Fast Lattice Polytopes using PPL.

The LatticePolytope_PPL() class is a thin wrapper around PPL polyhedra. Its main purpose is to be fast to construct, at the cost of being much less full-featured than the usual polyhedra. This makes it possible to iterate with it over the list of all 473800776 reflexive polytopes in 4 dimensions.

Note

For general lattice polyhedra you should use Polyhedron() with base_ring=ZZ.

The class derives from the PPL ppl.polyhedron.C_Polyhedron class, so you can work with the underlying generator and constraint objects. However, integral points are generally represented by \(\ZZ\)-vectors. In the following, we always use generator to refer the PPL generator objects and vertex (or integral point) for the corresponding \(\ZZ\)-vector.

EXAMPLES:

Sage

sage: vertices = [(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (-9, -6, -1, -1)]
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: P = LatticePolytope_PPL(vertices);  P
A 4-dimensional lattice polytope in ZZ^4 with 5 vertices
sage: P.integral_points()
((-9, -6, -1, -1), (-3, -2, 0, 0), (-2, -1, 0, 0), (-1, -1, 0, 0),
 (-1, 0, 0, 0), (0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0))
sage: P.integral_points_not_interior_to_facets()
((-9, -6, -1, -1), (-3, -2, 0, 0), (0, 0, 0, 0), (1, 0, 0, 0),
 (0, 1, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0))

Python

>>> from sage.all import *
>>> vertices = [(Integer(1), Integer(0), Integer(0), Integer(0)), (Integer(0), Integer(1), Integer(0), Integer(0)), (Integer(0), Integer(0), Integer(1), Integer(0)), (Integer(0), Integer(0), Integer(0), Integer(1)), (-Integer(9), -Integer(6), -Integer(1), -Integer(1))]
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> P = LatticePolytope_PPL(vertices);  P
A 4-dimensional lattice polytope in ZZ^4 with 5 vertices
>>> P.integral_points()
((-9, -6, -1, -1), (-3, -2, 0, 0), (-2, -1, 0, 0), (-1, -1, 0, 0),
 (-1, 0, 0, 0), (0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0))
>>> P.integral_points_not_interior_to_facets()
((-9, -6, -1, -1), (-3, -2, 0, 0), (0, 0, 0, 0), (1, 0, 0, 0),
 (0, 1, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0))

Fibrations of the lattice polytopes are defined as lattice sub-polytopes and give rise to fibrations of toric varieties for suitable fan refinements. We can compute them using fibration_generator()

Sage

sage: F = next(P.fibration_generator(2))
sage: F.vertices()
((1, 0, 0, 0), (0, 1, 0, 0), (-3, -2, 0, 0))

Python

>>> from sage.all import *
>>> F = next(P.fibration_generator(Integer(2)))
>>> F.vertices()
((1, 0, 0, 0), (0, 1, 0, 0), (-3, -2, 0, 0))

Finally, we can compute automorphisms and identify fibrations that only differ by a lattice automorphism:

Sage

sage: square = LatticePolytope_PPL((-1,-1), (-1,1), (1,-1), (1,1))
sage: fibers = [ f.vertices() for f in square.fibration_generator(1) ];  fibers
[((1, 0), (-1, 0)), ((0, 1), (0, -1)), ((-1, -1), (1, 1)), ((-1, 1), (1, -1))]
sage: square.pointsets_mod_automorphism(fibers)                                     # needs sage.groups
(frozenset({(-1, -1), (1, 1)}), frozenset({(-1, 0), (1, 0)}))

Python

>>> from sage.all import *
>>> square = LatticePolytope_PPL((-Integer(1),-Integer(1)), (-Integer(1),Integer(1)), (Integer(1),-Integer(1)), (Integer(1),Integer(1)))
>>> fibers = [ f.vertices() for f in square.fibration_generator(Integer(1)) ];  fibers
[((1, 0), (-1, 0)), ((0, 1), (0, -1)), ((-1, -1), (1, 1)), ((-1, 1), (1, -1))]
>>> square.pointsets_mod_automorphism(fibers)                                     # needs sage.groups
(frozenset({(-1, -1), (1, 1)}), frozenset({(-1, 0), (1, 0)}))

AUTHORS:

• Volker Braun: initial version, 2012

sage.geometry.polyhedron.ppl_lattice_polytope.LatticePolytope_PPL(*args)

Construct a new instance of the PPL-based lattice polytope class.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0), (1,0), (0,1))
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices

sage: from ppl import point, Generator_System, C_Polyhedron, Linear_Expression  # needs pplpy
sage: p = point(Linear_Expression([2,3],0));  p                                 # needs pplpy
point(2/1, 3/1)
sage: LatticePolytope_PPL(p)                                                    # needs pplpy
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex

sage: P = C_Polyhedron(Generator_System(p));  P                                 # needs pplpy
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
sage: LatticePolytope_PPL(P)                                                    # needs pplpy
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(0),Integer(1)))
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices

>>> from ppl import point, Generator_System, C_Polyhedron, Linear_Expression  # needs pplpy
>>> p = point(Linear_Expression([Integer(2),Integer(3)],Integer(0)));  p                                 # needs pplpy
point(2/1, 3/1)
>>> LatticePolytope_PPL(p)                                                    # needs pplpy
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex

>>> P = C_Polyhedron(Generator_System(p));  P                                 # needs pplpy
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
>>> LatticePolytope_PPL(P)                                                    # needs pplpy
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex

