H$-triangle $\mathrm{\Delta }$. In [3] we made a conjecture that for any $H$-triangle with $k$ interior $H$-points, we have $b(\mathrm{\Delta })\in \{3,4,\dots ,3k+4,3k+5,3k+7\}$. In this note, we prove the conjecture is true for $k=4$, but not true for $k=5$ because $b(\mathrm{\Delta })$ cannot equal $15$.