Base class for polyhedra: Graph-theoretic methods

Define methods relying on sage.graphs.

class sage.geometry.polyhedron.base4.Polyhedron_base4(parent, Vrep, Hrep, Vrep_minimal=None, Hrep_minimal=None, pref_rep=None, mutable=False, **kwds)

Bases: Polyhedron_base3

Methods relying on sage.graphs.

See sage.geometry.polyhedron.base.Polyhedron_base.

combinatorial_automorphism_group(vertex_graph_only=False)

Computes the combinatorial automorphism group.

If vertex_graph_only is True, the automorphism group of the vertex-edge graph of the polyhedron is returned. Otherwise the automorphism group of the vertex-facet graph, which is isomorphic to the automorphism group of the face lattice is returned.

INPUT:

• vertex_graph_only – boolean (default: False); whether to return the automorphism group of the vertex edges graph or of the lattice

OUTPUT:

A PermutationGroup that is isomorphic to the combinatorial automorphism group is returned.

• if vertex_graph_only is True: The automorphism group of the vertex-edge graph of the polyhedron

• if vertex_graph_only is False (default): The automorphism group of the vertex-facet graph of the polyhedron, see vertex_facet_graph(). This group is isomorphic to the automorphism group of the face lattice of the polyhedron.

NOTE:

Depending on vertex_graph_only, this method returns groups that are not necessarily isomorphic, see the examples below.

See also

is_combinatorially_isomorphic(), graph(), vertex_facet_graph().

EXAMPLES:

sage: quadrangle = Polyhedron(vertices=[(0,0),(1,0),(0,1),(2,3)])
sage: quadrangle.combinatorial_automorphism_group().is_isomorphic(
....:     groups.permutation.Dihedral(4))
True
sage: quadrangle.restricted_automorphism_group()
Permutation Group with generators [()]

Permutations of the vertex graph only exchange vertices with vertices:

sage: P = Polyhedron(vertices=[(1,0), (1,1)], rays=[(1,0)])
sage: P.combinatorial_automorphism_group(vertex_graph_only=True)
Permutation Group with generators [(A vertex at (1,0),A vertex at (1,1))]

This shows an example of two polytopes whose vertex-edge graphs are isomorphic, but their face lattices are not isomorphic:

sage: Q = Polyhedron([[-123984206864/2768850730773, -101701330976/922950243591, -64154618668/2768850730773, -2748446474675/2768850730773],
....:                 [-11083969050/98314591817, -4717557075/98314591817, -32618537490/98314591817, -91960210208/98314591817],
....:                 [-9690950/554883199, -73651220/554883199, 1823050/554883199, -549885101/554883199],
....:                 [-5174928/72012097, 5436288/72012097, -37977984/72012097, 60721345/72012097],
....:                 [-19184/902877, 26136/300959, -21472/902877, 899005/902877],
....:                 [53511524/1167061933, 88410344/1167061933, 621795064/1167061933, 982203941/1167061933],
....:                 [4674489456/83665171433, -4026061312/83665171433, 28596876672/83665171433, -78383796375/83665171433],
....:                 [857794884940/98972360190089, -10910202223200/98972360190089, 2974263671400/98972360190089, -98320463346111/98972360190089]])
sage: C = polytopes.cyclic_polytope(4,8)
sage: C.is_combinatorially_isomorphic(Q)
False
sage: C.combinatorial_automorphism_group(vertex_graph_only=True).is_isomorphic(
....:     Q.combinatorial_automorphism_group(vertex_graph_only=True))
True
sage: C.combinatorial_automorphism_group(vertex_graph_only=False).is_isomorphic(
....:     Q.combinatorial_automorphism_group(vertex_graph_only=False))
False

The automorphism group of the face lattice is isomorphic to the combinatorial automorphism group:

sage: CG = C.hasse_diagram().automorphism_group()
sage: C.combinatorial_automorphism_group().is_isomorphic(CG)
True
sage: QG = Q.hasse_diagram().automorphism_group()
sage: Q.combinatorial_automorphism_group().is_isomorphic(QG)
True

face_lattice()

Return the face-lattice poset.

OUTPUT:

A FinitePoset. Elements are given as PolyhedronFace.

