A triangulation¶

In Sage, the PointConfiguration and Triangulation satisfy a parent/element relationship. In particular, each triangulation refers back to its point configuration. If you want to triangulate a point configuration, you should construct a point configuration first and then use one of its methods to triangulate it according to your requirements. You should never have to construct a Triangulation object directly.

EXAMPLES:

First, we select the internal implementation for enumerating triangulations:

Sage

sage: PointConfiguration.set_engine('internal')   # to make doctests independent of TOPCOM

Python

>>> from sage.all import *
>>> PointConfiguration.set_engine('internal')   # to make doctests independent of TOPCOM

Sage Live

PointConfiguration.set_engine('internal')   # to make doctests independent of TOPCOM

Here is a simple example of how to triangulate a point configuration:

Sage

sage: p = [[0,-1,-1],[0,0,1],[0,1,0], [1,-1,-1],[1,0,1],[1,1,0]]
sage: points = PointConfiguration(p)
sage: triang = points.triangulate();  triang
(<0,1,2,5>, <0,1,3,5>, <1,3,4,5>)
sage: triang.plot(axes=False)                                                       # needs sage.plot
Graphics3d Object

Python

>>> from sage.all import *
>>> p = [[Integer(0),-Integer(1),-Integer(1)],[Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(0)], [Integer(1),-Integer(1),-Integer(1)],[Integer(1),Integer(0),Integer(1)],[Integer(1),Integer(1),Integer(0)]]
>>> points = PointConfiguration(p)
>>> triang = points.triangulate();  triang
(<0,1,2,5>, <0,1,3,5>, <1,3,4,5>)
>>> triang.plot(axes=False)                                                       # needs sage.plot
Graphics3d Object

Sage Live

p = [[0,-1,-1],[0,0,1],[0,1,0], [1,-1,-1],[1,0,1],[1,1,0]]
points = PointConfiguration(p)
triang = points.triangulate();  triang
triang.plot(axes=False)                                                       # needs sage.plot

See sage.geometry.triangulation.point_configuration for more details.

class sage.geometry.triangulation.element.Triangulation(triangulation, parent, check=True)[source]¶

Bases: Element

A triangulation of a PointConfiguration.

Warning

You should never create Triangulation objects manually. See triangulate() and triangulations() to triangulate point configurations.

adjacency_graph()[source]¶

Return a graph showing which simplices are adjacent in the triangulation.

OUTPUT:

A graph consisting of vertices referring to the simplices in the triangulation, and edges showing which simplices are adjacent to each other.

See also

To obtain the triangulation’s 1-skeleton, use SimplicialComplex.graph() through MyTriangulation.simplicial_complex().graph().

AUTHORS:

Stephen Farley (2013-08-10): initial version

EXAMPLES:

Sage

sage: p = PointConfiguration([[1,0,0], [0,1,0], [0,0,1], [-1,0,1],
....:                         [1,0,-1], [-1,0,0], [0,-1,0], [0,0,-1]])
sage: t = p.triangulate()
sage: t.adjacency_graph()                                                   # needs sage.graphs
Graph on 8 vertices

Python

>>> from sage.all import *
>>> p = PointConfiguration([[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)], [-Integer(1),Integer(0),Integer(1)],
...                         [Integer(1),Integer(0),-Integer(1)], [-Integer(1),Integer(0),Integer(0)], [Integer(0),-Integer(1),Integer(0)], [Integer(0),Integer(0),-Integer(1)]])
>>> t = p.triangulate()
>>> t.adjacency_graph()                                                   # needs sage.graphs
Graph on 8 vertices

Sage Live

p = PointConfiguration([[1,0,0], [0,1,0], [0,0,1], [-1,0,1],
                        [1,0,-1], [-1,0,0], [0,-1,0], [0,0,-1]])
t = p.triangulate()
t.adjacency_graph()                                                   # needs sage.graphs

boundary()[source]¶

Return the boundary of the triangulation.

OUTPUT:

The outward-facing boundary simplices (of dimension \(d-1\)) of the \(d\)-dimensional triangulation as a set. Each boundary is returned by a tuple of point indices.