A TypeError is raised if the arguments do not specify a lattice polytope:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0), (1/2,1))                                       # needs pplpy
Traceback (most recent call last):
...
TypeError: unable to convert rational 1/2 to an integer

sage: from ppl import point, Generator_System, C_Polyhedron, Linear_Expression  # needs pplpy
sage: p = point(Linear_Expression([2,3],0), 5);  p                              # needs pplpy
point(2/5, 3/5)
sage: LatticePolytope_PPL(p)                                                    # needs pplpy
Traceback (most recent call last):
 ...
TypeError: generator is not a lattice polytope generator

sage: P = C_Polyhedron(Generator_System(p));  P                                 # needs pplpy
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
sage: LatticePolytope_PPL(P)                                                    # needs pplpy
Traceback (most recent call last):
...
TypeError: polyhedron has non-integral generators

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(1)/Integer(2),Integer(1)))                                       # needs pplpy
Traceback (most recent call last):
...
TypeError: unable to convert rational 1/2 to an integer

>>> from ppl import point, Generator_System, C_Polyhedron, Linear_Expression  # needs pplpy
>>> p = point(Linear_Expression([Integer(2),Integer(3)],Integer(0)), Integer(5));  p                              # needs pplpy
point(2/5, 3/5)
>>> LatticePolytope_PPL(p)                                                    # needs pplpy
Traceback (most recent call last):
 ...
TypeError: generator is not a lattice polytope generator

>>> P = C_Polyhedron(Generator_System(p));  P                                 # needs pplpy
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
>>> LatticePolytope_PPL(P)                                                    # needs pplpy
Traceback (most recent call last):
...
TypeError: polyhedron has non-integral generators

class sage.geometry.polyhedron.ppl_lattice_polytope.LatticePolytope_PPL_class

Bases: C_Polyhedron

The lattice polytope class.

You should use LatticePolytope_PPL() to construct instances.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0), (1,0), (0,1))
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(0),Integer(1)))
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices

affine_lattice_polytope()

Return the lattice polytope restricted to affine_space().

OUTPUT:

A new, full-dimensional lattice polytope.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: poly_4d = LatticePolytope_PPL((-9,-6,0,0), (0,1,0,0), (1,0,0,0));  poly_4d
A 2-dimensional lattice polytope in ZZ^4 with 3 vertices
sage: poly_4d.space_dimension()
4
sage: poly_2d = poly_4d.affine_lattice_polytope();  poly_2d
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
sage: poly_2d.space_dimension()
2

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> poly_4d = LatticePolytope_PPL((-Integer(9),-Integer(6),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(0),Integer(0),Integer(0)));  poly_4d
A 2-dimensional lattice polytope in ZZ^4 with 3 vertices
>>> poly_4d.space_dimension()
4
>>> poly_2d = poly_4d.affine_lattice_polytope();  poly_2d
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
>>> poly_2d.space_dimension()
2

affine_space()

Return the affine space spanned by the polytope.

OUTPUT:

The free module \(\ZZ^n\), where \(n\) is the dimension of the affine space spanned by the points of the polytope.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: point = LatticePolytope_PPL((1,2,3))
sage: point.affine_space()
Free module of degree 3 and rank 0 over Integer Ring
Echelon basis matrix:
[]
sage: line = LatticePolytope_PPL((1,1,1), (1,2,3))
sage: line.affine_space()
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[0 1 2]

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> point = LatticePolytope_PPL((Integer(1),Integer(2),Integer(3)))
>>> point.affine_space()
Free module of degree 3 and rank 0 over Integer Ring
Echelon basis matrix:
[]
>>> line = LatticePolytope_PPL((Integer(1),Integer(1),Integer(1)), (Integer(1),Integer(2),Integer(3)))
>>> line.affine_space()
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[0 1 2]

ambient_space()

Return the ambient space.

OUTPUT:

The free module \(\ZZ^d\), where \(d\) is the ambient space dimension.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: point = LatticePolytope_PPL((1,2,3))
sage: point.ambient_space()
Ambient free module of rank 3 over the principal ideal domain Integer Ring

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> point = LatticePolytope_PPL((Integer(1),Integer(2),Integer(3)))
>>> point.ambient_space()
Ambient free module of rank 3 over the principal ideal domain Integer Ring

base_projection(fiber)

The projection that maps the sub-polytope fiber to a single point.

OUTPUT:

The quotient module of the ambient space modulo the affine_space() spanned by the fiber.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: poly = LatticePolytope_PPL((-9,-6,-1,-1),
....:                            (0,0,0,1), (0,0,1,0), (0,1,0,0), (1,0,0,0))
sage: fiber = next(poly.fibration_generator(2))
sage: poly.base_projection(fiber)
Finitely generated module V/W over Integer Ring with invariants (0, 0)

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> poly = LatticePolytope_PPL((-Integer(9),-Integer(6),-Integer(1),-Integer(1)),
...                            (Integer(0),Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(1),Integer(0)), (Integer(0),Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(0),Integer(0),Integer(0)))
>>> fiber = next(poly.fibration_generator(Integer(2)))
>>> poly.base_projection(fiber)
Finitely generated module V/W over Integer Ring with invariants (0, 0)

base_projection_matrix(fiber)

The projection that maps the sub-polytope fiber to a single point.

OUTPUT:

An integer matrix that represents the projection to the base.

