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# latticePoints -- computes the lattice points of a polytope

- Usage:
    `L = latticePoints P`
- Inputs:
  - P, a convex polyhedron, 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 a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldPolyhedra.m2:6341:0.