A Pick-Type Theorem in Archimedean Planar Tiling

Ren Ding , John Reay, and Jianjun Wang
College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050000, P.R. China

Abstract

We denote by L = L [3.6.3.6] the set of vertices generated by a face-to-face Archimedean planer tiling using regular triangles and regular hexagons of unit length. A polygon is called L - polygon if all corners of the polygon are in L = L [3.6.3.6]. After introducing two special parameters, boundary characteristic and side characteristic of an L-lattice polygon, we obtain the following result which is, in some sense, a generalization of the classical Pick theorem:
If P is an L- polygon, then the area of P is

where b =b (P) = , i = i (P) =|L int(P)|, c=c (P) is the boundary characteristic of P, and s=s (P) is the side characteristic of P.