EXAMPLES:

Sage

sage: triangulation = polytopes.cube().triangulate(engine='internal')
sage: triangulation
(<0,1,2,7>, <0,1,5,7>, <0,2,3,7>, <0,3,4,7>, <0,4,5,7>, <1,5,6,7>)
sage: triangulation.boundary()
frozenset({(0, 1, 2),
           (0, 1, 5),
           (0, 2, 3),
           (0, 3, 4),
           (0, 4, 5),
           (1, 2, 7),
           (1, 5, 6),
           (1, 6, 7),
           (2, 3, 7),
           (3, 4, 7),
           (4, 5, 7),
           (5, 6, 7)})
sage: triangulation.interior_facets()
frozenset({(0, 1, 7), (0, 2, 7), (0, 3, 7), (0, 4, 7), (0, 5, 7), (1, 5, 7)})

Python

>>> from sage.all import *
>>> triangulation = polytopes.cube().triangulate(engine='internal')
>>> triangulation
(<0,1,2,7>, <0,1,5,7>, <0,2,3,7>, <0,3,4,7>, <0,4,5,7>, <1,5,6,7>)
>>> triangulation.boundary()
frozenset({(0, 1, 2),
           (0, 1, 5),
           (0, 2, 3),
           (0, 3, 4),
           (0, 4, 5),
           (1, 2, 7),
           (1, 5, 6),
           (1, 6, 7),
           (2, 3, 7),
           (3, 4, 7),
           (4, 5, 7),
           (5, 6, 7)})
>>> triangulation.interior_facets()
frozenset({(0, 1, 7), (0, 2, 7), (0, 3, 7), (0, 4, 7), (0, 5, 7), (1, 5, 7)})

Sage Live

triangulation = polytopes.cube().triangulate(engine='internal')
triangulation
triangulation.boundary()
triangulation.interior_facets()

boundary_polyhedral_complex(**kwds)[source]¶

Return the boundary of self as a PolyhedralComplex.

OUTPUT:

A PolyhedralComplex whose maximal cells are the simplices of the boundary of self.

EXAMPLES:

Sage

sage: P = polytopes.cube()
sage: pc = PointConfiguration(P.vertices())
sage: T = pc.placing_triangulation(); T
(<0,1,2,7>, <0,1,5,7>, <0,2,3,7>, <0,3,4,7>, <0,4,5,7>, <1,5,6,7>)
sage: bd_C = T.boundary_polyhedral_complex(); bd_C                          # needs sage.graphs
Polyhedral complex with 12 maximal cells
sage: [P.vertices_list() for P in bd_C.maximal_cells_sorted()]              # needs sage.graphs
[[[-1, -1, -1], [-1, -1, 1], [-1, 1, 1]],
 [[-1, -1, -1], [-1, -1, 1], [1, -1, -1]],
 [[-1, -1, -1], [-1, 1, -1], [-1, 1, 1]],
 [[-1, -1, -1], [-1, 1, -1], [1, 1, -1]],
 [[-1, -1, -1], [1, -1, -1], [1, 1, -1]],
 [[-1, -1, 1], [-1, 1, 1], [1, -1, 1]],
 [[-1, -1, 1], [1, -1, -1], [1, -1, 1]],
 [[-1, 1, -1], [-1, 1, 1], [1, 1, -1]],
 [[-1, 1, 1], [1, -1, 1], [1, 1, 1]],
 [[-1, 1, 1], [1, 1, -1], [1, 1, 1]],
 [[1, -1, -1], [1, -1, 1], [1, 1, 1]],
 [[1, -1, -1], [1, 1, -1], [1, 1, 1]]]

Python

>>> from sage.all import *
>>> P = polytopes.cube()
>>> pc = PointConfiguration(P.vertices())
>>> T = pc.placing_triangulation(); T
(<0,1,2,7>, <0,1,5,7>, <0,2,3,7>, <0,3,4,7>, <0,4,5,7>, <1,5,6,7>)
>>> bd_C = T.boundary_polyhedral_complex(); bd_C                          # needs sage.graphs
Polyhedral complex with 12 maximal cells
>>> [P.vertices_list() for P in bd_C.maximal_cells_sorted()]              # needs sage.graphs
[[[-1, -1, -1], [-1, -1, 1], [-1, 1, 1]],
 [[-1, -1, -1], [-1, -1, 1], [1, -1, -1]],
 [[-1, -1, -1], [-1, 1, -1], [-1, 1, 1]],
 [[-1, -1, -1], [-1, 1, -1], [1, 1, -1]],
 [[-1, -1, -1], [1, -1, -1], [1, 1, -1]],
 [[-1, -1, 1], [-1, 1, 1], [1, -1, 1]],
 [[-1, -1, 1], [1, -1, -1], [1, -1, 1]],
 [[-1, 1, -1], [-1, 1, 1], [1, 1, -1]],
 [[-1, 1, 1], [1, -1, 1], [1, 1, 1]],
 [[-1, 1, 1], [1, 1, -1], [1, 1, 1]],
 [[1, -1, -1], [1, -1, 1], [1, 1, 1]],
 [[1, -1, -1], [1, 1, -1], [1, 1, 1]]]

Sage Live

P = polytopes.cube()
pc = PointConfiguration(P.vertices())
T = pc.placing_triangulation(); T
bd_C = T.boundary_polyhedral_complex(); bd_C                          # needs sage.graphs
[P.vertices_list() for P in bd_C.maximal_cells_sorted()]              # needs sage.graphs

It is a subcomplex of self as a polyhedral_complex():

Sage

sage: C = T.polyhedral_complex()                                            # needs sage.graphs
sage: bd_C.is_subcomplex(C)                                                 # needs sage.graphs
True

Python

>>> from sage.all import *
>>> C = T.polyhedral_complex()                                            # needs sage.graphs
>>> bd_C.is_subcomplex(C)                                                 # needs sage.graphs
True

Sage Live

C = T.polyhedral_complex()                                            # needs sage.graphs
bd_C.is_subcomplex(C)                                                 # needs sage.graphs

boundary_simplicial_complex()[source]¶

Return the boundary of self as an (abstract) simplicial complex.

