On the Number of \(H\)-points in a Circle

Abstract

Let an \(H\)-point be a vertex of a tiling of \(\mathbb{R}^2\) by regular hexagons of side length 1, and \(D(n)\) a circle of radius \(n\) (\(n \in \mathbb{Z}^+\)) centered at an \(H\)-point. In this paper, we present an algorithm to calculate the number, \(\mathcal{N}_H(D(n))\), of H-points that lie inside or on the boundary of \(D(n)\). Furthermore, we show that the ratio \(\mathcal{N}_H(D(n))/n^2\) tends to \(\frac{2\pi}{S}\) as \(n\) tends to \(\infty\), where \(S = \frac{3\sqrt{3}}{2}\) is the area of the regular hexagonal tiles.