Table of Contents

BigObject PolyhedralComplex<Scalar>

from application fan

A polyhedral complex. The derivation from PolyhedralFan works like the derivation of Polytope from Cone.

Type Parameters:

Scalar: numeric data type used for the coordinates, must be an ordered field. Default is Rational.

derived from:
PolyhedralFan
Example:

The following defines a subdivision of a square in the plane into two triangles. 

 > $c=new PolyhedralComplex(VERTICES=>[[1,0,0],[1,1,0],[1,0,1],[1,1,1]],MAXIMAL_POLYTOPES=>[[0,1,2],[1,2,3]]);

Properties

Input property

These properties are for input only. They allow redundant information.

INPUT_POLYTOPES

Maybe redundant list of not necessarily maximal polytopes. Indices refer to POINTS. Each polytope must list all vertices of POINTS it contains. The polytopes are allowed to contain lineality. An empty complex does not have any polytopes. Input section only. Ask for MAXIMAL_POLYTOPES if you want to know the maximal polytopes (indexed by VERTICES). Alias for property INPUT_CONES.

Type:
IncidenceMatrix<NonSymmetric>
Example:

We can define a polyhderal complex consisting of two distinct triangles with the following (note that additionally stating one side of one of these triangles does not affect our resulting complex): 

 > $c = new PolyhedralComplex(POINTS=>[[1,1,0],[1,1,1],[1,0,1],[1,-1,0],[1,-1,-1],[1,0,-1]],INPUT_POLYTOPES=>[[0,1,2],[3,4,5],[0,1]]);

print $c→MAXIMAL_POLYTOPES;

POINTS

Points in homogeneous coordinates from which the polytopes are formed. May be redundant. All vectors in the input must be non-zero. You also need to provide INPUT_POLYTOPES to define a complex completely. Input section only. Ask for VERTICES if you want a list of non-redundant points. Alias for property INPUT_RAYS.

Type:
Matrix<Scalar,NonSymmetric>
Example:

To obtain a complex consisting of two triangles we can do this (note that, contrary to a polytope, this complex is not convex): 

 > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,3,-1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,3],[1,2,3]]);

Combinatorics

These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

COMPACTIFICATION

The Hasse diagram of the compactification of the polyhedral complex. For a simplicial polyhedral complex, this is the cubical compactification (or cubical complex, see [Omid Amini: “The combinatorial Chow ring of products of graphs”]). For tropical varieties, this is the tropical compactification, as in [Brian Osserman and Joseph Rabinoff: “Lifting nonproper tropical intersections”]. The vertices of the compactification correspond to the faces of the original complex that have the same dimension as their recession cone. We call the face corresponding to a vertex the 'realisation' of the vertex. The decoration has four entries: 1. The face in the vertices of the compactification 2. The rank of the face 3. The realisation of the face. This is the union of the realisations of the new vertices. 4. The sedentarity of the face. This is the intersection of the sedentarities of the vertices.

Type:
PartiallyOrderedSet<SedentarityDecoration,Nonsequential>
Example:

The compactification of the positive orthant in three dimensions has the same Hasse diagram as the three dimensional cube. 

 > $pc1 = new PolyhedralComplex(POINTS=>[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], INPUT_POLYTOPES=>[[0,1,2,3]]);
 > print $pc1->COMPACTIFICATION->DECORATION;
 ({} 0 {} {})
 ({0} 1 {0 1 2 3} {1 2 3})
 ({1} 1 {0 2 3} {2 3})
 ({2} 1 {0 1 3} {1 3})
 ({3} 1 {0 1 2} {1 2})
 ({4} 1 {0 3} {3})
 ({5} 1 {0 2} {2})
 ({6} 1 {0 1} {1})
 ({7} 1 {0} {})
 ({0 1} 2 {0 1 2 3} {2 3})
 ({0 2} 2 {0 1 2 3} {1 3})
 ({0 3} 2 {0 1 2 3} {1 2})
 ({1 4} 2 {0 2 3} {3})
 ({1 5} 2 {0 2 3} {2})
 ({2 4} 2 {0 1 3} {3})
 ({2 6} 2 {0 1 3} {1})
 ({3 5} 2 {0 1 2} {2})
 ({3 6} 2 {0 1 2} {1})
 ({4 7} 2 {0 3} {})
 ({5 7} 2 {0 2} {})
 ({6 7} 2 {0 1} {})
 ({0 1 2 4} 3 {0 1 2 3} {3})
 ({0 1 3 5} 3 {0 1 2 3} {2})
 ({0 2 3 6} 3 {0 1 2 3} {1})
 ({1 4 5 7} 3 {0 2 3} {})
 ({2 4 6 7} 3 {0 1 3} {})
 ({3 5 6 7} 3 {0 1 2} {})
 ({0 1 2 3 4 5 6 7} 4 {0 1 2 3} {})
 ({-1} 5 {-1} {})

