Combinatorial face of a polyhedron#

This module provides the combinatorial type of a polyhedral face.

EXAMPLES:

Obtain a face from a face iterator:

sage: P = polytopes.cube()
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator()
sage: face = next(it); face
A 2-dimensional face of a 3-dimensional combinatorial polyhedron

Obtain a face from a face lattice index:

sage: P = polytopes.simplex(2)
sage: C = CombinatorialPolyhedron(P)
sage: sorted(C.face_lattice()._elements)                                            # needs sage.combinat
[0, 1, 2, 3, 4, 5, 6, 7]
sage: face = C.face_by_face_lattice_index(0); face
A -1-dimensional face of a 2-dimensional combinatorial polyhedron

Obtain further information regarding a face:

sage: P = polytopes.octahedron()
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator(2)
sage: face = next(it); face
A 2-dimensional face of a 3-dimensional combinatorial polyhedron
sage: face.ambient_Vrepresentation()
(A vertex at (0, 0, 1), A vertex at (0, 1, 0), A vertex at (1, 0, 0))
sage: face.n_ambient_Vrepresentation()
3
sage: face.ambient_H_indices()
(5,)
sage: face.dimension()
2
sage: face.ambient_dimension()
3

AUTHOR:

Jonathan Kliem (2019-05)

class sage.geometry.polyhedron.combinatorial_polyhedron.combinatorial_face.CombinatorialFace#

Bases: SageObject

A class of the combinatorial type of a polyhedral face.

EXAMPLES:

Obtain a combinatorial face from a face iterator:

sage: P = polytopes.cyclic_polytope(5,8)
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator()
sage: next(it)
A 0-dimensional face of a 5-dimensional combinatorial polyhedron

Obtain a combinatorial face from an index of the face lattice:

sage: F = C.face_lattice()                                                      # needs sage.combinat
sage: F._elements[3]                                                            # needs sage.combinat
34
sage: C.face_by_face_lattice_index(29)
A 1-dimensional face of a 5-dimensional combinatorial polyhedron

Obtain the dimension of a combinatorial face:

sage: face = next(it)
sage: face.dimension()
0

The dimension of the polyhedron:

sage: face.ambient_dimension()
5

The Vrepresentation:

sage: face.ambient_Vrepresentation()
(A vertex at (6, 36, 216, 1296, 7776),)
sage: face.ambient_V_indices()
(6,)
sage: face.n_ambient_Vrepresentation()
1

The Hrepresentation:

sage: face.ambient_Hrepresentation()
(An inequality (60, -112, 65, -14, 1) x + 0 >= 0,
 An inequality (180, -216, 91, -16, 1) x + 0 >= 0,
 An inequality (360, -342, 119, -18, 1) x + 0 >= 0,
 An inequality (840, -638, 179, -22, 1) x + 0 >= 0,
 An inequality (-2754, 1175, -245, 25, -1) x + 2520 >= 0,
 An inequality (504, -450, 145, -20, 1) x + 0 >= 0,
 An inequality (-1692, 853, -203, 23, -1) x + 1260 >= 0,
 An inequality (252, -288, 113, -18, 1) x + 0 >= 0,
 An inequality (-844, 567, -163, 21, -1) x + 420 >= 0,
 An inequality (84, -152, 83, -16, 1) x + 0 >= 0,
 An inequality (-210, 317, -125, 19, -1) x + 0 >= 0)
sage: face.ambient_H_indices()
(3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 19)
sage: face.n_ambient_Hrepresentation()
11

ambient_H_indices(add_equations=True)#

Return the indices of the Hrepresentation objects of the ambient polyhedron defining the face.

INPUT:

add_equations – boolean (default: True); whether or not to include the equations

EXAMPLES:

sage: # needs sage.combinat
sage: P = polytopes.permutahedron(5)
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator(2)
sage: face = next(it)
sage: face.ambient_H_indices(add_equations=False)
(28, 29)
sage: face2 = next(it)
sage: face2.ambient_H_indices(add_equations=False)
(25, 29)

Add the indices of the equation:

sage: face.ambient_H_indices(add_equations=True)                            # needs sage.combinat
(28, 29, 30)
sage: face2.ambient_H_indices(add_equations=True)                           # needs sage.combinat
(25, 29, 30)

Another example:

sage: P = polytopes.cyclic_polytope(4,6)
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator()
sage: _ = next(it); _ = next(it)
sage: next(it).ambient_H_indices()
(0, 1, 2, 4, 5, 7)
sage: next(it).ambient_H_indices()
(0, 1, 5, 6, 7, 8)
sage: next(it).ambient_H_indices()
(0, 1, 2, 3, 6, 8)
sage: [next(it).dimension() for _ in range(2)]
[0, 1]
sage: face = next(it)
sage: face.ambient_H_indices()
(4, 5, 7)

See also

ambient_Hrepresentation().

ambient_Hrepresentation()#

Return the Hrepresentation objects of the ambient polyhedron defining the face.