In the case of a full-dimensional polytope, the faces are pairs (vertices, inequalities) of the spanning vertices and corresponding saturated inequalities. In general, a face is defined by a pair (V-rep. objects, H-rep. objects). The V-representation objects span the face, and the corresponding H-representation objects are those inequalities and equations that are saturated on the face.

The bottom-most element of the face lattice is the “empty face”. It contains no V-representation object. All H-representation objects are incident.

The top-most element is the “full face”. It is spanned by all V-representation objects. The incident H-representation objects are all equations and no inequalities.

In the case of a full-dimensional polytope, the “empty face” and the “full face” are the empty set (no vertices, all inequalities) and the full polytope (all vertices, no inequalities), respectively.

ALGORITHM:

See sage.geometry.polyhedron.combinatorial_polyhedron.face_iterator.

Note

The face lattice is not cached, as long as this creates a memory leak, see github issue #28982.

EXAMPLES:

sage: square = polytopes.hypercube(2)
sage: fl = square.face_lattice();fl
Finite lattice containing 10 elements
sage: list(f.ambient_V_indices() for f in fl)
[(), (0,), (1,), (0, 1), (2,), (1, 2), (3,), (0, 3), (2, 3), (0, 1, 2, 3)]
sage: poset_element = fl[5]
sage: a_face = poset_element
sage: a_face
A 1-dimensional face of a Polyhedron in ZZ^2 defined as the convex hull of 2 vertices
sage: a_face.ambient_V_indices()
(1, 2)
sage: set(a_face.ambient_Vrepresentation()) ==             ....: set([square.Vrepresentation(1), square.Vrepresentation(2)])
True
sage: a_face.ambient_Vrepresentation()
(A vertex at (1, 1), A vertex at (-1, 1))
sage: a_face.ambient_Hrepresentation()
(An inequality (0, -1) x + 1 >= 0,)

A more complicated example:

sage: c5_10 = Polyhedron(vertices = [[i,i^2,i^3,i^4,i^5] for i in range(1,11)])
sage: c5_10_fl = c5_10.face_lattice()
sage: [len(x) for x in c5_10_fl.level_sets()]
[1, 10, 45, 100, 105, 42, 1]

Note that if the polyhedron contains lines then there is a dimension gap between the empty face and the first non-empty face in the face lattice:

sage: line = Polyhedron(vertices=[(0,)], lines=[(1,)])
sage: [ fl.dim() for fl in line.face_lattice() ]
[-1, 1]

flag_f_vector(*args)

Return the flag f-vector.

For each \(-1 < i_0 < \dots < i_n < d\) the flag f-vector counts the number of flags \(F_0 \subset \dots \subset F_n\) with \(F_j\) of dimension \(i_j\) for each \(0 \leq j \leq n\), where \(d\) is the dimension of the polyhedron.

INPUT:

• args – integers (optional); specify an entry of the flag-f-vector; must be an increasing sequence of integers

OUTPUT:

• a dictionary, if no arguments were given

• an Integer, if arguments were given

EXAMPLES:

Obtain the entire flag-f-vector:

sage: P = polytopes.twenty_four_cell()
sage: P.flag_f_vector()
    {(-1,): 1,
     (0,): 24,
     (0, 1): 192,
     (0, 1, 2): 576,
     (0, 1, 2, 3): 1152,
     (0, 1, 3): 576,
     (0, 2): 288,
     (0, 2, 3): 576,
     (0, 3): 144,
     (1,): 96,
     (1, 2): 288,
     (1, 2, 3): 576,
     (1, 3): 288,
     (2,): 96,
     (2, 3): 192,
     (3,): 24,
     (4,): 1}

Specify an entry:

sage: P.flag_f_vector(0,3)
144
sage: P.flag_f_vector(2)
96

Leading -1 and trailing entry of dimension are allowed:

sage: P.flag_f_vector(-1,0,3)
144
sage: P.flag_f_vector(-1,0,3,4)
144

One can get the number of trivial faces:

sage: P.flag_f_vector(-1)
1
sage: P.flag_f_vector(4)
1

Polyhedra with lines, have 0 entries accordingly:

sage: P = (Polyhedron(lines=[[1]]) * polytopes.cross_polytope(3))
sage: P.flag_f_vector()
{(-1,): 1,
 (0, 1): 0,
 (0, 1, 2): 0,
 (0, 1, 3): 0,
 (0, 2): 0,
 (0, 2, 3): 0,
 (0, 3): 0,
 (0,): 0,
 (1, 2): 24,
 (1, 2, 3): 48,
 (1, 3): 24,
 (1,): 6,
 (2, 3): 24,
 (2,): 12,
 (3,): 8,
 4: 1}