MAXIMAL_POLYTOPES

Non redundant list of maximal polytopes. Indices refer to VERTICES. An empty complex does not have any polytopes. Alias for property MAXIMAL_CONES.

Type:
IncidenceMatrix<NonSymmetric>
Example:

After creating a complex via the INPUT_POLYTOPES property, we can display all maximal polytopes rising from that definition: 

 > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[2]]);
 > print $c->MAXIMAL_POLYTOPES;
 {0 1 2}
 {0 3}

MAXIMAL_POLYTOPES_COMBINATORIAL_DIMS

The combinatorial dimensions of the maximal polytopes. The i-th entry refers to the i-th entry of MAXIMAL_POLYTOPES. Alias for property MAXIMAL_CONES_COMBINATORIAL_DIMS.

Type:
Array<Int>
Example:

When connecting two vertices of a triangle to a vertex distinct from that triangle we receive a polyhedral complex with maximal polytopes of dimensions 2, 1 and 1, respectively: 

 > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[2,3]]);
 > print $c->MAXIMAL_POLYTOPES;
 {0 1 2}
 {0 3}
 {2 3}
 > print $c->MAXIMAL_POLYTOPES_COMBINATORIAL_DIMS;
 2 1 1

MAXIMAL_POLYTOPES_INCIDENCES

Array of incidence matrices of all maximal polytopes. Alias for property MAXIMAL_CONES_INCIDENCES.

Type:
Array<IncidenceMatrix<NonSymmetric>>
Example:

Here we construct a polyhedral complex made of two triangles which share a side; this fact can afterwards be read from this property: 

 > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[1,2,3]]);
 > print $c->MAXIMAL_POLYTOPES;
 {0 1 2}
 {1 2 3}
 > print $c->MAXIMAL_POLYTOPES_INCIDENCES;
 <{1 2}
 {0 2}
 {0 1}
 >
 <{1 2}
 {2 3}
 {1 3}
 >

MAXIMAL_POLYTOPES_THRU_VERTICES

Alias for property MAXIMAL_CONES_THRU_RAYS.

Type:
IncidenceMatrix<NonSymmetric>

N_MAXIMAL_POLYTOPES

Number of MAXIMAL_POLYTOPES. Alias for property N_MAXIMAL_CONES.

Type:
Int
Example:

The number of maximal polytopes of a planar net of a polytope is the number of facets of that polytope; here we see this for the dodecahedron: 

 > $c = planar_net(dodecahedron());
 > print $c->N_MAXIMAL_POLYTOPES;
 12

N_POLYTOPES

Alias for property N_CONES.

Type:
Int

POLYTOPES

List of all polytopes of the complex of each dimension. Indices refer to VERTICES. Alias for property CONES.

Type:
Array<IncidenceMatrix<NonSymmetric>>
Example:

A complex whose only maximal polytope is a triangle also contains 3 line segments and 3 points: 

 > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2]]);
 > print $c->POLYTOPES;
 <{1}
 {2}
 {0}
 >
 <{1 2}
 {0 2}
 {0 1}
 >
 <{0 1 2}
 >

Geometry

These properties capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.

AFFINE_HULL

# The possible linear span normals of all maximal polytopes. Empty if PURE and FULL_DIM, i.e. each maximal polytope has the same dimension as the ambient space. Alias for property LINEAR_SPAN_NORMALS.

Type:
Matrix<Scalar,NonSymmetric>
Example:

In the plane, when we construct a polyhedral complex with a 2-dimensional and two 1-dimensional maximal polytopes only the latter two will have a linear span with a normal in this ambient space: 

 > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,-1,0],[1,-1,-1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[0,4]]);
 > print $c->AFFINE_HULL;
 0 0 1
 0 -1 1

BOUNDED

True if each object in MAXIMAL_POLYTOPES is bounded.

Type:
Bool

FAR_VERTICES

Indices of vertices that are rays.