OUTPUT: a SimplicialComplex

EXAMPLES:

Sage

sage: p = polytopes.cuboctahedron()
sage: triangulation = p.triangulate(engine='internal')
sage: bd_sc = triangulation.boundary_simplicial_complex(); bd_sc            # needs sage.graphs
Simplicial complex with 12 vertices and 20 facets

Python

>>> from sage.all import *
>>> p = polytopes.cuboctahedron()
>>> triangulation = p.triangulate(engine='internal')
>>> bd_sc = triangulation.boundary_simplicial_complex(); bd_sc            # needs sage.graphs
Simplicial complex with 12 vertices and 20 facets

Sage Live

p = polytopes.cuboctahedron()
triangulation = p.triangulate(engine='internal')
bd_sc = triangulation.boundary_simplicial_complex(); bd_sc            # needs sage.graphs

The boundary of every convex set is a topological sphere, so it has spherical homology:

Sage

sage: bd_sc.homology()                                                      # needs sage.graphs
{0: 0, 1: 0, 2: Z}

Python

>>> from sage.all import *
>>> bd_sc.homology()                                                      # needs sage.graphs
{0: 0, 1: 0, 2: Z}

Sage Live

bd_sc.homology()                                                      # needs sage.graphs

It is a subcomplex of self as a simplicial_complex():

Sage

sage: sc = triangulation.simplicial_complex()                               # needs sage.graphs
sage: all(f in sc for f in bd_sc.maximal_faces())                           # needs sage.graphs
True

Python

>>> from sage.all import *
>>> sc = triangulation.simplicial_complex()                               # needs sage.graphs
>>> all(f in sc for f in bd_sc.maximal_faces())                           # needs sage.graphs
True

Sage Live

sc = triangulation.simplicial_complex()                               # needs sage.graphs
all(f in sc for f in bd_sc.maximal_faces())                           # needs sage.graphs

enumerate_simplices()[source]¶

Return the enumerated simplices.

OUTPUT: a tuple of integers that uniquely specifies the triangulation

EXAMPLES:

Sage

sage: pc = PointConfiguration(matrix([
....:    [ 0, 0, 0, 0, 0, 2, 4,-1, 1, 1, 0, 0, 1, 0],
....:    [ 0, 0, 0, 1, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0],
....:    [ 0, 2, 0, 0, 0, 0,-1, 0, 1, 0, 1, 0, 0, 1],
....:    [ 0, 1, 1, 0, 0, 1, 0,-2, 1, 0, 0,-1, 1, 1],
....:    [ 0, 0, 0, 0, 1, 0,-1, 0, 0, 0, 0, 0, 0, 0]
....: ]).columns())
sage: triangulation = pc.lexicographic_triangulation()
sage: triangulation.enumerate_simplices()
(1678, 1688, 1769, 1779, 1895, 1905, 2112, 2143, 2234, 2360, 2555, 2580,
 2610, 2626, 2650, 2652, 2654, 2661, 2663, 2667, 2685, 2755, 2757, 2759,
 2766, 2768, 2772, 2811, 2881, 2883, 2885, 2892, 2894, 2898)

Python

>>> from sage.all import *
>>> pc = PointConfiguration(matrix([
...    [ Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(2), Integer(4),-Integer(1), Integer(1), Integer(1), Integer(0), Integer(0), Integer(1), Integer(0)],
...    [ Integer(0), Integer(0), Integer(0), Integer(1), Integer(0), Integer(0),-Integer(1), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0)],
...    [ Integer(0), Integer(2), Integer(0), Integer(0), Integer(0), Integer(0),-Integer(1), Integer(0), Integer(1), Integer(0), Integer(1), Integer(0), Integer(0), Integer(1)],
...    [ Integer(0), Integer(1), Integer(1), Integer(0), Integer(0), Integer(1), Integer(0),-Integer(2), Integer(1), Integer(0), Integer(0),-Integer(1), Integer(1), Integer(1)],
...    [ Integer(0), Integer(0), Integer(0), Integer(0), Integer(1), Integer(0),-Integer(1), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0)]
... ]).columns())
>>> triangulation = pc.lexicographic_triangulation()
>>> triangulation.enumerate_simplices()
(1678, 1688, 1769, 1779, 1895, 1905, 2112, 2143, 2234, 2360, 2555, 2580,
 2610, 2626, 2650, 2652, 2654, 2661, 2663, 2667, 2685, 2755, 2757, 2759,
 2766, 2768, 2772, 2811, 2881, 2883, 2885, 2892, 2894, 2898)

