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OldPolyhedra :: latticePoints

latticePoints -- computes the lattice points of a polytope

Synopsis

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 :

For the programmer

The object latticePoints is a method function.