Type:
Set<Int>
Example:

We construct a PolyhedralComplex consisting only of one unbounded Polytope which is the Minkowski sum of an interval and a cone orthogonal to this line. Such a Minkowski sum always has a far vertex: 

 > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[0,0,1]],INPUT_POLYTOPES=>[[0,1,2]]);
 > print rows_numbered($c->VERTICES);
 0:1 0 0
 1:1 1 0
 2:0 0 1
 > print $c->FAR_VERTICES;
 {2}

MAXIMAL_POLYTOPES_AFFINE_HULL_NORMALS

A basis of the normal space for each maximal polytope. This uniquely determines the affine hull of the corresponding maximal polytope. Indices refer to AFFINE_HULL. Rows correspond to MAXIMAL_POLYTOPES. An empty row corresponds to a full-dimensional cone. Alias for property MAXIMAL_CONES_LINEAR_SPAN_NORMALS.

Type:
IncidenceMatrix<NonSymmetric>
Example:

In the plane, when we construct a polyhedral complex with a 2-dimensional and two 1-dimensional maximal polytopes only the latter will have a linear span with a normal in this ambient space: 

 > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,-1,0],[1,-1,-1]],INPUT_POLYTOPES=>[[0,1,2],[0,3],[0,4]]);
 > print $c->MAXIMAL_POLYTOPES;
 {0 1 2}
 {0 3}
 {0 4}
 > print rows_numbered($c->AFFINE_HULL);
 0:0 0 1
 1:0 -1 1
 > print $c->MAXIMAL_POLYTOPES_AFFINE_HULL_NORMALS;
 {}
 {0}
 {1}

MAXIMAL_POLYTOPES_FACETS

Tells for each maximal polytope what are its facet normals, thus implying the facets. Each row corresponds to a maximal polytope and each column to the row with the same index of AFFINE_HULL. A negative number means that the corresponding row of AFFINE_HULL has to be negated. Alias for property MAXIMAL_CONES_FACETS.

Type:
SparseMatrix<Int,NonSymmetric>
Example:

Here we see the facet normals of the maximal polytopes of a complex made of two triangles (note that some facet normal appear to be redundant due to usage of homogeneous coordinates and the derivation from PolyhedralFan): 

 > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,1,1],[1,0,1]],INPUT_POLYTOPES=>[[0,1,2],[0,2,3]]);
 > print $c->MAXIMAL_POLYTOPES;
 {0 1 2}
 {0 2 3}
 > print rows_numbered($c->FACET_NORMALS);
 0:1 -1 0
 1:0 1 -1
 2:0 0 1
 3:1 0 -1
 4:0 1 0
 > print $c->MAXIMAL_POLYTOPES_FACETS;
 1 1 1 0 0
 0 -1 0 1 1

N_POINTS

Number of POINTS. Alias for property N_INPUT_RAYS.

Type:
Int
Example:

In the plane, glueing two triangles together along one side gives us a complex with four vertices; nevertheless we can specify these two triangles using six points with redundancies: 

 > $c = new PolyhedralComplex(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,1,0],[2,0,2],[1,1,1]],INPUT_POLYTOPES=>[[0,1,2],[3,4,5]]);
 > print $c->N_VERTICES;
 4
 > print $c->N_POINTS;
 6

N_VERTICES

Number of VERTICES. Alias for property N_RAYS.

Type:
Int

VERTICES

Vertices from which the polytopes are formed. Non-redundant. Co-exists with LINEALITY_SPACE. Alias for property RAYS.

Type:
Matrix<Scalar,NonSymmetric>

Symmetry

These properties capture information of the object that is concerned with the action of permutation groups.

GROUP
derived from:
GROUP
Type:
Group
Properties of GROUP:
COORDINATE_ACTION
Type:
PermutationAction<Int,Rational>
Properties of COORDINATE_ACTION:
N_POINTS_GENERATORS

Alias for property N_INPUT_RAYS_GENERATORS.

Type:
Int
N_VERTICES_GENERATORS

Alias for property N_RAYS_GENERATORS.

Type:
Int
POINTS_GENERATORS

Alias for property INPUT_RAYS_GENERATORS.

Type:
Matrix<Rational,NonSymmetric>
VERTICES_GENERATORS

Alias for property RAYS_GENERATORS.