See also

The base_projection() yields equivalent information, and is easier to use. However, just returning the matrix has lower overhead.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: poly = LatticePolytope_PPL((-9,-6,-1,-1),
....:                            (0,0,0,1), (0,0,1,0), (0,1,0,0), (1,0,0,0))
sage: fiber = next(poly.fibration_generator(2))
sage: poly.base_projection_matrix(fiber)
[ 0  0 -1  0]
[ 0  0  0 -1]

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> poly = LatticePolytope_PPL((-Integer(9),-Integer(6),-Integer(1),-Integer(1)),
...                            (Integer(0),Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(1),Integer(0)), (Integer(0),Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(0),Integer(0),Integer(0)))
>>> fiber = next(poly.fibration_generator(Integer(2)))
>>> poly.base_projection_matrix(fiber)
[ 0  0 -1  0]
[ 0  0  0 -1]

Note that the basis choice in base_projection() for the quotient is usually different:

Sage

sage: proj = poly.base_projection(fiber)
sage: proj_matrix = poly.base_projection_matrix(fiber)
sage: [proj(p) for p in poly.integral_points()]
[(-1, -1), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 1), (1, 0)]
sage: [proj_matrix*p for p in poly.integral_points()]
[(1, 1), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, -1), (-1, 0)]

Python

>>> from sage.all import *
>>> proj = poly.base_projection(fiber)
>>> proj_matrix = poly.base_projection_matrix(fiber)
>>> [proj(p) for p in poly.integral_points()]
[(-1, -1), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 1), (1, 0)]
>>> [proj_matrix*p for p in poly.integral_points()]
[(1, 1), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, 0), (0, -1), (-1, 0)]

base_rays(fiber, points)

Return the primitive lattice vectors that generate the direction given by the base projection of points.

INPUT:

• fiber – a sub-lattice polytope defining the base_projection()

• points – the points to project to the base

OUTPUT: a tuple of primitive \(\ZZ\)-vectors

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: poly = LatticePolytope_PPL((-9,-6,-1,-1),
....:                            (0,0,0,1), (0,0,1,0), (0,1,0,0), (1,0,0,0))
sage: fiber = next(poly.fibration_generator(2))
sage: poly.base_rays(fiber, poly.integral_points_not_interior_to_facets())
((-1, -1), (0, 1), (1, 0))

sage: p = LatticePolytope_PPL((1,0), (1,2), (-1,0))
sage: f = LatticePolytope_PPL((1,0), (-1,0))
sage: p.base_rays(f, p.integral_points())
((1),)

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> poly = LatticePolytope_PPL((-Integer(9),-Integer(6),-Integer(1),-Integer(1)),
...                            (Integer(0),Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(1),Integer(0)), (Integer(0),Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(0),Integer(0),Integer(0)))
>>> fiber = next(poly.fibration_generator(Integer(2)))
>>> poly.base_rays(fiber, poly.integral_points_not_interior_to_facets())
((-1, -1), (0, 1), (1, 0))

>>> p = LatticePolytope_PPL((Integer(1),Integer(0)), (Integer(1),Integer(2)), (-Integer(1),Integer(0)))
>>> f = LatticePolytope_PPL((Integer(1),Integer(0)), (-Integer(1),Integer(0)))
>>> p.base_rays(f, p.integral_points())
((1),)

bounding_box()

Return the coordinates of a rectangular box containing the non-empty polytope.

OUTPUT:

A pair of tuples (box_min, box_max) where box_min are the coordinates of a point bounding the coordinates of the polytope from below and box_max bounds the coordinates from above.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0), (1,0), (0,1)).bounding_box()
((0, 0), (1, 1))

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(0),Integer(1))).bounding_box()
((0, 0), (1, 1))

contains(point_coordinates)

Test whether point is contained in the polytope.

INPUT:

• point_coordinates – list/tuple/iterable of rational numbers; the coordinates of the point

OUTPUT: boolean

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: line = LatticePolytope_PPL((1,2,3), (-1,-2,-3))
sage: line.contains([0,0,0])
True
sage: line.contains([1,0,0])
False

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> line = LatticePolytope_PPL((Integer(1),Integer(2),Integer(3)), (-Integer(1),-Integer(2),-Integer(3)))
>>> line.contains([Integer(0),Integer(0),Integer(0)])
True
>>> line.contains([Integer(1),Integer(0),Integer(0)])
False

contains_origin()

Test whether the polytope contains the origin.

OUTPUT: boolean

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((1,2,3), (-1,-2,-3)).contains_origin()
True
sage: LatticePolytope_PPL((1,2,5), (-1,-2,-3)).contains_origin()
False

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> LatticePolytope_PPL((Integer(1),Integer(2),Integer(3)), (-Integer(1),-Integer(2),-Integer(3))).contains_origin()
True
>>> LatticePolytope_PPL((Integer(1),Integer(2),Integer(5)), (-Integer(1),-Integer(2),-Integer(3))).contains_origin()
False

embed_in_reflexive_polytope(output='hom')

Find an embedding as a sub-polytope of a maximal reflexive polytope.

INPUT:

• hom – string. One of 'hom' (default), 'polytope', or points. How the embedding is returned. See the output section for details.

OUTPUT:

An embedding into a reflexive polytope. Depending on the output option slightly different data is returned.

• If output='hom', a map from a reflexive polytope onto self is returned.

• If output='polytope', a reflexive polytope that contains self (up to a lattice linear transformation) is returned. That is, the domain of the output='hom' map is returned. If the affine span of self is less or equal 2-dimensional, the output is one of the following three possibilities:

  polar_P2_polytope(), polar_P1xP1_polytope(), or polar_P2_112_polytope().

• If output='points', a dictionary containing the integral points of self as keys and the corresponding integral point of the reflexive polytope as value.