If the arguments are not stricly increasing or out of range, a key error is raised:

sage: P.flag_f_vector(-1,0,3,6)
Traceback (most recent call last):
...
KeyError: (0, 3, 6)
sage: P.flag_f_vector(-1,3,0)
Traceback (most recent call last):
...
KeyError: (3, 0)

graph(**kwds)

Return a graph in which the vertices correspond to vertices of the polyhedron, and edges to edges.

INPUT:

• names – boolean (default: True); if False, then the nodes of the graph are labeld by the indices of the Vrepresentation

• algorithm – string (optional); specify whether the face generator starts with facets or vertices:

  • 'primal' – start with the facets

  • 'dual' – start with the vertices

  • None – choose automatically

Note

The graph of a polyhedron with lines has no vertices, as the polyhedron has no vertices (\(0\)-faces).

The method vertices() returns the defining points in this case.

EXAMPLES:

sage: g3 = polytopes.hypercube(3).vertex_graph(); g3
Graph on 8 vertices
sage: g3.automorphism_group().cardinality()
48
sage: s4 = polytopes.simplex(4).vertex_graph(); s4
Graph on 5 vertices
sage: s4.is_eulerian()
True

The graph of an unbounded polyhedron is the graph of the bounded complex:

sage: open_triangle = Polyhedron(vertices=[[1,0], [0,1]],
....:                            rays    =[[1,1]])
sage: open_triangle.vertex_graph()
Graph on 2 vertices

The graph of a polyhedron with lines has no vertices:

sage: line = Polyhedron(lines=[[0,1]])
sage: line.vertex_graph()
Graph on 0 vertices

hasse_diagram()

Return the Hasse diagram of the face lattice of self.

This is the Hasse diagram of the poset of the faces of self.

OUTPUT: a directed graph

EXAMPLES:

sage: # needs sage.rings.number_field
sage: P = polytopes.regular_polygon(4).pyramid()
sage: D = P.hasse_diagram(); D
Digraph on 20 vertices
sage: D.degree_polynomial()
x^5 + x^4*y + x*y^4 + y^5 + 4*x^3*y + 8*x^2*y^2 + 4*x*y^3

Faces of a mutable polyhedron are not hashable. Hence those are not suitable as vertices of the hasse diagram. Use the combinatorial polyhedron instead:

sage: # needs sage.rings.number_field
sage: P = polytopes.regular_polygon(4).pyramid()
sage: parent = P.parent()
sage: parent = parent.change_ring(QQ, backend='ppl')
sage: Q = parent._element_constructor_(P, mutable=True)
sage: Q.hasse_diagram()
Traceback (most recent call last):
...
TypeError: mutable polyhedra are unhashable
sage: C = Q.combinatorial_polyhedron()
sage: D = C.hasse_diagram()
sage: set(D.vertices(sort=False)) == set(range(20))
True
sage: def index_to_combinatorial_face(n):
....:     return C.face_by_face_lattice_index(n)
sage: D.relabel(index_to_combinatorial_face, inplace=True)
sage: D.vertices(sort=True)
[A -1-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 0-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 0-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 0-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 0-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 0-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 1-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 1-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 1-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 1-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 1-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 1-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 1-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 1-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 2-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 2-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 2-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 2-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 2-dimensional face of a 3-dimensional combinatorial polyhedron,
 A 3-dimensional face of a 3-dimensional combinatorial polyhedron]
sage: D.degree_polynomial()
x^5 + x^4*y + x*y^4 + y^5 + 4*x^3*y + 8*x^2*y^2 + 4*x*y^3

is_combinatorially_isomorphic(other, algorithm='bipartite_graph')

Return whether the polyhedron is combinatorially isomorphic to another polyhedron.

We only consider bounded polyhedra. By definition, they are combinatorially isomorphic if their face lattices are isomorphic.

