Combinatorial face of a polyhedron

This module provides the combinatorial type of a polyhedral face.

See also

sage.geometry.polyhedron.combinatorial_polyhedron.base, sage.geometry.polyhedron.combinatorial_polyhedron.face_iterator.

EXAMPLES:

Obtain a face from a face iterator:

Sage

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

Python

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

Sage Live

P = polytopes.cube()
C = CombinatorialPolyhedron(P)
it = C.face_generator()
face = next(it); face

Obtain a face from a face lattice index:

Sage

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

Python

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

Sage Live

P = polytopes.simplex(2)
C = CombinatorialPolyhedron(P)
sorted(C.face_lattice()._elements)                                            # needs sage.combinat
face = C.face_by_face_lattice_index(0); face

Obtain further information regarding a face:

Sage

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

Python

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

Sage Live

P = polytopes.octahedron()
C = CombinatorialPolyhedron(P)
it = C.face_generator(2)
face = next(it); face
face.ambient_Vrepresentation()
face.n_ambient_Vrepresentation()
face.ambient_H_indices()
face.dimension()
face.ambient_dimension()

See also

sage.geometry.polyhedron.combinatorial_polyhedron.base.CombinatorialPolyhedron.

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

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

Python

>>> from sage.all import *
>>> P = polytopes.cyclic_polytope(Integer(5),Integer(8))
>>> C = CombinatorialPolyhedron(P)
>>> it = C.face_generator()
>>> next(it)
A 0-dimensional face of a 5-dimensional combinatorial polyhedron

Sage Live

P = polytopes.cyclic_polytope(5,8)
C = CombinatorialPolyhedron(P)
it = C.face_generator()
next(it)

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

Sage

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

Python

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

Sage Live

F = C.face_lattice()                                                      # needs sage.combinat
F._elements[3]                                                            # needs sage.combinat
C.face_by_face_lattice_index(29)

Obtain the dimension of a combinatorial face:

Sage

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

Python

>>> from sage.all import *
>>> face = next(it)
>>> face.dimension()
0

Sage Live

face = next(it)
face.dimension()

The dimension of the polyhedron:

Sage

sage: face.ambient_dimension()
5

Python

>>> from sage.all import *
>>> face.ambient_dimension()
5

Sage Live

face.ambient_dimension()

The Vrepresentation:

Sage

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

Python

>>> from sage.all import *
>>> face.ambient_Vrepresentation()
(A vertex at (6, 36, 216, 1296, 7776),)
>>> face.ambient_V_indices()
(6,)
>>> face.n_ambient_Vrepresentation()
1

Sage Live

face.ambient_Vrepresentation()
face.ambient_V_indices()
face.n_ambient_Vrepresentation()

The Hrepresentation:

Sage

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

Python

>>> from sage.all import *
>>> 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)
>>> face.ambient_H_indices()
(3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 19)
>>> face.n_ambient_Hrepresentation()
11

Sage Live

face.ambient_Hrepresentation()
face.ambient_H_indices()
face.n_ambient_Hrepresentation()

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

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)

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> P = polytopes.permutahedron(Integer(5))
>>> C = CombinatorialPolyhedron(P)
>>> it = C.face_generator(Integer(2))
>>> face = next(it)
>>> face.ambient_H_indices(add_equations=False)
(28, 29)
>>> face2 = next(it)
>>> face2.ambient_H_indices(add_equations=False)
(25, 29)

Sage Live

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

Add the indices of the equation:

Sage

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)

Python

>>> from sage.all import *
>>> face.ambient_H_indices(add_equations=True)                            # needs sage.combinat
(28, 29, 30)
>>> face2.ambient_H_indices(add_equations=True)                           # needs sage.combinat
(25, 29, 30)

Sage Live

face.ambient_H_indices(add_equations=True)                            # needs sage.combinat
face2.ambient_H_indices(add_equations=True)                           # needs sage.combinat

Another example:

Sage

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)

Python

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

Sage Live

P = polytopes.cyclic_polytope(4,6)
C = CombinatorialPolyhedron(P)
it = C.face_generator()
_ = next(it); _ = next(it)
next(it).ambient_H_indices()
next(it).ambient_H_indices()
next(it).ambient_H_indices()
[next(it).dimension() for _ in range(2)]
face = next(it)
face.ambient_H_indices()

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

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)

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> P = polytopes.permutahedron(Integer(5))
>>> C = CombinatorialPolyhedron(P)
>>> it = C.face_generator(Integer(2))
>>> 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)
>>> 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)