It consists of the facets/inequalities that contain the face and the equations defining the ambient polyhedron.

EXAMPLES:

sage: # needs sage.combinat
sage: P = polytopes.permutahedron(5)
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator(2)
sage: next(it).ambient_Hrepresentation()
(An inequality (1, 1, 1, 0, 0) x - 6 >= 0,
 An inequality (0, 0, 0, -1, 0) x + 5 >= 0,
 An equation (1, 1, 1, 1, 1) x - 15 == 0)
sage: next(it).ambient_Hrepresentation()
(An inequality (0, 0, -1, -1, 0) x + 9 >= 0,
 An inequality (0, 0, 0, -1, 0) x + 5 >= 0,
 An equation (1, 1, 1, 1, 1) x - 15 == 0)

sage: P = polytopes.cyclic_polytope(4,6)
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator()
sage: next(it).ambient_Hrepresentation()
(An inequality (-20, 29, -10, 1) x + 0 >= 0,
 An inequality (60, -47, 12, -1) x + 0 >= 0,
 An inequality (30, -31, 10, -1) x + 0 >= 0,
 An inequality (10, -17, 8, -1) x + 0 >= 0,
 An inequality (-154, 71, -14, 1) x + 120 >= 0,
 An inequality (-78, 49, -12, 1) x + 40 >= 0)
sage: next(it).ambient_Hrepresentation()
(An inequality (-50, 35, -10, 1) x + 24 >= 0,
 An inequality (-12, 19, -8, 1) x + 0 >= 0,
 An inequality (-20, 29, -10, 1) x + 0 >= 0,
 An inequality (60, -47, 12, -1) x + 0 >= 0,
 An inequality (-154, 71, -14, 1) x + 120 >= 0,
 An inequality (-78, 49, -12, 1) x + 40 >= 0)

See also

ambient_H_indices().

ambient_V_indices()#

Return the indices of the Vrepresentation objects of the ambient polyhedron defining the face.

EXAMPLES:

sage: # needs sage.combinat
sage: P = polytopes.permutahedron(5)
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator(dimension=2)
sage: face = next(it)
sage: next(it).ambient_V_indices()
(32, 91, 92, 93, 94, 95)
sage: next(it).ambient_V_indices()
(32, 89, 90, 94)

sage: C = CombinatorialPolyhedron([[0,1,2],[0,1,3],[0,2,3],[1,2,3]])
sage: it = C.face_generator()
sage: for face in it: (face.dimension(), face.ambient_V_indices())
(2, (1, 2, 3))
(2, (0, 2, 3))
(2, (0, 1, 3))
(2, (0, 1, 2))
(1, (2, 3))
(1, (1, 3))
(1, (1, 2))
(0, (3,))
(0, (2,))
(0, (1,))
(1, (0, 3))
(1, (0, 2))
(0, (0,))
(1, (0, 1))

See also

ambient_Vrepresentation().

ambient_Vrepresentation()#

Return the Vrepresentation objects of the ambient polyhedron defining the face.

It consists of the vertices/rays/lines that face contains.

EXAMPLES:

sage: # needs sage.combinat
sage: P = polytopes.permutahedron(5)
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator(dimension=2)
sage: face = next(it)
sage: face.ambient_Vrepresentation()
(A vertex at (1, 3, 2, 5, 4),
 A vertex at (2, 3, 1, 5, 4),
 A vertex at (3, 1, 2, 5, 4),
 A vertex at (3, 2, 1, 5, 4),
 A vertex at (2, 1, 3, 5, 4),
 A vertex at (1, 2, 3, 5, 4))
sage: face = next(it)
sage: face.ambient_Vrepresentation()
(A vertex at (2, 1, 4, 5, 3),
 A vertex at (3, 2, 4, 5, 1),
 A vertex at (3, 1, 4, 5, 2),
 A vertex at (1, 3, 4, 5, 2),
 A vertex at (1, 2, 4, 5, 3),
 A vertex at (2, 3, 4, 5, 1))

sage: C = CombinatorialPolyhedron([[0,1,2],[0,1,3],[0,2,3],[1,2,3]])
sage: it = C.face_generator()
sage: for face in it: (face.dimension(), face.ambient_Vrepresentation())
(2, (1, 2, 3))
(2, (0, 2, 3))
(2, (0, 1, 3))
(2, (0, 1, 2))
(1, (2, 3))
(1, (1, 3))
(1, (1, 2))
(0, (3,))
(0, (2,))
(0, (1,))
(1, (0, 3))
(1, (0, 2))
(0, (0,))
(1, (0, 1))

See also

ambient_V_indices().

ambient_dimension()#

Return the dimension of the polyhedron.