If there is no such embedding, a LatticePolytopeNoEmbeddingError is raised. Even if it exists, the ambient reflexive polytope is usually not uniquely determined and a random but fixed choice will be returned.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: polygon = LatticePolytope_PPL((0,0,2,1), (0,1,2,0), (2,3,0,0), (2,0,0,3))
sage: polygon.embed_in_reflexive_polytope()
The map A*x+b with
  A=
    [ 1  1]
    [ 0  1]
    [-1 -1]
    [ 1  0]
  b = (-1, 0, 3, 0)
sage: polygon.embed_in_reflexive_polytope('polytope')
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
sage: polygon.embed_in_reflexive_polytope('points')
{(0, 0, 2, 1): (1, 0),
 (0, 1, 2, 0): (0, 1),
 (1, 0, 1, 2): (2, 0),
 (1, 1, 1, 1): (1, 1),
 (1, 2, 1, 0): (0, 2),
 (2, 0, 0, 3): (3, 0),
 (2, 1, 0, 2): (2, 1),
 (2, 2, 0, 1): (1, 2),
 (2, 3, 0, 0): (0, 3)}

sage: LatticePolytope_PPL((0,0), (4,0), (0,4)).embed_in_reflexive_polytope()
Traceback (most recent call last):
...
LatticePolytopeNoEmbeddingError: not a sub-polytope of a reflexive polygon

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> polygon = LatticePolytope_PPL((Integer(0),Integer(0),Integer(2),Integer(1)), (Integer(0),Integer(1),Integer(2),Integer(0)), (Integer(2),Integer(3),Integer(0),Integer(0)), (Integer(2),Integer(0),Integer(0),Integer(3)))
>>> polygon.embed_in_reflexive_polytope()
The map A*x+b with
  A=
    [ 1  1]
    [ 0  1]
    [-1 -1]
    [ 1  0]
  b = (-1, 0, 3, 0)
>>> polygon.embed_in_reflexive_polytope('polytope')
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
>>> polygon.embed_in_reflexive_polytope('points')
{(0, 0, 2, 1): (1, 0),
 (0, 1, 2, 0): (0, 1),
 (1, 0, 1, 2): (2, 0),
 (1, 1, 1, 1): (1, 1),
 (1, 2, 1, 0): (0, 2),
 (2, 0, 0, 3): (3, 0),
 (2, 1, 0, 2): (2, 1),
 (2, 2, 0, 1): (1, 2),
 (2, 3, 0, 0): (0, 3)}

>>> LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(4),Integer(0)), (Integer(0),Integer(4))).embed_in_reflexive_polytope()
Traceback (most recent call last):
...
LatticePolytopeNoEmbeddingError: not a sub-polytope of a reflexive polygon

fibration_generator(dim)

Generate the lattice polytope fibrations.

For the purposes of this function, a lattice polytope fiber is a sub-lattice polytope. Projecting the plane spanned by the subpolytope to a point yields another lattice polytope, the base of the fibration.

INPUT:

• dim – integer; the dimension of the lattice polytope fiber

OUTPUT:

A generator yielding the distinct lattice polytope fibers of given dimension.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: p = LatticePolytope_PPL((-9,-6,-1,-1),
....:                         (0,0,0,1), (0,0,1,0), (0,1,0,0), (1,0,0,0))
sage: list(p.fibration_generator(2))
[A 2-dimensional lattice polytope in ZZ^4 with 3 vertices]

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> p = LatticePolytope_PPL((-Integer(9),-Integer(6),-Integer(1),-Integer(1)),
...                         (Integer(0),Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(1),Integer(0)), (Integer(0),Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(0),Integer(0),Integer(0)))
>>> list(p.fibration_generator(Integer(2)))
[A 2-dimensional lattice polytope in ZZ^4 with 3 vertices]

has_IP_property()

Whether the lattice polytope has the IP property.

That is, the polytope is full-dimensional and the origin is a interior point not on the boundary.

OUTPUT: boolean

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((-1,-1), (0,1), (1,0)).has_IP_property()
True
sage: LatticePolytope_PPL((-1,-1), (1,1)).has_IP_property()
False

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> LatticePolytope_PPL((-Integer(1),-Integer(1)), (Integer(0),Integer(1)), (Integer(1),Integer(0))).has_IP_property()
True
>>> LatticePolytope_PPL((-Integer(1),-Integer(1)), (Integer(1),Integer(1))).has_IP_property()
False

integral_points()

Return the integral points in the polyhedron.

Uses the naive algorithm (iterate over a rectangular bounding box).

OUTPUT:

The list of integral points in the polyhedron. If the polyhedron is not compact, a ValueError is raised.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((-1,-1), (1,0), (1,1), (0,1)).integral_points()
((-1, -1), (0, 0), (0, 1), (1, 0), (1, 1))

sage: simplex = LatticePolytope_PPL((1,2,3), (2,3,7), (-2,-3,-11))
sage: simplex.integral_points()
((-2, -3, -11), (0, 0, -2), (1, 2, 3), (2, 3, 7))

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> LatticePolytope_PPL((-Integer(1),-Integer(1)), (Integer(1),Integer(0)), (Integer(1),Integer(1)), (Integer(0),Integer(1))).integral_points()
((-1, -1), (0, 0), (0, 1), (1, 0), (1, 1))

>>> simplex = LatticePolytope_PPL((Integer(1),Integer(2),Integer(3)), (Integer(2),Integer(3),Integer(7)), (-Integer(2),-Integer(3),-Integer(11)))
>>> simplex.integral_points()
((-2, -3, -11), (0, 0, -2), (1, 2, 3), (2, 3, 7))

The polyhedron need not be full-dimensional:

Sage

sage: simplex = LatticePolytope_PPL((1,2,3,5), (2,3,7,5), (-2,-3,-11,5))
sage: simplex.integral_points()
((-2, -3, -11, 5), (0, 0, -2, 5), (1, 2, 3, 5), (2, 3, 7, 5))

sage: point = LatticePolytope_PPL((2,3,7))
sage: point.integral_points()
((2, 3, 7),)

sage: empty = LatticePolytope_PPL()
sage: empty.integral_points()
()

Python

>>> from sage.all import *
>>> simplex = LatticePolytope_PPL((Integer(1),Integer(2),Integer(3),Integer(5)), (Integer(2),Integer(3),Integer(7),Integer(5)), (-Integer(2),-Integer(3),-Integer(11),Integer(5)))
>>> simplex.integral_points()
((-2, -3, -11, 5), (0, 0, -2, 5), (1, 2, 3, 5), (2, 3, 7, 5))

