# latticePoints -- computes the lattice points of a polytope

## Synopsis

• Usage:
L = latticePoints P
• Inputs:
• P, , which must be compact
• Outputs:
• L, a list, containing the lattice points as matrices over ZZ with only one column

## Description

latticePoints can only be applied to polytopes, i.e. compact polyhedra. It embeds the polytope on height 1 in a space of dimension plus 1 and takes the Cone over this polytope. Then it projects the elements of height 1 of the Hilbert basis back again.

 i1 : P = crossPolytope 3 o1 = {ambient dimension => 3 } dimension of lineality space => 0 dimension of polyhedron => 3 number of facets => 8 number of rays => 0 number of vertices => 6 o1 : Polyhedron i2 : latticePoints P o2 = {| -1 |, | 0 |, | 0 |, 0, | 0 |, | 0 |, | 1 |} | 0 | | -1 | | 0 | | 0 | | 1 | | 0 | | 0 | | 0 | | -1 | | 1 | | 0 | | 0 | o2 : List i3 : Q = cyclicPolytope(2,4) o3 = {ambient dimension => 2 } dimension of lineality space => 0 dimension of polyhedron => 2 number of facets => 4 number of rays => 0 number of vertices => 4 o3 : Polyhedron i4 : latticePoints Q o4 = {0, | 1 |, | 1 |, | 1 |, | 2 |, | 2 |, | 2 |, | 3 |} | 1 | | 2 | | 3 | | 4 | | 5 | | 6 | | 9 | o4 : List

## Ways to use latticePoints :

• "latticePoints(Polyhedron)"

## For the programmer

The object latticePoints is .