EXAMPLES:

sage: P = polytopes.cube()
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator()
sage: face = next(it)
sage: face.ambient_dimension()
3

as_combinatorial_polyhedron(quotient=False)#

Return self as combinatorial polyhedron.

If quotient is True, return the quotient of the polyhedron by self. Let G be the face corresponding to self in the dual/polar polytope. The quotient is the dual/polar of G.

Let \([\hat{0}, \hat{1}]\) be the face lattice of the ambient polyhedron and \(F\) be self as element of the face lattice. The face lattice of self as polyhedron corresponds to \([\hat{0}, F]\) and the face lattice of the quotient by self corresponds to \([F, \hat{1}]\).

EXAMPLES:

sage: P = polytopes.cyclic_polytope(7,11)
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator(4)
sage: f = next(it); f
A 4-dimensional face of a 7-dimensional combinatorial polyhedron
sage: F = f.as_combinatorial_polyhedron(); F
A 4-dimensional combinatorial polyhedron with 5 facets
sage: F.f_vector()
(1, 5, 10, 10, 5, 1)
sage: F_alt = polytopes.cyclic_polytope(4,5).combinatorial_polyhedron()
sage: F_alt.vertex_facet_graph().is_isomorphic(F.vertex_facet_graph())      # needs sage.graphs
True

Obtaining the quotient:

sage: Q = f.as_combinatorial_polyhedron(quotient=True); Q
A 2-dimensional combinatorial polyhedron with 6 facets
sage: Q
A 2-dimensional combinatorial polyhedron with 6 facets
sage: Q.f_vector()
(1, 6, 6, 1)

The Vrepresentation of the face as polyhedron is given by the ambient Vrepresentation of the face in that order:

sage: P = polytopes.cube()
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator(2)
sage: f = next(it)
sage: F = f.as_combinatorial_polyhedron()
sage: C.Vrepresentation()
(A vertex at (1, -1, -1),
A vertex at (1, 1, -1),
A vertex at (1, 1, 1),
A vertex at (1, -1, 1),
A vertex at (-1, -1, 1),
A vertex at (-1, -1, -1),
A vertex at (-1, 1, -1),
A vertex at (-1, 1, 1))
sage: f.ambient_Vrepresentation()
(A vertex at (1, -1, -1),
A vertex at (1, -1, 1),
A vertex at (-1, -1, 1),
A vertex at (-1, -1, -1))
sage: F.Vrepresentation()
(0, 1, 2, 3)

To obtain the facets of the face as polyhedron, we compute the meet of each facet with the face. The first representative of each element strictly contained in the face is kept:

sage: C.facets(names=False)
((0, 1, 2, 3),
 (1, 2, 6, 7),
 (2, 3, 4, 7),
 (4, 5, 6, 7),
 (0, 1, 5, 6),
 (0, 3, 4, 5))
sage: F.facets(names=False)
((0, 1), (1, 2), (2, 3), (0, 3))

The Hrepresentation of the quotient by the face is given by the ambient Hrepresentation of the face in that order:

sage: it = C.face_generator(1)
sage: f = next(it)
sage: Q = f.as_combinatorial_polyhedron(quotient=True)
sage: C.Hrepresentation()
(An inequality (-1, 0, 0) x + 1 >= 0,
An inequality (0, -1, 0) x + 1 >= 0,
An inequality (0, 0, -1) x + 1 >= 0,
An inequality (1, 0, 0) x + 1 >= 0,
An inequality (0, 0, 1) x + 1 >= 0,
An inequality (0, 1, 0) x + 1 >= 0)
sage: f.ambient_Hrepresentation()
(An inequality (0, 0, 1) x + 1 >= 0, An inequality (0, 1, 0) x + 1 >= 0)
sage: Q.Hrepresentation()
(0, 1)

