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]]);

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

INPUT_POLYTOPES

Alias for property INPUT_CONES.

Type:

IncidenceMatrix<NonSymmetric>

POINTS

Alias for property INPUT_RAYS.

Type:

Matrix<Scalar,NonSymmetric>

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:

Lattice<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

Alias for property MAXIMAL_CONES.

Type:

IncidenceMatrix<NonSymmetric>

MAXIMAL_POLYTOPES_COMBINATORIAL_DIMS

Alias for property MAXIMAL_CONES_COMBINATORIAL_DIMS.

Type:

Array<Int>

MAXIMAL_POLYTOPES_INCIDENCES

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

Type:

Array<IncidenceMatrix<NonSymmetric>>

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

N_POLYTOPES

Alias for property N_CONES.

Type:

Int

POLYTOPES

Alias for property CONES.

Type:

Array<IncidenceMatrix<NonSymmetric>>

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

Alias for property LINEAR_SPAN_NORMALS.

Type:

Matrix<Scalar,NonSymmetric>

BOUNDED

True if each object in MAXIMAL_POLYTOPES is bounded.

Type:

Bool

FAR_VERTICES

Indices of vertices that are rays.

Type:

Set<Int>

MAXIMAL_POLYTOPES_AFFINE_HULL_NORMALS

Alias for property MAXIMAL_CONES_LINEAR_SPAN_NORMALS.

Type:

IncidenceMatrix<NonSymmetric>

MAXIMAL_POLYTOPES_FACETS

Alias for property MAXIMAL_CONES_FACETS.

Type:

SparseMatrix<Int,NonSymmetric>

N_POINTS

Number of POINTS. Alias for property N_INPUT_RAYS.

Type:

Int

N_VERTICES

Number of VERTICES. Alias for property N_RAYS.

Type:

Int

VERTICES

Alias for property RAYS.

Type:

Matrix<Scalar,NonSymmetric>

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>

REPRESENTATIVE_VERTICES

Alias for property REPRESENTATIVE_RAYS.

Type:

Matrix<Scalar,NonSymmetric>

VERTICES_ACTION

Alias for property RAYS_ACTION.

Type:

PermutationAction<Int,Rational>

INPUT_POLYTOPES_REPS

Alias for property INPUT_CONES_REPS.

Type:

IncidenceMatrix<NonSymmetric>

MAXIMAL_POLYTOPES_IN_ORBITS

Alias for property MAXIMAL_CONES_IN_ORBITS.

Type:

Array<Set<Int>>

MAXIMAL_POLYTOPES_ORBIT_SIZES

Alias for property MAXIMAL_CONES_ORBIT_SIZES.

Type:

Array<Int>

MAXIMAL_POLYTOPES_REPS

Alias for property MAXIMAL_CONES_REPS.

Type:

Array<Set<Int>>

MAXIMAL_POLYTOPES_REPS_AFFINE_SPAN_NORMALS

Alias for property MAXIMAL_CONES_REPS_LINEAR_SPAN_NORMALS.

Type:

IncidenceMatrix<NonSymmetric>

MAXIMAL_POLYTOPES_REPS_DIMS

Alias for property MAXIMAL_CONES_REPS_DIMS.

Type:

Array<Int>

MAXIMAL_POLYTOPES_REPS_FACETS

Alias for property MAXIMAL_CONES_REPS_FACETS.

Type:

SparseMatrix<Int,NonSymmetric>

N_MAXIMAL_POLYTOPES_ORBITS

Alias for property N_MAXIMAL_CONE_ORBITS.

Type:

Int

N_VERTICES_ORBITS

Alias for property N_RAY_ORBITS.

Type:

Int

POINTS_REPS

Alias for property INPUT_RAYS_REPS.

Type:

Matrix<Scalar,NonSymmetric>

POLYTOPES_ORBIT_SIZES

Alias for property CONES_ORBIT_SIZES.

Type:

Array<Array<Int>>

POLYTOPES_REPS

Alias for property CONES_REPS.

Type:

Array<Array<Set<Int>>>

REPS_AFFINE_SPAN_NORMALS

Alias for property REPS_LINEAR_SPAN_NORMALS.

Type:

Matrix<Scalar,NonSymmetric>

VERTICES_IMAGES

Alias for property RAYS_IMAGES.

Type:

Array<Array<Int>>

VERTICES_IN_ORBITS

Alias for property RAYS_IN_ORBITS.

Type:

Array<Set<Int>>

VERTICES_ORBIT_SIZES

Alias for property RAYS_ORBIT_SIZES.

Type:

Array<Int>

VERTICES_REPS

Alias for property RAYS_REPS.

Type:

Matrix<Scalar,NonSymmetric>

VERTICES_REPS_LABELS

Alias for property RAYS_REPS_LABELS.

Type:

Array<String>

These properties are for visualization.

POINT_LABELS

Alias for property INPUT_RAY_LABELS.

Type:

Array<String>

VERTEX_LABELS

Alias for property RAY_LABELS.

Type:

Array<String>

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

DIM()

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

Returns:

Int

polytope(Int i)

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

Parameters:

Int i

Returns:

Polytope

These methods are for visualization.

VISUAL_METRIC_TIGHT_SPAN()

This is a variation of VISUAL_BOUNDED_GRAPH for the special case of a tight span. The vertices are embedded according to the metric, the others are hung in between. This only produces meaningful results for extended tight spans produced from metrics, e.g. through metric_extended_tight_span.

Options:

Array<String> Taxa: Labels for the taxa of the metric.

Int seed: random seed value for the string embedder

String norm: which norm to use when calculating the distances between metric vectors (“max” or “square”)

Returns:

Visual::Graph

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

VISUAL()

Visualizes the polyhedral complex.

Options:

option list geometric_options

Returns:

Visual::PolyhedralFan