>>> P = polytopes.cyclic_polytope(Integer(4),Integer(6))
>>> C = CombinatorialPolyhedron(P)
>>> it = C.face_generator()
>>> 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)
>>> 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)

Sage Live

# needs sage.combinat
P = polytopes.permutahedron(5)
C = CombinatorialPolyhedron(P)
it = C.face_generator(2)
next(it).ambient_Hrepresentation()
next(it).ambient_Hrepresentation()
P = polytopes.cyclic_polytope(4,6)
C = CombinatorialPolyhedron(P)
it = C.face_generator()
next(it).ambient_Hrepresentation()
next(it).ambient_Hrepresentation()

See also

ambient_H_indices().

ambient_V_indices()

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

EXAMPLES:

Sage

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))

Python

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

>>> C = CombinatorialPolyhedron([[Integer(0),Integer(1),Integer(2)],[Integer(0),Integer(1),Integer(3)],[Integer(0),Integer(2),Integer(3)],[Integer(1),Integer(2),Integer(3)]])
>>> it = C.face_generator()
>>> 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))

Sage Live

# needs sage.combinat
P = polytopes.permutahedron(5)
C = CombinatorialPolyhedron(P)
it = C.face_generator(dimension=2)
face = next(it)
next(it).ambient_V_indices()
next(it).ambient_V_indices()
C = CombinatorialPolyhedron([[0,1,2],[0,1,3],[0,2,3],[1,2,3]])
it = C.face_generator()
for face in it: (face.dimension(), face.ambient_V_indices())

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

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))

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> P = polytopes.permutahedron(Integer(5))
>>> C = CombinatorialPolyhedron(P)
>>> it = C.face_generator(dimension=Integer(2))
>>> face = next(it)
>>> 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))
>>> face = next(it)
>>> 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))

>>> C = CombinatorialPolyhedron([[Integer(0),Integer(1),Integer(2)],[Integer(0),Integer(1),Integer(3)],[Integer(0),Integer(2),Integer(3)],[Integer(1),Integer(2),Integer(3)]])
>>> it = C.face_generator()
>>> 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))

Sage Live

# needs sage.combinat
P = polytopes.permutahedron(5)
C = CombinatorialPolyhedron(P)
it = C.face_generator(dimension=2)
face = next(it)
face.ambient_Vrepresentation()
face = next(it)
face.ambient_Vrepresentation()
C = CombinatorialPolyhedron([[0,1,2],[0,1,3],[0,2,3],[1,2,3]])
it = C.face_generator()
for face in it: (face.dimension(), face.ambient_Vrepresentation())

See also

ambient_V_indices().

ambient_dimension()

Return the dimension of the polyhedron.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> P = polytopes.cube()
>>> C = CombinatorialPolyhedron(P)
>>> it = C.face_generator()
>>> face = next(it)
>>> face.ambient_dimension()
3

Sage Live

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

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

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

Python

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

Sage Live

P = polytopes.cyclic_polytope(7,11)
C = CombinatorialPolyhedron(P)
it = C.face_generator(4)
f = next(it); f
F = f.as_combinatorial_polyhedron(); F
F.f_vector()
F_alt = polytopes.cyclic_polytope(4,5).combinatorial_polyhedron()
F_alt.vertex_facet_graph().is_isomorphic(F.vertex_facet_graph())      # needs sage.graphs

Obtaining the quotient:

Sage

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)

Python

>>> from sage.all import *
>>> Q = f.as_combinatorial_polyhedron(quotient=True); Q
A 2-dimensional combinatorial polyhedron with 6 facets
>>> Q
A 2-dimensional combinatorial polyhedron with 6 facets
>>> Q.f_vector()
(1, 6, 6, 1)

Sage Live

Q = f.as_combinatorial_polyhedron(quotient=True); Q
Q
Q.f_vector()

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

Sage

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)

Python

>>> from sage.all import *
>>> P = polytopes.cube()
>>> C = CombinatorialPolyhedron(P)
>>> it = C.face_generator(Integer(2))
>>> f = next(it)
>>> F = f.as_combinatorial_polyhedron()
>>> 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))
>>> 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))
>>> F.Vrepresentation()
(0, 1, 2, 3)

Sage Live

P = polytopes.cube()
C = CombinatorialPolyhedron(P)
it = C.face_generator(2)
f = next(it)
F = f.as_combinatorial_polyhedron()
C.Vrepresentation()
f.ambient_Vrepresentation()
F.Vrepresentation()

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

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))