>>> point = LatticePolytope_PPL((Integer(2),Integer(3),Integer(7)))
>>> point.integral_points()
((2, 3, 7),)

>>> empty = LatticePolytope_PPL()
>>> empty.integral_points()
()

Here is a simplex where the naive algorithm of running over all points in a rectangular bounding box no longer works fast enough:

Sage

sage: v = [(1,0,7,-1), (-2,-2,4,-3), (-1,-1,-1,4), (2,9,0,-5), (-2,-1,5,1)]
sage: simplex = LatticePolytope_PPL(v); simplex
A 4-dimensional lattice polytope in ZZ^4 with 5 vertices
sage: len(simplex.integral_points())
49

Python

>>> from sage.all import *
>>> v = [(Integer(1),Integer(0),Integer(7),-Integer(1)), (-Integer(2),-Integer(2),Integer(4),-Integer(3)), (-Integer(1),-Integer(1),-Integer(1),Integer(4)), (Integer(2),Integer(9),Integer(0),-Integer(5)), (-Integer(2),-Integer(1),Integer(5),Integer(1))]
>>> simplex = LatticePolytope_PPL(v); simplex
A 4-dimensional lattice polytope in ZZ^4 with 5 vertices
>>> len(simplex.integral_points())
49

Finally, the 3-d reflexive polytope number 4078:

Sage

sage: v = [(1,0,0), (0,1,0), (0,0,1), (0,0,-1), (0,-2,1),
....:      (-1,2,-1), (-1,2,-2), (-1,1,-2), (-1,-1,2), (-1,-3,2)]
sage: P = LatticePolytope_PPL(*v)
sage: pts1 = P.integral_points()            # Sage's own code
sage: pts2 = LatticePolytope(v).points()                                    # needs palp
sage: for p in pts1: p.set_immutable()
sage: set(pts1) == set(pts2)                                                # needs palp
True

sage: len(Polyhedron(v).integral_points())  # takes about 1 ms
23
sage: len(LatticePolytope(v).points())      # takes about 13 ms             # needs palp
23
sage: len(LatticePolytope_PPL(*v).integral_points())  # takes about 0.5 ms
23

Python

>>> from sage.all import *
>>> v = [(Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0)), (Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),-Integer(1)), (Integer(0),-Integer(2),Integer(1)),
...      (-Integer(1),Integer(2),-Integer(1)), (-Integer(1),Integer(2),-Integer(2)), (-Integer(1),Integer(1),-Integer(2)), (-Integer(1),-Integer(1),Integer(2)), (-Integer(1),-Integer(3),Integer(2))]
>>> P = LatticePolytope_PPL(*v)
>>> pts1 = P.integral_points()            # Sage's own code
>>> pts2 = LatticePolytope(v).points()                                    # needs palp
>>> for p in pts1: p.set_immutable()
>>> set(pts1) == set(pts2)                                                # needs palp
True

>>> len(Polyhedron(v).integral_points())  # takes about 1 ms
23
>>> len(LatticePolytope(v).points())      # takes about 13 ms             # needs palp
23
>>> len(LatticePolytope_PPL(*v).integral_points())  # takes about 0.5 ms
23

integral_points_not_interior_to_facets()

Return the integral points not interior to facets.

OUTPUT:

A tuple whose entries are the coordinate vectors of integral points not interior to facets (codimension one faces) of the lattice polytope.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: square = LatticePolytope_PPL((-1,-1), (-1,1), (1,-1), (1,1))
sage: square.n_integral_points()
9
sage: square.integral_points_not_interior_to_facets()
((-1, -1), (-1, 1), (0, 0), (1, -1), (1, 1))

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> square = LatticePolytope_PPL((-Integer(1),-Integer(1)), (-Integer(1),Integer(1)), (Integer(1),-Integer(1)), (Integer(1),Integer(1)))
>>> square.n_integral_points()
9
>>> square.integral_points_not_interior_to_facets()
((-1, -1), (-1, 1), (0, 0), (1, -1), (1, 1))

is_bounded()

Return whether the lattice polytope is compact.

OUTPUT: always True, since polytopes are by definition compact

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0), (1,0), (0,1)).is_bounded()
True

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(0),Integer(1))).is_bounded()
True

is_full_dimensional()

Return whether the lattice polytope is full dimensional.

OUTPUT: boolean; whether the affine_dimension() equals the ambient space dimension

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: p = LatticePolytope_PPL((0,0), (0,1))
sage: p.is_full_dimensional()
False
sage: q = LatticePolytope_PPL((0,0), (0,1), (1,0))
sage: q.is_full_dimensional()
True

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> p = LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(0),Integer(1)))
>>> p.is_full_dimensional()
False
>>> q = LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(0),Integer(1)), (Integer(1),Integer(0)))
>>> q.is_full_dimensional()
True

is_simplex()

Return whether the polyhedron is a simplex.

OUTPUT:

Boolean, whether the polyhedron is a simplex (possibly of strictly smaller dimension than the ambient space).

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0,0), (1,0,0), (0,1,0)).is_simplex()
True

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> LatticePolytope_PPL((Integer(0),Integer(0),Integer(0)), (Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0))).is_simplex()
True

lattice_automorphism_group(points=None, point_labels=None)

The integral subgroup of the restricted automorphism group.

INPUT:

• points – tuple of coordinate vectors or None (default). If specified, the points must form complete orbits under the lattice automorphism group. If None all vertices are used.

• point_labels – tuple of labels for the points or None (default). These will be used as labels for the do permutation group. If None, the points will be used themselves.