To obtain the vertices of the face as polyhedron, we compute the join of each vertex with the face. The first representative of each element strictly containing the face is kept:

sage: [g.ambient_H_indices() for g in C.face_generator(0)]
[(3, 4, 5),
(0, 4, 5),
(2, 3, 5),
(0, 2, 5),
(1, 3, 4),
(0, 1, 4),
(1, 2, 3),
(0, 1, 2)]
sage: [g.ambient_H_indices() for g in Q.face_generator(0)]
[(1,), (0,)]

The method is not implemented for unbounded polyhedra:

sage: P = Polyhedron(rays=[[0,1]])*polytopes.cube()
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator(2)
sage: f = next(it)
sage: f.as_combinatorial_polyhedron()
Traceback (most recent call last):
...
NotImplementedError: only implemented for bounded polyhedra

REFERENCES:

For more information, see Exercise 2.9 of [Zie2007].

Note

This method is tested in _test_combinatorial_face_as_combinatorial_polyhedron().

dim()#

Return the dimension of the face.

EXAMPLES:

sage: # needs sage.combinat
sage: P = polytopes.associahedron(['A', 3])
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator()
sage: face = next(it)
sage: face.dimension()
2

dim is an alias:

sage: face.dim()                                                            # needs sage.combinat
2

dimension()#

Return the dimension of the face.

EXAMPLES:

sage: # needs sage.combinat
sage: P = polytopes.associahedron(['A', 3])
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator()
sage: face = next(it)
sage: face.dimension()
2

dim is an alias:

sage: face.dim()                                                            # needs sage.combinat
2

is_subface(other)#

Return whether self is contained in other.

EXAMPLES:

sage: P = polytopes.cube()
sage: C = P.combinatorial_polyhedron()
sage: it = C.face_generator()
sage: face = next(it)
sage: face.ambient_V_indices()
(0, 3, 4, 5)
sage: face2 = next(it)
sage: face2.ambient_V_indices()
(0, 1, 5, 6)
sage: face.is_subface(face2)
False
sage: face2.is_subface(face)
False
sage: it.only_subfaces()
sage: face3 = next(it)
sage: face3.ambient_V_indices()
(0, 5)
sage: face3.is_subface(face2)
True
sage: face3.is_subface(face)
True

Works for faces of the same combinatorial polyhedron; also from different iterators:

sage: it = C.face_generator(algorithm='dual')
sage: v7 = next(it); v7.ambient_V_indices()
(7,)
sage: v6 = next(it); v6.ambient_V_indices()
(6,)
sage: v5 = next(it); v5.ambient_V_indices()
(5,)
sage: face.ambient_V_indices()
(0, 3, 4, 5)
sage: face.is_subface(v7)
False
sage: v7.is_subface(face)
False
sage: v6.is_subface(face)
False
sage: v5.is_subface(face)
True
sage: face2.ambient_V_indices()
(0, 1, 5, 6)
sage: face2.is_subface(v7)
False
sage: v7.is_subface(face2)
False
sage: v6.is_subface(face2)
True
sage: v5.is_subface(face2)
True

Only implemented for faces of the same combinatorial polyhedron:

sage: P1 = polytopes.cube()
sage: C1 = P1.combinatorial_polyhedron()
sage: it = C1.face_generator()
sage: other_face = next(it)
sage: other_face.ambient_V_indices()
(0, 3, 4, 5)
sage: face.ambient_V_indices()
(0, 3, 4, 5)
sage: C is C1
False
sage: face.is_subface(other_face)
Traceback (most recent call last):
...
NotImplementedError: is_subface only implemented for faces of the same polyhedron

n_ambient_Hrepresentation(add_equations=True)#

Return the length of the CombinatorialFace.ambient_H_indices().

Might be faster than then using len.

INPUT:

add_equations – boolean (default: True); whether or not to count the equations

EXAMPLES:

sage: P = polytopes.cube()
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator()
sage: all(face.n_ambient_Hrepresentation() == len(face.ambient_Hrepresentation()) for face in it)
True

Specifying whether to count the equations or not:

sage: # needs sage.combinat
sage: P = polytopes.permutahedron(5)
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator(2)
sage: f = next(it)
sage: f.n_ambient_Hrepresentation(add_equations=True)
3
sage: f.n_ambient_Hrepresentation(add_equations=False)
2

n_ambient_Vrepresentation()#

Return the length of the CombinatorialFace.ambient_V_indices().

Might be faster than using len.

EXAMPLES:

sage: P = polytopes.cube()
sage: C = CombinatorialPolyhedron(P)
sage: it = C.face_generator()
sage: all(face.n_ambient_Vrepresentation() == len(face.ambient_Vrepresentation()) for face in it)
True