Sage Live

pc = PointConfiguration(matrix([
   [ 0, 0, 0, 0, 0, 2, 4,-1, 1, 1, 0, 0, 1, 0],
   [ 0, 0, 0, 1, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0],
   [ 0, 2, 0, 0, 0, 0,-1, 0, 1, 0, 1, 0, 0, 1],
   [ 0, 1, 1, 0, 0, 1, 0,-2, 1, 0, 0,-1, 1, 1],
   [ 0, 0, 0, 0, 1, 0,-1, 0, 0, 0, 0, 0, 0, 0]
]).columns())
triangulation = pc.lexicographic_triangulation()
triangulation.enumerate_simplices()

You can recreate the triangulation from this list by passing it to the constructor:

Sage

sage: from sage.geometry.triangulation.point_configuration import Triangulation
sage: Triangulation([1678, 1688, 1769, 1779, 1895, 1905, 2112, 2143,
....:  2234, 2360, 2555, 2580, 2610, 2626, 2650, 2652, 2654, 2661, 2663,
....:  2667, 2685, 2755, 2757, 2759, 2766, 2768, 2772, 2811, 2881, 2883,
....:  2885, 2892, 2894, 2898], pc)
(<1,3,4,7,10,13>, <1,3,4,8,10,13>, <1,3,6,7,10,13>, <1,3,6,8,10,13>,
 <1,4,6,7,10,13>, <1,4,6,8,10,13>, <2,3,4,6,7,12>, <2,3,4,7,12,13>,
 <2,3,6,7,12,13>, <2,4,6,7,12,13>, <3,4,5,6,9,12>, <3,4,5,8,9,12>,
 <3,4,6,7,11,12>, <3,4,6,9,11,12>, <3,4,7,10,11,13>, <3,4,7,11,12,13>,
 <3,4,8,9,10,12>, <3,4,8,10,12,13>, <3,4,9,10,11,12>, <3,4,10,11,12,13>,
 <3,5,6,8,9,12>, <3,6,7,10,11,13>, <3,6,7,11,12,13>, <3,6,8,9,10,12>,
 <3,6,8,10,12,13>, <3,6,9,10,11,12>, <3,6,10,11,12,13>, <4,5,6,8,9,12>,
 <4,6,7,10,11,13>, <4,6,7,11,12,13>, <4,6,8,9,10,12>, <4,6,8,10,12,13>,
 <4,6,9,10,11,12>, <4,6,10,11,12,13>)

Python

>>> from sage.all import *
>>> from sage.geometry.triangulation.point_configuration import Triangulation
>>> Triangulation([Integer(1678), Integer(1688), Integer(1769), Integer(1779), Integer(1895), Integer(1905), Integer(2112), Integer(2143),
...  Integer(2234), Integer(2360), Integer(2555), Integer(2580), Integer(2610), Integer(2626), Integer(2650), Integer(2652), Integer(2654), Integer(2661), Integer(2663),
...  Integer(2667), Integer(2685), Integer(2755), Integer(2757), Integer(2759), Integer(2766), Integer(2768), Integer(2772), Integer(2811), Integer(2881), Integer(2883),
...  Integer(2885), Integer(2892), Integer(2894), Integer(2898)], pc)
(<1,3,4,7,10,13>, <1,3,4,8,10,13>, <1,3,6,7,10,13>, <1,3,6,8,10,13>,
 <1,4,6,7,10,13>, <1,4,6,8,10,13>, <2,3,4,6,7,12>, <2,3,4,7,12,13>,
 <2,3,6,7,12,13>, <2,4,6,7,12,13>, <3,4,5,6,9,12>, <3,4,5,8,9,12>,
 <3,4,6,7,11,12>, <3,4,6,9,11,12>, <3,4,7,10,11,13>, <3,4,7,11,12,13>,
 <3,4,8,9,10,12>, <3,4,8,10,12,13>, <3,4,9,10,11,12>, <3,4,10,11,12,13>,
 <3,5,6,8,9,12>, <3,6,7,10,11,13>, <3,6,7,11,12,13>, <3,6,8,9,10,12>,
 <3,6,8,10,12,13>, <3,6,9,10,11,12>, <3,6,10,11,12,13>, <4,5,6,8,9,12>,
 <4,6,7,10,11,13>, <4,6,7,11,12,13>, <4,6,8,9,10,12>, <4,6,8,10,12,13>,
 <4,6,9,10,11,12>, <4,6,10,11,12,13>)

Sage Live

from sage.geometry.triangulation.point_configuration import Triangulation
Triangulation([1678, 1688, 1769, 1779, 1895, 1905, 2112, 2143,
 2234, 2360, 2555, 2580, 2610, 2626, 2650, 2652, 2654, 2661, 2663,
 2667, 2685, 2755, 2757, 2759, 2766, 2768, 2772, 2811, 2881, 2883,
 2885, 2892, 2894, 2898], pc)

fan(origin=None)[source]¶

Construct the fan of cones over the simplices of the triangulation.

