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: 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
>>> 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
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
>>> 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
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
>>> 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
AUTHOR:
- Jonathan Kliem (2019-05) 
- class sage.geometry.polyhedron.combinatorial_polyhedron.combinatorial_face.CombinatorialFace[source]¶
- 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 - >>> 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 - 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 - >>> 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 - Obtain the dimension of a combinatorial face: - sage: face = next(it) sage: face.dimension() 0 - >>> from sage.all import * >>> face = next(it) >>> face.dimension() 0 - The dimension of the polyhedron: - sage: face.ambient_dimension() 5 - >>> from sage.all import * >>> 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 - >>> 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 - 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 - >>> 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 - ambient_H_indices(add_equations=True)[source]¶
- 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) - >>> 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) - 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) - >>> 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) - 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) - >>> 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) - See also 
 - ambient_Hrepresentation()[source]¶
- 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) - >>> 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) - See also 
 - ambient_V_indices()[source]¶
- 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)) - >>> 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)) - See also 
 - ambient_Vrepresentation()[source]¶
- 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)) - >>> 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)) - See also 
 - ambient_dimension()[source]¶
- 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 - >>> from sage.all import * >>> P = polytopes.cube() >>> C = CombinatorialPolyhedron(P) >>> it = C.face_generator() >>> face = next(it) >>> face.ambient_dimension() 3 
 - as_combinatorial_polyhedron(quotient=False)[source]¶
- Return - selfas combinatorial polyhedron.- If - quotientis- True, return the quotient of the polyhedron by- self. Let- Gbe the face corresponding to- selfin the dual/polar polytope. The- quotientis the dual/polar of- G.- Let \([\hat{0}, \hat{1}]\) be the face lattice of the ambient polyhedron and \(F\) be - selfas element of the face lattice. The face lattice of- selfas polyhedron corresponds to \([\hat{0}, F]\) and the face lattice of the quotient by- selfcorresponds 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 - >>> 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 - 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) - >>> 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) - 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) - >>> 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) - 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)) - >>> 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)) - 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) - >>> 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) - 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,)] - >>> 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,)] - 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 - >>> 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 - REFERENCES: - For more information, see Exercise 2.9 of [Zie2007]. - Note - This method is tested in - _test_combinatorial_face_as_combinatorial_polyhedron().
 - dim()[source]¶
- 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 - >>> 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 - dimis an alias:- sage: face.dim() # needs sage.combinat 2 - >>> from sage.all import * >>> face.dim() # needs sage.combinat 2 
 - dimension()[source]¶
- 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 - >>> 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 - dimis an alias:- sage: face.dim() # needs sage.combinat 2 - >>> from sage.all import * >>> face.dim() # needs sage.combinat 2 
 - is_subface(other)[source]¶
- Return whether - selfis 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 - >>> 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 - 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 - >>> 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 - 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 - >>> 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 
 - n_ambient_Hrepresentation(add_equations=True)[source]¶
- 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 - >>> 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 - 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 - >>> 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 
 - n_ambient_Vrepresentation()[source]¶
- 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 - >>> 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