Python

>>> from sage.all import *
>>> 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))
>>> F.facets(names=False)
((0, 1), (1, 2), (2, 3), (0, 3))

Sage Live

C.facets(names=False)
F.facets(names=False)

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

Sage

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)

Python

>>> from sage.all import *
>>> it = C.face_generator(Integer(1))
>>> f = next(it)
>>> Q = f.as_combinatorial_polyhedron(quotient=True)
>>> 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)
>>> f.ambient_Hrepresentation()
(An inequality (0, 0, 1) x + 1 >= 0, An inequality (0, 1, 0) x + 1 >= 0)
>>> Q.Hrepresentation()
(0, 1)

Sage Live

it = C.face_generator(1)
f = next(it)
Q = f.as_combinatorial_polyhedron(quotient=True)
C.Hrepresentation()
f.ambient_Hrepresentation()
Q.Hrepresentation()

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

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,)]

Python

>>> from sage.all import *
>>> [g.ambient_H_indices() for g in C.face_generator(Integer(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)]
>>> [g.ambient_H_indices() for g in Q.face_generator(Integer(0))]
[(1,), (0,)]

Sage Live

[g.ambient_H_indices() for g in C.face_generator(0)]
[g.ambient_H_indices() for g in Q.face_generator(0)]

The method is not implemented for unbounded polyhedra:

Sage

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

Python

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

Sage Live

P = Polyhedron(rays=[[0,1]])*polytopes.cube()
C = CombinatorialPolyhedron(P)
it = C.face_generator(2)
f = next(it)
f.as_combinatorial_polyhedron()

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

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

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> P = polytopes.associahedron(['A', Integer(3)])
>>> C = CombinatorialPolyhedron(P)
>>> it = C.face_generator()
>>> face = next(it)
>>> face.dimension()
2

Sage Live

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

dim is an alias:

Sage

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

Python

>>> from sage.all import *
>>> face.dim()                                                            # needs sage.combinat
2

Sage Live

face.dim()                                                            # needs sage.combinat

dimension()

Return the dimension of the face.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> P = polytopes.associahedron(['A', Integer(3)])
>>> C = CombinatorialPolyhedron(P)
>>> it = C.face_generator()
>>> face = next(it)
>>> face.dimension()
2

Sage Live

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

dim is an alias:

Sage

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

Python

>>> from sage.all import *
>>> face.dim()                                                            # needs sage.combinat
2

Sage Live

face.dim()                                                            # needs sage.combinat

is_subface(other)

Return whether self is contained in other.

EXAMPLES:

Sage

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

Python

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

Sage Live

P = polytopes.cube()
C = P.combinatorial_polyhedron()
it = C.face_generator()
face = next(it)
face.ambient_V_indices()
face2 = next(it)
face2.ambient_V_indices()
face.is_subface(face2)
face2.is_subface(face)
it.only_subfaces()
face3 = next(it)
face3.ambient_V_indices()
face3.is_subface(face2)
face3.is_subface(face)

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

Sage

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

Python

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

Sage Live

it = C.face_generator(algorithm='dual')
v7 = next(it); v7.ambient_V_indices()
v6 = next(it); v6.ambient_V_indices()
v5 = next(it); v5.ambient_V_indices()
face.ambient_V_indices()
face.is_subface(v7)
v7.is_subface(face)
v6.is_subface(face)
v5.is_subface(face)
face2.ambient_V_indices()
face2.is_subface(v7)
v7.is_subface(face2)
v6.is_subface(face2)
v5.is_subface(face2)

Only implemented for faces of the same combinatorial polyhedron:

Sage

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

Python

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

Sage Live

P1 = polytopes.cube()
C1 = P1.combinatorial_polyhedron()
it = C1.face_generator()
other_face = next(it)
other_face.ambient_V_indices()
face.ambient_V_indices()
C is C1
face.is_subface(other_face)

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

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

Python

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

Sage Live

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

Specifying whether to count the equations or not:

Sage

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

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> P = polytopes.permutahedron(Integer(5))
>>> C = CombinatorialPolyhedron(P)
>>> it = C.face_generator(Integer(2))
>>> f = next(it)
>>> f.n_ambient_Hrepresentation(add_equations=True)
3
>>> f.n_ambient_Hrepresentation(add_equations=False)
2

Sage Live

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

n_ambient_Vrepresentation()

Return the length of the CombinatorialFace.ambient_V_indices().

Might be faster than using len.

EXAMPLES:

Sage

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

Python

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

Sage Live

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

Make live