INPUT:

• other – a polyhedron object

• algorithm (default = 'bipartite_graph') – the algorithm to use. The other possible value is 'face_lattice'.

OUTPUT:

• True if the two polyhedra are combinatorially isomorphic

• False otherwise

See also

combinatorial_automorphism_group(), vertex_facet_graph().

REFERENCES:

For the equivalence of the two algorithms see [KK1995], p. 877-878

EXAMPLES:

The square is combinatorially isomorphic to the 2-dimensional cube:

sage: polytopes.hypercube(2).is_combinatorially_isomorphic(polytopes.regular_polygon(4))
True

All the faces of the 3-dimensional permutahedron are either combinatorially isomorphic to a square or a hexagon:

sage: H = polytopes.regular_polygon(6)                                      # needs sage.rings.number_field
sage: S = polytopes.hypercube(2)
sage: P = polytopes.permutahedron(4)
sage: all(F.as_polyhedron().is_combinatorially_isomorphic(S)                # needs sage.rings.number_field
....:       or F.as_polyhedron().is_combinatorially_isomorphic(H)
....:     for F in P.faces(2))
True

Checking that a regular simplex intersected with its reflection through the origin is combinatorially isomorphic to the intersection of a cube with a hyperplane perpendicular to its long diagonal:

sage: def simplex_intersection(k):
....:   S1 = Polyhedron([vector(v)-vector(polytopes.simplex(k).center()) for v in polytopes.simplex(k).vertices_list()])
....:   S2 = Polyhedron([-vector(v) for v in S1.vertices_list()])
....:   return S1.intersection(S2)
sage: def cube_intersection(k):
....:    C = polytopes.hypercube(k+1)
....:    H = Polyhedron(eqns=[[0]+[1 for i in range(k+1)]])
....:    return C.intersection(H)
sage: [simplex_intersection(k).is_combinatorially_isomorphic(cube_intersection(k)) for k in range(2,5)]
[True, True, True]
sage: simplex_intersection(2).is_combinatorially_isomorphic(polytopes.regular_polygon(6))                   # needs sage.rings.number_field
True
sage: simplex_intersection(3).is_combinatorially_isomorphic(polytopes.octahedron())
True

Two polytopes with the same \(f\)-vector, but different combinatorial types:

sage: P = Polyhedron([[-605520/1525633, -605520/1525633, -1261500/1525633, -52200/1525633, 11833/1525633],\
 [-720/1769, -600/1769, 1500/1769, 0, -31/1769], [-216/749, 240/749, -240/749, -432/749, 461/749], \
 [-50/181, 50/181, 60/181, -100/181, -119/181], [-32/51, -16/51, -4/51, 12/17, 1/17],\
 [1, 0, 0, 0, 0], [16/129, 128/129, 0, 0, 1/129], [64/267, -128/267, 24/89, -128/267, 57/89],\
 [1200/3953, -1200/3953, -1440/3953, -360/3953, -3247/3953], [1512/5597, 1512/5597, 588/5597, 4704/5597, 2069/5597]])
sage: C = polytopes.cyclic_polytope(5,10)
sage: C.f_vector() == P.f_vector(); C.f_vector()
True
(1, 10, 45, 100, 105, 42, 1)
sage: C.is_combinatorially_isomorphic(P)
False

sage: S = polytopes.simplex(3)
sage: S = S.face_truncation(S.faces(0)[3])
sage: S = S.face_truncation(S.faces(0)[4])
sage: S = S.face_truncation(S.faces(0)[5])
sage: T = polytopes.simplex(3)
sage: T = T.face_truncation(T.faces(0)[3])
sage: T = T.face_truncation(T.faces(0)[4])
sage: T = T.face_truncation(T.faces(0)[4])
sage: T.is_combinatorially_isomorphic(S)
False
sage: T.f_vector(), S.f_vector()
((1, 10, 15, 7, 1), (1, 10, 15, 7, 1))

sage: C = polytopes.hypercube(5)
sage: C.is_combinatorially_isomorphic(C)
True
sage: C.is_combinatorially_isomorphic(C, algorithm='magic')
Traceback (most recent call last):
...
AssertionError: `algorithm` must be 'bipartite graph' or 'face_lattice'

sage: G = Graph()
sage: C.is_combinatorially_isomorphic(G)
Traceback (most recent call last):
...
AssertionError: input `other` must be a polyhedron

sage: H = Polyhedron(eqns=[[0,1,1,1,1]]); H
A 3-dimensional polyhedron in QQ^4 defined as the convex hull of 1 vertex and 3 lines
sage: C.is_combinatorially_isomorphic(H)
Traceback (most recent call last):
...
AssertionError: polyhedron `other` must be bounded

is_self_dual()

Return whether the polytope is self-dual.

