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

Abstract

Let $$\mathcal{T}$$ be the tiling of R 3 with unit cubes whose vertices belong to the fundamental lattice L 1 of points with integer coordinates. Denote by L n the lattice consisting of all points x in R 3 such that nx belongs to L 1. When the vertices of a polyhedron P in R 3 are restricted to lie in L 1 then there is a formula which relates the volume of P to the numbers of all points of two lattices L 1 and L n lying in the interior and on the boundary of P. In the simplest case of the lattices L 1 and L 2 there are 27 points in each cube from $$\mathcal{T}$$ 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 $$\mathcal{T}$$ : one from L 2 and eight belonging to L 4. Another virtue of our formula is that it does not employ any additional parameters, such as the Euler characteristic.