INPUT:

origin – None (default) or coordinates of a point. The common apex of all cones of the fan. If None, the triangulation must be a star triangulation and the distinguished central point is used as the origin.

OUTPUT:

A RationalPolyhedralFan. The coordinates of the points are shifted so that the apex of the fan is the origin of the coordinate system.

Note

If the set of cones over the simplices is not a fan, a suitable exception is raised.

EXAMPLES:

Sage

sage: pc = PointConfiguration([(0,0), (1,0), (0,1), (-1,-1)], star=0, fine=True)
sage: triangulation = pc.triangulate()
sage: fan = triangulation.fan(); fan
Rational polyhedral fan in 2-d lattice N
sage: fan.is_equivalent(toric_varieties.P2().fan())                         # needs palp sage.graphs
True

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(0),Integer(1)), (-Integer(1),-Integer(1))], star=Integer(0), fine=True)
>>> triangulation = pc.triangulate()
>>> fan = triangulation.fan(); fan
Rational polyhedral fan in 2-d lattice N
>>> fan.is_equivalent(toric_varieties.P2().fan())                         # needs palp sage.graphs
True

Sage Live

pc = PointConfiguration([(0,0), (1,0), (0,1), (-1,-1)], star=0, fine=True)
triangulation = pc.triangulate()
fan = triangulation.fan(); fan
fan.is_equivalent(toric_varieties.P2().fan())                         # needs palp sage.graphs

Toric diagrams (the \(\ZZ_5\) hyperconifold):

Sage

sage: vertices=[(0, 1, 0), (0, 3, 1), (0, 2, 3), (0, 0, 2)]
sage: interior=[(0, 1, 1), (0, 1, 2), (0, 2, 1), (0, 2, 2)]
sage: points = vertices + interior
sage: pc = PointConfiguration(points, fine=True)
sage: triangulation = pc.triangulate()
sage: fan = triangulation.fan((-1,0,0)); fan
Rational polyhedral fan in 3-d lattice N
sage: fan.rays()
N(1, 1, 0),
N(1, 3, 1),
N(1, 2, 3),
N(1, 0, 2),
N(1, 1, 1),
N(1, 1, 2),
N(1, 2, 1),
N(1, 2, 2)
in 3-d lattice N

Python

>>> from sage.all import *
>>> vertices=[(Integer(0), Integer(1), Integer(0)), (Integer(0), Integer(3), Integer(1)), (Integer(0), Integer(2), Integer(3)), (Integer(0), Integer(0), Integer(2))]
>>> interior=[(Integer(0), Integer(1), Integer(1)), (Integer(0), Integer(1), Integer(2)), (Integer(0), Integer(2), Integer(1)), (Integer(0), Integer(2), Integer(2))]
>>> points = vertices + interior
>>> pc = PointConfiguration(points, fine=True)
>>> triangulation = pc.triangulate()
>>> fan = triangulation.fan((-Integer(1),Integer(0),Integer(0))); fan
Rational polyhedral fan in 3-d lattice N
>>> fan.rays()
N(1, 1, 0),
N(1, 3, 1),
N(1, 2, 3),
N(1, 0, 2),
N(1, 1, 1),
N(1, 1, 2),
N(1, 2, 1),
N(1, 2, 2)
in 3-d lattice N

Sage Live

vertices=[(0, 1, 0), (0, 3, 1), (0, 2, 3), (0, 0, 2)]
interior=[(0, 1, 1), (0, 1, 2), (0, 2, 1), (0, 2, 2)]
points = vertices + interior
pc = PointConfiguration(points, fine=True)
triangulation = pc.triangulate()
fan = triangulation.fan((-1,0,0)); fan
fan.rays()

gkz_phi()[source]¶

Calculate the GKZ phi vector of the triangulation.

The phi vector is a vector of length equals to the number of points in the point configuration. For a fixed triangulation \(T\), the entry corresponding to the \(i\)-th point \(p_i\) is

\[\phi_T(p_i) = \sum_{t\in T, t\owns p_i} Vol(t)\]

that is, the total volume of all simplices containing \(p_i\). See also [GKZ1994] page 220 equation 1.4.