OUTPUT:

The integral subgroup of the restricted automorphism group acting on the given points, or all vertices if not specified.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: Z3square = LatticePolytope_PPL((0,0), (1,2), (2,1), (3,3))
sage: Z3square.lattice_automorphism_group()                                 # needs sage.graphs sage.groups
Permutation Group with generators [(), ((1,2),(2,1)),
((0,0),(3,3)), ((0,0),(3,3))((1,2),(2,1))]

sage: G1 = Z3square.lattice_automorphism_group(point_labels=(1,2,3,4))      # needs sage.graphs sage.groups
sage: G1                                                                    # needs sage.graphs sage.groups
Permutation Group with generators [(), (2,3), (1,4), (1,4)(2,3)]
sage: G1.cardinality()                                                      # needs sage.graphs sage.groups
4

sage: G2 = Z3square.restricted_automorphism_group(vertex_labels=(1,2,3,4))  # needs sage.graphs sage.groups
sage: G2 == PermutationGroup([[(2,3)], [(1,2),(3,4)], [(1,4)]])             # needs sage.graphs sage.groups
True
sage: G2.cardinality()                                                      # needs sage.graphs sage.groups
8

sage: points = Z3square.integral_points(); points
((0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3))
sage: Z3square.lattice_automorphism_group(points,                           # needs sage.graphs sage.groups
....:                                     point_labels=(1,2,3,4,5,6))
Permutation Group with generators [(), (3,4), (1,6)(2,5), (1,6)(2,5)(3,4)]

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> Z3square = LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(1),Integer(2)), (Integer(2),Integer(1)), (Integer(3),Integer(3)))
>>> Z3square.lattice_automorphism_group()                                 # needs sage.graphs sage.groups
Permutation Group with generators [(), ((1,2),(2,1)),
((0,0),(3,3)), ((0,0),(3,3))((1,2),(2,1))]

>>> G1 = Z3square.lattice_automorphism_group(point_labels=(Integer(1),Integer(2),Integer(3),Integer(4)))      # needs sage.graphs sage.groups
>>> G1                                                                    # needs sage.graphs sage.groups
Permutation Group with generators [(), (2,3), (1,4), (1,4)(2,3)]
>>> G1.cardinality()                                                      # needs sage.graphs sage.groups
4

>>> G2 = Z3square.restricted_automorphism_group(vertex_labels=(Integer(1),Integer(2),Integer(3),Integer(4)))  # needs sage.graphs sage.groups
>>> G2 == PermutationGroup([[(Integer(2),Integer(3))], [(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(4))]])             # needs sage.graphs sage.groups
True
>>> G2.cardinality()                                                      # needs sage.graphs sage.groups
8

>>> points = Z3square.integral_points(); points
((0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3))
>>> Z3square.lattice_automorphism_group(points,                           # needs sage.graphs sage.groups
...                                     point_labels=(Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)))
Permutation Group with generators [(), (3,4), (1,6)(2,5), (1,6)(2,5)(3,4)]

Point labels also work for lattice polytopes that are not full-dimensional, see Issue #16669:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: lp = LatticePolytope_PPL((1,0,0), (0,1,0), (-1,-1,0))
sage: lp.lattice_automorphism_group(point_labels=(0,1,2))                   # needs sage.graphs sage.groups
Permutation Group with generators [(), (1,2), (0,1), (0,1,2), (0,2,1), (0,2)]

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> lp = LatticePolytope_PPL((Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0)), (-Integer(1),-Integer(1),Integer(0)))
>>> lp.lattice_automorphism_group(point_labels=(Integer(0),Integer(1),Integer(2)))                   # needs sage.graphs sage.groups
Permutation Group with generators [(), (1,2), (0,1), (0,1,2), (0,2,1), (0,2)]

n_integral_points()

Return the number of integral points.

OUTPUT:

Integer. The number of integral points contained in the lattice polytope.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0), (1,0), (0,1)).n_integral_points()
3

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(0),Integer(1))).n_integral_points()
3

n_vertices()

Return the number of vertices.

OUTPUT: integer; the number of vertices

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0,0), (1,0,0), (0,1,0)).n_vertices()
3

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> LatticePolytope_PPL((Integer(0),Integer(0),Integer(0)), (Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0))).n_vertices()
3

pointsets_mod_automorphism(pointsets)

Return pointsets modulo the automorphisms of self.

INPUT:

• polytopes – tuple/list/iterable of subsets of the integral points of self

OUTPUT:

Representatives of the point sets modulo the lattice_automorphism_group() of self.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: square = LatticePolytope_PPL((-1,-1), (-1,1), (1,-1), (1,1))
sage: fibers = [f.vertices() for f in square.fibration_generator(1)]
sage: square.pointsets_mod_automorphism(fibers)                             # needs sage.graphs sage.groups
(frozenset({(-1, -1), (1, 1)}), frozenset({(-1, 0), (1, 0)}))

sage: cell24 = LatticePolytope_PPL(
....:     (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (1,-1,-1,1), (0,0,-1,1),
....:     (0,-1,0,1), (-1,0,0,1), (1,0,0,-1), (0,1,0,-1), (0,0,1,-1), (-1,1,1,-1),
....:     (1,-1,-1,0), (0,0,-1,0), (0,-1,0,0), (-1,0,0,0), (1,-1,0,0), (1,0,-1,0),
....:     (0,1,1,-1), (-1,1,1,0), (-1,1,0,0), (-1,0,1,0), (0,-1,-1,1), (0,0,0,-1))
sage: fibers = [f.vertices() for f in cell24.fibration_generator(2)]
sage: cell24.pointsets_mod_automorphism(fibers)     # long time             # needs sage.graphs sage.groups
(frozenset({(-1, 0, 0, 0),
            (-1, 0, 0, 1),
            (0, 0, 0, -1),
            (0, 0, 0, 1),
            (1, 0, 0, -1),
            (1, 0, 0, 0)}),
 frozenset({(-1, 0, 0, 0), (-1, 1, 1, 0), (1, -1, -1, 0), (1, 0, 0, 0)}))

