A note on area of lattice polygons in an Archimedean tiling
Let \(L\)[3.6.3.6] be the set of vertices generated by the edge-to-edge Archimedean planar tiling using regular triangles and regular hexagons of unit-length, a point \(x\in L\)[3.6.3.6] be an \(L-\)point. Suppose the corners of a planar polygon \(P\) are \(L-\)points of \(L[3.6.3.6]\), then the area of \(P\) is \(A(P)=\frac{\sqrt{3}}{3}[b(P)+2i(P)+\frac{c(P)}{8}\) \(-3]-\frac{\sqrt{3}}{12}sc(P)\), where \(b(P)\) is the number of \(L-\)points on the boundary of \(P, i(P)\) is the number of \(L-\)points in the interior of \(P, c(P)\) is the boundary characteristic of \(P\), and \(sc(P)\) is the side characteristic of \(P\).
archimedean tiling side characteristic primitive segment discrete geometry