OUTPUT: the phi vector of self

EXAMPLES:

Sage

sage: p = PointConfiguration([[0,0],[1,0],[2,1],[1,2],[0,1]])
sage: p.triangulate().gkz_phi()
(3, 1, 5, 2, 4)
sage: p.lexicographic_triangulation().gkz_phi()
(1, 3, 4, 2, 5)

Python

>>> from sage.all import *
>>> p = PointConfiguration([[Integer(0),Integer(0)],[Integer(1),Integer(0)],[Integer(2),Integer(1)],[Integer(1),Integer(2)],[Integer(0),Integer(1)]])
>>> p.triangulate().gkz_phi()
(3, 1, 5, 2, 4)
>>> p.lexicographic_triangulation().gkz_phi()
(1, 3, 4, 2, 5)

Sage Live

p = PointConfiguration([[0,0],[1,0],[2,1],[1,2],[0,1]])
p.triangulate().gkz_phi()
p.lexicographic_triangulation().gkz_phi()

interior_facets()[source]¶

Return the interior facets of the triangulation.

OUTPUT:

The inward-facing boundary simplices (of dimension \(d-1\)) of the \(d\)-dimensional triangulation as a set. Each boundary is returned by a tuple of point indices.

EXAMPLES:

Sage

sage: triangulation = polytopes.cube().triangulate(engine='internal')
sage: triangulation
(<0,1,2,7>, <0,1,5,7>, <0,2,3,7>, <0,3,4,7>, <0,4,5,7>, <1,5,6,7>)
sage: triangulation.boundary()
frozenset({(0, 1, 2),
           (0, 1, 5),
           (0, 2, 3),
           (0, 3, 4),
           (0, 4, 5),
           (1, 2, 7),
           (1, 5, 6),
           (1, 6, 7),
           (2, 3, 7),
           (3, 4, 7),
           (4, 5, 7),
           (5, 6, 7)})
sage: triangulation.interior_facets()
frozenset({(0, 1, 7), (0, 2, 7), (0, 3, 7), (0, 4, 7), (0, 5, 7), (1, 5, 7)})

Python

>>> from sage.all import *
>>> triangulation = polytopes.cube().triangulate(engine='internal')
>>> triangulation
(<0,1,2,7>, <0,1,5,7>, <0,2,3,7>, <0,3,4,7>, <0,4,5,7>, <1,5,6,7>)
>>> triangulation.boundary()
frozenset({(0, 1, 2),
           (0, 1, 5),
           (0, 2, 3),
           (0, 3, 4),
           (0, 4, 5),
           (1, 2, 7),
           (1, 5, 6),
           (1, 6, 7),
           (2, 3, 7),
           (3, 4, 7),
           (4, 5, 7),
           (5, 6, 7)})
>>> triangulation.interior_facets()
frozenset({(0, 1, 7), (0, 2, 7), (0, 3, 7), (0, 4, 7), (0, 5, 7), (1, 5, 7)})

Sage Live

triangulation = polytopes.cube().triangulate(engine='internal')
triangulation
triangulation.boundary()
triangulation.interior_facets()

normal_cone()[source]¶

Return the (closure of the) normal cone of the triangulation.

Recall that a regular triangulation is one that equals the “crease lines” of a convex piecewise-linear function. This support function is not unique, for example, you can scale it by a positive constant. The set of all piecewise-linear functions with fixed creases forms an open cone. This cone can be interpreted as the cone of normal vectors at a point of the secondary polytope, which is why we call it normal cone. See [GKZ1994] Section 7.1 for details.

OUTPUT:

The closure of the normal cone. The \(i\)-th entry equals the value of the piecewise-linear function at the \(i\)-th point of the configuration.

For an irregular triangulation, the normal cone is empty. In this case, a single point (the origin) is returned.

EXAMPLES:

Sage

sage: triangulation = polytopes.hypercube(2).triangulate(engine='internal')
sage: triangulation
(<0,1,3>, <1,2,3>)
sage: N = triangulation.normal_cone();  N
4-d cone in 4-d lattice
sage: N.rays()
( 0,  0,  0, -1),
( 0,  0,  1,  1),
( 0,  0, -1, -1),
( 1,  0,  0,  1),
(-1,  0,  0, -1),
( 0,  1,  0, -1),
( 0, -1,  0,  1)
in Ambient free module of rank 4
over the principal ideal domain Integer Ring
sage: N.dual().rays()
(1, -1, 1, -1)
in Ambient free module of rank 4
over the principal ideal domain Integer Ring

Python

>>> from sage.all import *
>>> triangulation = polytopes.hypercube(Integer(2)).triangulate(engine='internal')
>>> triangulation
(<0,1,3>, <1,2,3>)
>>> N = triangulation.normal_cone();  N
4-d cone in 4-d lattice
>>> N.rays()
( 0,  0,  0, -1),
( 0,  0,  1,  1),
( 0,  0, -1, -1),
( 1,  0,  0,  1),
(-1,  0,  0, -1),
( 0,  1,  0, -1),
( 0, -1,  0,  1)
in Ambient free module of rank 4
over the principal ideal domain Integer Ring
>>> N.dual().rays()
(1, -1, 1, -1)
in Ambient free module of rank 4
over the principal ideal domain Integer Ring

Sage Live

triangulation = polytopes.hypercube(2).triangulate(engine='internal')
triangulation
N = triangulation.normal_cone();  N
N.rays()
N.dual().rays()

plot(**kwds)[source]¶

Produce a graphical representation of the triangulation.