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> square = LatticePolytope_PPL((-Integer(1),-Integer(1)), (-Integer(1),Integer(1)), (Integer(1),-Integer(1)), (Integer(1),Integer(1)))
>>> fibers = [f.vertices() for f in square.fibration_generator(Integer(1))]
>>> square.pointsets_mod_automorphism(fibers)                             # needs sage.graphs sage.groups
(frozenset({(-1, -1), (1, 1)}), frozenset({(-1, 0), (1, 0)}))

>>> cell24 = LatticePolytope_PPL(
...     (Integer(1),Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1),Integer(0)), (Integer(0),Integer(0),Integer(0),Integer(1)), (Integer(1),-Integer(1),-Integer(1),Integer(1)), (Integer(0),Integer(0),-Integer(1),Integer(1)),
...     (Integer(0),-Integer(1),Integer(0),Integer(1)), (-Integer(1),Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(0),-Integer(1)), (Integer(0),Integer(1),Integer(0),-Integer(1)), (Integer(0),Integer(0),Integer(1),-Integer(1)), (-Integer(1),Integer(1),Integer(1),-Integer(1)),
...     (Integer(1),-Integer(1),-Integer(1),Integer(0)), (Integer(0),Integer(0),-Integer(1),Integer(0)), (Integer(0),-Integer(1),Integer(0),Integer(0)), (-Integer(1),Integer(0),Integer(0),Integer(0)), (Integer(1),-Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(0),-Integer(1),Integer(0)),
...     (Integer(0),Integer(1),Integer(1),-Integer(1)), (-Integer(1),Integer(1),Integer(1),Integer(0)), (-Integer(1),Integer(1),Integer(0),Integer(0)), (-Integer(1),Integer(0),Integer(1),Integer(0)), (Integer(0),-Integer(1),-Integer(1),Integer(1)), (Integer(0),Integer(0),Integer(0),-Integer(1)))
>>> fibers = [f.vertices() for f in cell24.fibration_generator(Integer(2))]
>>> cell24.pointsets_mod_automorphism(fibers)     # long time             # needs sage.graphs sage.groups
(frozenset({(-1, 0, 0, 0),
            (-1, 0, 0, 1),
            (0, 0, 0, -1),
            (0, 0, 0, 1),
            (1, 0, 0, -1),
            (1, 0, 0, 0)}),
 frozenset({(-1, 0, 0, 0), (-1, 1, 1, 0), (1, -1, -1, 0), (1, 0, 0, 0)}))

restricted_automorphism_group(vertex_labels=None)

Return the restricted automorphism group.

First, let the linear automorphism group be the subgroup of the Euclidean group \(E(d) = GL(d,\RR) \ltimes \RR^d\) preserving the \(d\)-dimensional polyhedron. The Euclidean group acts in the usual way \(\vec{x}\mapsto A\vec{x}+b\) on the ambient space. The restricted automorphism group is the subgroup of the linear automorphism group generated by permutations of vertices. If the polytope is full-dimensional, it is equal to the full (unrestricted) automorphism group.

INPUT:

• vertex_labels – tuple or None (default). The labels of the vertices that will be used in the output permutation group. By default, the vertices are used themselves.

OUTPUT:

A PermutationGroup acting on the vertices (or the vertex_labels, if specified).

REFERENCES:

[BSS2009]

EXAMPLES:

Sage

sage: # needs sage.graphs sage.groups
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: Z3square = LatticePolytope_PPL((0,0), (1,2), (2,1), (3,3))
sage: G1234 = Z3square.restricted_automorphism_group(
....:     vertex_labels=(1,2,3,4))
sage: G1234 == PermutationGroup([[(2,3)], [(1,2),(3,4)]])
True
sage: G = Z3square.restricted_automorphism_group()
sage: G == PermutationGroup([[((1,2),(2,1))], [((0,0),(1,2)),
....:                         ((2,1),(3,3))], [((0,0),(3,3))]])
True
sage: set(G.domain()) == set(Z3square.vertices())
True
sage: (set(tuple(x) for x in G.orbit(Z3square.vertices()[0]))
....:   == set([(0, 0), (1, 2), (3, 3), (2, 1)]))
True
sage: cell24 = LatticePolytope_PPL(
....:     (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (1,-1,-1,1), (0,0,-1,1),
....:     (0,-1,0,1), (-1,0,0,1), (1,0,0,-1), (0,1,0,-1), (0,0,1,-1), (-1,1,1,-1),
....:     (1,-1,-1,0), (0,0,-1,0), (0,-1,0,0), (-1,0,0,0), (1,-1,0,0), (1,0,-1,0),
....:     (0,1,1,-1), (-1,1,1,0), (-1,1,0,0), (-1,0,1,0), (0,-1,-1,1), (0,0,0,-1))
sage: cell24.restricted_automorphism_group().cardinality()
1152