A polytope is self-dual if its face lattice is isomorphic to the face lattice of its dual polytope.

EXAMPLES:

sage: polytopes.simplex().is_self_dual()
True
sage: polytopes.twenty_four_cell().is_self_dual()
True
sage: polytopes.cube().is_self_dual()
False
sage: polytopes.hypersimplex(5,2).is_self_dual()                            # needs sage.combinat
False
sage: P = Polyhedron(vertices=[[1/2, 1/3]], rays=[[1, 1]]).is_self_dual()
Traceback (most recent call last):
...
ValueError: polyhedron has to be compact

restricted_automorphism_group(output='abstract')

Return the restricted automorphism group.

First, let the linear automorphism group be the subgroup of the affine group \(AGL(d,\RR) = GL(d,\RR) \ltimes \RR^d\) preserving the \(d\)-dimensional polyhedron. The affine 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 the generators of the same type. That is, vertices can only be permuted with vertices, ray generators with ray generators, and line generators with line generators.

For example, take the first quadrant

\[Q = \Big\{ (x,y) \Big| x\geq 0,\; y\geq0 \Big\} \subset \QQ^2\]

Then the linear automorphism group is

\[\begin{split}\mathrm{Aut}(Q) = \left\{ \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} ,~ \begin{pmatrix} 0 & c \\ d & 0 \end{pmatrix} :~ a, b, c, d \in \QQ_{>0} \right\} \subset GL(2,\QQ) \subset E(d)\end{split}\]

Note that there are no translations that map the quadrant \(Q\) to itself, so the linear automorphism group is contained in the general linear group (the subgroup of transformations preserving the origin). The restricted automorphism group is

\[\begin{split}\mathrm{Aut}(Q) = \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} ,~ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \right\} \simeq \ZZ_2\end{split}\]

INPUT:

• output – how the group should be represented:

  • "abstract" (default) – return an abstract permutation group without further meaning.

  • "permutation" – return a permutation group on the indices of the polyhedron generators. For example, the permutation (0,1) would correspond to swapping self.Vrepresentation(0) and self.Vrepresentation(1).

  • "matrix" – return a matrix group representing affine transformations. When acting on affine vectors, you should append a \(1\) to every vector. If the polyhedron is not full dimensional, the returned matrices act as the identity on the orthogonal complement of the affine space spanned by the polyhedron.

  • "matrixlist" – like matrix, but return the list of elements of the matrix group. Useful for fields without a good implementation of matrix groups or to avoid the overhead of creating the group.

OUTPUT:

• For output="abstract" and output="permutation": a PermutationGroup.

• For output="matrix": a MatrixGroup().

• For output="matrixlist": a list of matrices.

REFERENCES:

• [BSS2009]

EXAMPLES:

A cross-polytope example:

sage: P = polytopes.cross_polytope(3)
sage: P.restricted_automorphism_group() == PermutationGroup([[(3,4)], [(2,3),(4,5)],[(2,5)],[(1,2),(5,6)],[(1,6)]])
True
sage: P.restricted_automorphism_group(output="permutation") == PermutationGroup([[(2,3)],[(1,2),(3,4)],[(1,4)],[(0,1),(4,5)],[(0,5)]])
True
sage: mgens = [[[1,0,0,0],[0,1,0,0],[0,0,-1,0],[0,0,0,1]], [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]], [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]]

We test groups for equality in a fool-proof way; they can have different generators, etc:

sage: poly_g = P.restricted_automorphism_group(output="matrix")
sage: matrix_g = MatrixGroup([matrix(QQ,t) for t in mgens])
sage: all(t.matrix() in poly_g for t in matrix_g.gens())
True
sage: all(t.matrix() in matrix_g for t in poly_g.gens())
True

