Areas of lattice figures in the planar tilings with congruent regular polygons

Areas of GeneralizedH-Polygons

“…In either case these terms cancel each other in the sum (8). With similar reasoning all terms of the form u(zv j ) cancel in pairs in the sum (8). Also *(e)=&1 precisely when the orientation of side e in T e is the opposite orientation of e in P. Therefore…”

“…If *(e)=*( f ) then c(x, T e )=&c(x, T f ). In either case these terms cancel each other in the sum (8). With similar reasoning all terms of the form u(zv j ) cancel in pairs in the sum (8).…”

“…Note that for each term in (8), the 3 sharp corners v j , v j+1 , and z of triangle T e contribute 3Â4 to b(T e )Â4, and c(z, T e )Â12 contributes 3Â12, so these are exactly canceled by the &1 term.…”

“…An additional useful parameter (see [1] and [8] for further references) is the boundary characteristic. The following definition, valid for any H-polygon, reduces to the definitions of [1] and [2] in the special case when P is simple and thus has a positive (counter-clockwise) orientation in the plane.…”

A new formula for the volume of lattice polyhedra

“…The purpose of this note is to show that there exists a formula for the volume of a proper lattice polyhedron P which differs from Macdonald's formulae (3) and (4) in that this new formula uses only the numbers I~,..., I N of the interior lattice points in P but it does not use the numbers of points of the lattices Lk, 1 <~ k ~< N, lying on the boundary of P. Namely, we will show that the following formula is valid…”