Type:
Matrix<Rational,NonSymmetric>
INPUT_POLYTOPES_ACTION

Alias for property INPUT_CONES_ACTION.

Type:
PermutationAction<Int,Rational>
MATRIX_ACTION_ON_COMPLEX
Type:
MatrixActionOnVectors<Scalar>
Properties of MATRIX_ACTION_ON_COMPLEX:
VERTICES_GENERATORS

Alias for property RAYS_GENERATORS.

Type:
Matrix<OrbitGeneratorScalarType,NonSymmetric>
MAXIMAL_POLYTOPES_ACTION
derived from:
MAXIMAL_CONES_ACTION
Type:
PermutationAction<Int,Rational>
Properties of MAXIMAL_POLYTOPES_ACTION:
MAXIMAL_POLYTOPES_GENERATORS

Alias for property MAXIMAL_CONES_GENERATORS.

Type:
IncidenceMatrix<NonSymmetric>
N_POINTS_GENERATORS

Alias for property N_INPUT_RAYS_GENERATORS.

Type:
Int
N_VERTICES_GENERATORS

Alias for property N_RAYS_GENERATORS.

Type:
Int
POINTS_GENERATORS

Alias for property INPUT_RAYS_GENERATORS.

Type:
Matrix<Rational,NonSymmetric>
VERTICES_GENERATORS

Alias for property RAYS_GENERATORS.

Type:
Matrix<Rational,NonSymmetric>
POINTS_ACTION

Alias for property INPUT_RAYS_ACTION.

Type:
PermutationAction<Int,Rational>
POLYTOPES_ORBIT_SIZES

Alias for property CONES_ORBIT_SIZES.

Type:
Array<Array<Int>>
REPRESENTATIVE_MAXIMAL_POLYTOPES

Alias for property REPRESENTATIVE_MAXIMAL_CONES.

Type:
IncidenceMatrix<NonSymmetric>
REPRESENTATIVE_VERTICES

Alias for property REPRESENTATIVE_RAYS.

Type:
Matrix<Scalar,NonSymmetric>
VERTICES_ACTION

Alias for property RAYS_ACTION.

Type:
PermutationAction<Int,Rational>

Visualization

These properties are for visualization.

POINT_LABELS

Unique names assigned to the POINTS. Similar to VERTEX_LABELS for VERTICES. Alias for property INPUT_RAY_LABELS.

Type:
Array<String>

VERTEX_LABELS

Unique names assigned to the VERTICES. If specified, they are shown by visualization tools instead of vertex indices. For a polyhedral complex built from scratch, you should create this property by yourself, either manually in a text editor, or with a client program. Alias for property RAY_LABELS.

Type:
Array<String>

Methods

Geometry

These methods capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.

AMBIENT_DIM()

Returns the dimension of the ambient space.

Returns:
Int
Example:

The ambient dimension of a point in the line is 1: 

 > $c = new PolyhedralComplex(POINTS=>[[1,0]],INPUT_POLYTOPES=>[[0]]);
 > print $c->AMBIENT_DIM;
 1

DIM()

Returns the dimension of the linear space spanned by the complex.

Returns:
Int
Example:

The dimension of a point in the line is 0: 

 > $c = new PolyhedralComplex(POINTS=>[[1,0]],INPUT_POLYTOPES=>[[0]]);
 > print $c->DIM;
 0

polytope(Int i)

Returns the i-th facet of the complex as a Polytope.

Parameters:

Int i

Returns:
Polytope
Example:

The planar net of the 3-dimensional cross polytope consists only of triangles (and the according adjacent lines and vertices); asking for any of its polytopes thus gives us a triangle: 

 > $c = planar_net(cross(3));
 > $p = $c->polytope(5);
 > print rows_numbered($p->VERTICES);
 0:1 0 0
 1:1 0.707106781186547 -1.22474487139159
 2:1 -0.707106781186549 -1.22474487139159

Visualization

These methods are for visualization.

VISUAL_ORBIT_COLORED_GRAPH()

Visualizes the graph of a symmetric cone: All nodes belonging to one orbit get the same color.

Options:
option list Visual::Graph::decorations
Returns:
Visual::PolytopeGraph

no category

VISUAL()

Visualizes the polyhedral complex.

Options:
option list geometric_options
Returns:
Visual::PolyhedralFan

VISUAL_METRIC_TIGHT_SPAN()

UNDOCUMENTED

Options:
option list Visual::Graph::TightSpanDecorations