24-cell example:

sage: P24 = polytopes.twenty_four_cell()
sage: AutP24 = P24.restricted_automorphism_group()
sage: PermutationGroup([
....:     '(1,20,2,24,5,23)(3,18,10,19,4,14)(6,21,11,22,7,15)(8,12,16,17,13,9)',
....:     '(1,21,8,24,4,17)(2,11,6,15,9,13)(3,20)(5,22)(10,16,12,23,14,19)'
....: ]).is_isomorphic(AutP24)
True
sage: AutP24.order()
1152

Here is the quadrant example mentioned in the beginning:

sage: P = Polyhedron(rays=[(1,0),(0,1)])
sage: P.Vrepresentation()
(A vertex at (0, 0), A ray in the direction (0, 1), A ray in the direction (1, 0))
sage: P.restricted_automorphism_group(output="permutation")
Permutation Group with generators [(1,2)]

Also, the polyhedron need not be full-dimensional:

sage: P = Polyhedron(vertices=[(1,2,3,4,5),(7,8,9,10,11)])
sage: P.restricted_automorphism_group()
Permutation Group with generators [(1,2)]
sage: G = P.restricted_automorphism_group(output="matrixlist")
sage: G
(
[1 0 0 0 0 0]  [ -87/55  -82/55    -2/5   38/55   98/55   12/11]
[0 1 0 0 0 0]  [-142/55  -27/55    -2/5   38/55   98/55   12/11]
[0 0 1 0 0 0]  [-142/55  -82/55     3/5   38/55   98/55   12/11]
[0 0 0 1 0 0]  [-142/55  -82/55    -2/5   93/55   98/55   12/11]
[0 0 0 0 1 0]  [-142/55  -82/55    -2/5   38/55  153/55   12/11]
[0 0 0 0 0 1], [      0       0       0       0       0       1]
)
sage: g = AffineGroup(5, QQ)(G[1])
sage: g
      [ -87/55  -82/55    -2/5   38/55   98/55]     [12/11]
      [-142/55  -27/55    -2/5   38/55   98/55]     [12/11]
x |-> [-142/55  -82/55     3/5   38/55   98/55] x + [12/11]
      [-142/55  -82/55    -2/5   93/55   98/55]     [12/11]
      [-142/55  -82/55    -2/5   38/55  153/55]     [12/11]
sage: g^2
      [1 0 0 0 0]     [0]
      [0 1 0 0 0]     [0]
x |-> [0 0 1 0 0] x + [0]
      [0 0 0 1 0]     [0]
      [0 0 0 0 1]     [0]
sage: g(list(P.vertices()[0]))
(7, 8, 9, 10, 11)
sage: g(list(P.vertices()[1]))
(1, 2, 3, 4, 5)

Affine transformations do not change the restricted automorphism group. For example, any non-degenerate triangle has the dihedral group with 6 elements, \(D_6\), as its automorphism group:

sage: initial_points = [vector([1,0]), vector([0,1]), vector([-2,-1])]
sage: points = initial_points
sage: Polyhedron(vertices=points).restricted_automorphism_group()
Permutation Group with generators [(2,3), (1,2)]
sage: points = [pt - initial_points[0] for pt in initial_points]
sage: Polyhedron(vertices=points).restricted_automorphism_group()
Permutation Group with generators [(2,3), (1,2)]
sage: points = [pt - initial_points[1] for pt in initial_points]
sage: Polyhedron(vertices=points).restricted_automorphism_group()
Permutation Group with generators [(2,3), (1,2)]
sage: points = [pt - 2*initial_points[1] for pt in initial_points]
sage: Polyhedron(vertices=points).restricted_automorphism_group()
Permutation Group with generators [(2,3), (1,2)]

The output="matrixlist" can be used over fields without a complete implementation of matrix groups:

sage: P = polytopes.dodecahedron(); P
A 3-dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?)^3 defined as the convex hull of 20 vertices
sage: G = P.restricted_automorphism_group(output="matrixlist")
sage: len(G)
120

Floating-point computations are supported with a simple fuzzy zero implementation:

sage: P = Polyhedron(vertices=[(1/3,0,0,1),(0,1/4,0,1),(0,0,1/5,1)], base_ring=RDF)
sage: P.restricted_automorphism_group()
Permutation Group with generators [(2,3), (1,2)]
sage: len(P.restricted_automorphism_group(output="matrixlist"))
6

vertex_digraph(f, increasing=True)

Return the directed graph of the polyhedron according to a linear form.