EXAMPLES:

Sage

sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: triangulation = p.triangulate()
sage: triangulation
(<1,3,4>, <2,3,4>)
sage: triangulation.plot(axes=False)                                        # needs sage.plot
Graphics object consisting of 12 graphics primitives

Python

>>> from sage.all import *
>>> p = PointConfiguration([[Integer(0),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(0)],[Integer(1),Integer(1)],[-Integer(1),-Integer(1)]])
>>> triangulation = p.triangulate()
>>> triangulation
(<1,3,4>, <2,3,4>)
>>> triangulation.plot(axes=False)                                        # needs sage.plot
Graphics object consisting of 12 graphics primitives

Sage Live

p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
triangulation = p.triangulate()
triangulation
triangulation.plot(axes=False)                                        # needs sage.plot

point_configuration()[source]¶

Return the point configuration underlying the triangulation.

EXAMPLES:

Sage

sage: pconfig = PointConfiguration([[0,0],[0,1],[1,0]])
sage: pconfig
A point configuration in affine 2-space over Integer Ring
consisting of 3 points. The triangulations of this point
configuration are assumed to be connected, not necessarily
fine, not necessarily regular.
sage: triangulation = pconfig.triangulate()
sage: triangulation
(<0,1,2>)
sage: triangulation.point_configuration()
A point configuration in affine 2-space over Integer Ring
consisting of 3 points. The triangulations of this point
configuration are assumed to be connected, not necessarily
fine, not necessarily regular.
sage: pconfig == triangulation.point_configuration()
True

Python

>>> from sage.all import *
>>> pconfig = PointConfiguration([[Integer(0),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(0)]])
>>> pconfig
A point configuration in affine 2-space over Integer Ring
consisting of 3 points. The triangulations of this point
configuration are assumed to be connected, not necessarily
fine, not necessarily regular.
>>> triangulation = pconfig.triangulate()
>>> triangulation
(<0,1,2>)
>>> triangulation.point_configuration()
A point configuration in affine 2-space over Integer Ring
consisting of 3 points. The triangulations of this point
configuration are assumed to be connected, not necessarily
fine, not necessarily regular.
>>> pconfig == triangulation.point_configuration()
True

Sage Live

pconfig = PointConfiguration([[0,0],[0,1],[1,0]])
pconfig
triangulation = pconfig.triangulate()
triangulation
triangulation.point_configuration()
pconfig == triangulation.point_configuration()

polyhedral_complex(**kwds)[source]¶

Return self as a PolyhedralComplex.

OUTPUT:

A PolyhedralComplex whose maximal cells are the simplices of the triangulation.

EXAMPLES:

Sage

sage: P = polytopes.cube()
sage: pc = PointConfiguration(P.vertices())
sage: T = pc.placing_triangulation(); T
(<0,1,2,7>, <0,1,5,7>, <0,2,3,7>, <0,3,4,7>, <0,4,5,7>, <1,5,6,7>)
sage: C = T.polyhedral_complex(); C                                         # needs sage.graphs
Polyhedral complex with 6 maximal cells
sage: [P.vertices_list() for P in C.maximal_cells_sorted()]                 # needs sage.graphs
[[[-1, -1, -1], [-1, -1, 1], [-1, 1, 1], [1, -1, -1]],
 [[-1, -1, -1], [-1, 1, -1], [-1, 1, 1], [1, 1, -1]],
 [[-1, -1, -1], [-1, 1, 1], [1, -1, -1], [1, 1, -1]],
 [[-1, -1, 1], [-1, 1, 1], [1, -1, -1], [1, -1, 1]],
 [[-1, 1, 1], [1, -1, -1], [1, -1, 1], [1, 1, 1]],
 [[-1, 1, 1], [1, -1, -1], [1, 1, -1], [1, 1, 1]]]