Python

>>> from sage.all import *
>>> # needs sage.graphs sage.groups
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> Z3square = LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(1),Integer(2)), (Integer(2),Integer(1)), (Integer(3),Integer(3)))
>>> G1234 = Z3square.restricted_automorphism_group(
...     vertex_labels=(Integer(1),Integer(2),Integer(3),Integer(4)))
>>> G1234 == PermutationGroup([[(Integer(2),Integer(3))], [(Integer(1),Integer(2)),(Integer(3),Integer(4))]])
True
>>> G = Z3square.restricted_automorphism_group()
>>> G == PermutationGroup([[((Integer(1),Integer(2)),(Integer(2),Integer(1)))], [((Integer(0),Integer(0)),(Integer(1),Integer(2))),
...                         ((Integer(2),Integer(1)),(Integer(3),Integer(3)))], [((Integer(0),Integer(0)),(Integer(3),Integer(3)))]])
True
>>> set(G.domain()) == set(Z3square.vertices())
True
>>> (set(tuple(x) for x in G.orbit(Z3square.vertices()[Integer(0)]))
...   == set([(Integer(0), Integer(0)), (Integer(1), Integer(2)), (Integer(3), Integer(3)), (Integer(2), Integer(1))]))
True
>>> cell24 = LatticePolytope_PPL(
...     (Integer(1),Integer(0),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(1),Integer(0)), (Integer(0),Integer(0),Integer(0),Integer(1)), (Integer(1),-Integer(1),-Integer(1),Integer(1)), (Integer(0),Integer(0),-Integer(1),Integer(1)),
...     (Integer(0),-Integer(1),Integer(0),Integer(1)), (-Integer(1),Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(0),-Integer(1)), (Integer(0),Integer(1),Integer(0),-Integer(1)), (Integer(0),Integer(0),Integer(1),-Integer(1)), (-Integer(1),Integer(1),Integer(1),-Integer(1)),
...     (Integer(1),-Integer(1),-Integer(1),Integer(0)), (Integer(0),Integer(0),-Integer(1),Integer(0)), (Integer(0),-Integer(1),Integer(0),Integer(0)), (-Integer(1),Integer(0),Integer(0),Integer(0)), (Integer(1),-Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(0),-Integer(1),Integer(0)),
...     (Integer(0),Integer(1),Integer(1),-Integer(1)), (-Integer(1),Integer(1),Integer(1),Integer(0)), (-Integer(1),Integer(1),Integer(0),Integer(0)), (-Integer(1),Integer(0),Integer(1),Integer(0)), (Integer(0),-Integer(1),-Integer(1),Integer(1)), (Integer(0),Integer(0),Integer(0),-Integer(1)))
>>> cell24.restricted_automorphism_group().cardinality()
1152

sub_polytope_generator()

Generate the maximal lattice sub-polytopes.

OUTPUT:

A generator yielding the maximal (with respect to inclusion) lattice sub polytopes. That is, each can be gotten as the convex hull of the integral points of self with one vertex removed.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: P = LatticePolytope_PPL((1,0,0), (0,1,0), (0,0,1), (-1,-1,-1))
sage: for p in P.sub_polytope_generator():
....:     print(p.vertices())
((0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0))
((-1, -1, -1), (0, 0, 0), (0, 1, 0), (1, 0, 0))
((-1, -1, -1), (0, 0, 0), (0, 0, 1), (1, 0, 0))
((-1, -1, -1), (0, 0, 0), (0, 0, 1), (0, 1, 0))

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> P = LatticePolytope_PPL((Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0)), (Integer(0),Integer(0),Integer(1)), (-Integer(1),-Integer(1),-Integer(1)))
>>> for p in P.sub_polytope_generator():
...     print(p.vertices())
((0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0))
((-1, -1, -1), (0, 0, 0), (0, 1, 0), (1, 0, 0))
((-1, -1, -1), (0, 0, 0), (0, 0, 1), (1, 0, 0))
((-1, -1, -1), (0, 0, 0), (0, 0, 1), (0, 1, 0))

vertices()

Return the vertices as a tuple of \(\ZZ\)-vectors.

OUTPUT:

A tuple of \(\ZZ\)-vectors. Each entry is the coordinate vector of an integral points of the lattice polytope.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: p = LatticePolytope_PPL((-9,-6,-1,-1),
....:                         (0,0,0,1), (0,0,1,0), (0,1,0,0), (1,0,0,0))
sage: p.vertices()
((-9, -6, -1, -1), (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0))
sage: p.minimized_generators()
Generator_System {point(-9/1, -6/1, -1/1, -1/1), point(0/1, 0/1, 0/1, 1/1),
point(0/1, 0/1, 1/1, 0/1), point(0/1, 1/1, 0/1, 0/1), point(1/1, 0/1, 0/1, 0/1)}

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> p = LatticePolytope_PPL((-Integer(9),-Integer(6),-Integer(1),-Integer(1)),
...                         (Integer(0),Integer(0),Integer(0),Integer(1)), (Integer(0),Integer(0),Integer(1),Integer(0)), (Integer(0),Integer(1),Integer(0),Integer(0)), (Integer(1),Integer(0),Integer(0),Integer(0)))
>>> p.vertices()
((-9, -6, -1, -1), (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0))
>>> p.minimized_generators()
Generator_System {point(-9/1, -6/1, -1/1, -1/1), point(0/1, 0/1, 0/1, 1/1),
point(0/1, 0/1, 1/1, 0/1), point(0/1, 1/1, 0/1, 0/1), point(1/1, 0/1, 0/1, 0/1)}

vertices_saturating(constraint)

Return the vertices saturating the constraint.

INPUT:

• constraint – a constraint (inequality or equation) of the polytope

OUTPUT:

The tuple of vertices saturating the constraint. The vertices are returned as \(\ZZ\)-vectors, as in vertices().

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: p = LatticePolytope_PPL((0,0), (0,1), (1,0))
sage: ieq = next(iter(p.constraints()));  ieq
x0>=0
sage: p.vertices_saturating(ieq)
((0, 0), (0, 1))

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> p = LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(0),Integer(1)), (Integer(1),Integer(0)))
>>> ieq = next(iter(p.constraints()));  ieq
x0>=0
>>> p.vertices_saturating(ieq)
((0, 0), (0, 1))