The underlying undirected graph is the graph of vertices and edges.

INPUT:

• f – a linear form. The linear form can be provided as:

  • a vector space morphism with one-dimensional codomain, (see sage.modules.vector_space_morphism.linear_transformation() and sage.modules.vector_space_morphism.VectorSpaceMorphism)

  • a vector ; in this case the linear form is obtained by duality using the dot product: f(v) = v.dot_product(f).

• increasing – boolean (default True) whether to orient edges in the increasing or decreasing direction.

By default, an edge is oriented from \(v\) to \(w\) if \(f(v) \leq f(w)\).

If \(f(v)=f(w)\), then two opposite edges are created.

EXAMPLES:

sage: penta = Polyhedron([[0,0],[1,0],[0,1],[1,2],[3,2]])
sage: G = penta.vertex_digraph(vector([1,1])); G
Digraph on 5 vertices
sage: G.sinks()
[A vertex at (3, 2)]

sage: A = matrix(ZZ, [[1], [-1]])
sage: f = linear_transformation(A)
sage: G = penta.vertex_digraph(f) ; G
Digraph on 5 vertices
sage: G.is_directed_acyclic()
False

See also

vertex_graph()

vertex_facet_graph(labels=True)

Return the vertex-facet graph.

This function constructs a directed bipartite graph. The nodes of the graph correspond to the vertices of the polyhedron and the facets of the polyhedron. There is a directed edge from a vertex to a face if and only if the vertex is incident to the face.

INPUT:

• labels – boolean (default: True); decide how the nodes of the graph are labelled. Either with the original vertices/facets of the Polyhedron or with integers.

OUTPUT:

• a bipartite DiGraph. If labels is True, then the nodes of the graph will actually be the vertices and facets of self, otherwise they will be integers.

See also

combinatorial_automorphism_group(), is_combinatorially_isomorphic().

EXAMPLES:

sage: P = polytopes.cube()
sage: G = P.vertex_facet_graph(); G
Digraph on 14 vertices
sage: G.vertices(sort=True, key=lambda v: str(v))
[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),
 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 (0, 0, 1) x + 1 >= 0,
 An inequality (0, 1, 0) x + 1 >= 0,
 An inequality (1, 0, 0) x + 1 >= 0]
sage: G.automorphism_group().is_isomorphic(P.hasse_diagram().automorphism_group())
True
sage: O = polytopes.octahedron(); O
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices
sage: O.vertex_facet_graph()
Digraph on 14 vertices
sage: H = O.vertex_facet_graph()
sage: G.is_isomorphic(H)
False
sage: G2 = copy(G)
sage: G2.reverse_edges(G2.edges(sort=True))
sage: G2.is_isomorphic(H)
True

vertex_graph(**kwds)

Return a graph in which the vertices correspond to vertices of the polyhedron, and edges to edges.

INPUT:

• names – boolean (default: True); if False, then the nodes of the graph are labeld by the indices of the Vrepresentation

• algorithm – string (optional); specify whether the face generator starts with facets or vertices:

  • 'primal' – start with the facets

  • 'dual' – start with the vertices

  • None – choose automatically

Note

The graph of a polyhedron with lines has no vertices, as the polyhedron has no vertices (\(0\)-faces).

The method vertices() returns the defining points in this case.

EXAMPLES:

sage: g3 = polytopes.hypercube(3).vertex_graph(); g3
Graph on 8 vertices
sage: g3.automorphism_group().cardinality()
48
sage: s4 = polytopes.simplex(4).vertex_graph(); s4
Graph on 5 vertices
sage: s4.is_eulerian()
True

The graph of an unbounded polyhedron is the graph of the bounded complex:

sage: open_triangle = Polyhedron(vertices=[[1,0], [0,1]],
....:                            rays    =[[1,1]])
sage: open_triangle.vertex_graph()
Graph on 2 vertices

The graph of a polyhedron with lines has no vertices:

sage: line = Polyhedron(lines=[[0,1]])
sage: line.vertex_graph()
Graph on 0 vertices