Python

>>> from sage.all import *
>>> P = polytopes.cube()
>>> pc = PointConfiguration(P.vertices())
>>> T = pc.placing_triangulation(); T
(<0,1,2,7>, <0,1,5,7>, <0,2,3,7>, <0,3,4,7>, <0,4,5,7>, <1,5,6,7>)
>>> C = T.polyhedral_complex(); C                                         # needs sage.graphs
Polyhedral complex with 6 maximal cells
>>> [P.vertices_list() for P in C.maximal_cells_sorted()]                 # needs sage.graphs
[[[-1, -1, -1], [-1, -1, 1], [-1, 1, 1], [1, -1, -1]],
 [[-1, -1, -1], [-1, 1, -1], [-1, 1, 1], [1, 1, -1]],
 [[-1, -1, -1], [-1, 1, 1], [1, -1, -1], [1, 1, -1]],
 [[-1, -1, 1], [-1, 1, 1], [1, -1, -1], [1, -1, 1]],
 [[-1, 1, 1], [1, -1, -1], [1, -1, 1], [1, 1, 1]],
 [[-1, 1, 1], [1, -1, -1], [1, 1, -1], [1, 1, 1]]]

Sage Live

P = polytopes.cube()
pc = PointConfiguration(P.vertices())
T = pc.placing_triangulation(); T
C = T.polyhedral_complex(); C                                         # needs sage.graphs
[P.vertices_list() for P in C.maximal_cells_sorted()]                 # needs sage.graphs

simplicial_complex()[source]¶

Return self as an (abstract) simplicial complex.

OUTPUT: a SimplicialComplex

EXAMPLES:

Sage

sage: p = polytopes.cuboctahedron()
sage: sc = p.triangulate(engine='internal').simplicial_complex(); sc        # needs sage.graphs
Simplicial complex with 12 vertices and 16 facets

Python

>>> from sage.all import *
>>> p = polytopes.cuboctahedron()
>>> sc = p.triangulate(engine='internal').simplicial_complex(); sc        # needs sage.graphs
Simplicial complex with 12 vertices and 16 facets

Sage Live

p = polytopes.cuboctahedron()
sc = p.triangulate(engine='internal').simplicial_complex(); sc        # needs sage.graphs

Any convex set is contractable, so its reduced homology groups vanish:

Sage

sage: sc.homology()                                                         # needs sage.graphs
{0: 0, 1: 0, 2: 0, 3: 0}

Python

>>> from sage.all import *
>>> sc.homology()                                                         # needs sage.graphs
{0: 0, 1: 0, 2: 0, 3: 0}

Sage Live

sc.homology()                                                         # needs sage.graphs

sage.geometry.triangulation.element.triangulation_render_2d(triangulation, **kwds)[source]¶

Return a graphical representation of a 2-d triangulation.

INPUT:

triangulation – a Triangulation
**kwds – keywords that are passed on to the graphics primitives

OUTPUT: a 2-d graphics object

EXAMPLES:

Sage

sage: points = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: triang = points.triangulate()
sage: triang.plot(axes=False, aspect_ratio=1)   # indirect doctest              # needs sage.plot
Graphics object consisting of 12 graphics primitives

Python

>>> from sage.all import *
>>> points = PointConfiguration([[Integer(0),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(0)],[Integer(1),Integer(1)],[-Integer(1),-Integer(1)]])
>>> triang = points.triangulate()
>>> triang.plot(axes=False, aspect_ratio=Integer(1))   # indirect doctest              # needs sage.plot
Graphics object consisting of 12 graphics primitives

Sage Live

points = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
triang = points.triangulate()
triang.plot(axes=False, aspect_ratio=1)   # indirect doctest              # needs sage.plot

sage.geometry.triangulation.element.triangulation_render_3d(triangulation, **kwds)[source]¶

Return a graphical representation of a 3-d triangulation.

INPUT:

triangulation – a Triangulation
**kwds – keywords that are passed on to the graphics primitives

OUTPUT: a 3-d graphics object

EXAMPLES:

Sage

sage: p = [[0,-1,-1],[0,0,1],[0,1,0], [1,-1,-1],[1,0,1],[1,1,0]]
sage: points = PointConfiguration(p)
sage: triang = points.triangulate()
sage: triang.plot(axes=False)     # indirect doctest                            # needs sage.plot
Graphics3d Object

Python

>>> from sage.all import *
>>> p = [[Integer(0),-Integer(1),-Integer(1)],[Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(0)], [Integer(1),-Integer(1),-Integer(1)],[Integer(1),Integer(0),Integer(1)],[Integer(1),Integer(1),Integer(0)]]
>>> points = PointConfiguration(p)
>>> triang = points.triangulate()
>>> triang.plot(axes=False)     # indirect doctest                            # needs sage.plot
Graphics3d Object

Sage Live

p = [[0,-1,-1],[0,0,1],[0,1,0], [1,-1,-1],[1,0,1],[1,1,0]]
points = PointConfiguration(p)
triang = points.triangulate()
triang.plot(axes=False)     # indirect doctest                            # needs sage.plot