An ‘odd’ formula for the volume of three-dimensional lattice polyhedra

Abstract

Let T be the filing of R 3 with unit cubes whose vertices belong to the fundamental lattice Ll of points with integer coordinates. Denote by L,, the lattice consisting of all points z in R s such that nz belongs to L1. When the vertices of a polyhedron P in R 3 are restricted to lie in LI then there is a formula which relates the volume of P to the numbers of all points of two lattices Ll and Ln lying in the interior and on the boundary of P. In the simplest case of the lattices L1 and L2 there are 27 points in each cube from T whose relationships to the polyhedron P must be examined. In this note we present a new formula for the volume of lattice polyhedra in R 3 which involves only nine points in each cube of T: one from L2 and eight belonging to L4.