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Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2
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 isRational
.- derived from:
- 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
Alias for property
INPUT_CONES
.- Type:
-
POINTS
Alias for property
INPUT_RAYS
.- Type:
Matrix<Scalar,NonSymmetric>
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:
- 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:
-
MAXIMAL_POLYTOPES_COMBINATORIAL_DIMS
Alias for property
MAXIMAL_CONES_COMBINATORIAL_DIMS
.- Type:
-
MAXIMAL_POLYTOPES_INCIDENCES
Array of incidence matrices of all maximal polytopes. Alias for property
MAXIMAL_CONES_INCIDENCES
.- Type:
-
MAXIMAL_POLYTOPES_THRU_VERTICES
Alias for property
MAXIMAL_CONES_THRU_RAYS
.- Type:
-
N_MAXIMAL_POLYTOPES
Number of
MAXIMAL_POLYTOPES
. Alias for propertyN_MAXIMAL_CONES
.- Type:
-
N_POLYTOPES
Alias for property
N_CONES
.- Type:
-
POLYTOPES
Alias for property
CONES
.- Type:
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
Alias for property
LINEAR_SPAN_NORMALS
.- Type:
Matrix<Scalar,NonSymmetric>
-
BOUNDED
True if each object in
MAXIMAL_POLYTOPES
is bounded.- Type:
-
FAR_VERTICES
Indices of vertices that are rays.
- Type:
-
MAXIMAL_POLYTOPES_AFFINE_HULL_NORMALS
Alias for property
MAXIMAL_CONES_LINEAR_SPAN_NORMALS
.- Type:
-
MAXIMAL_POLYTOPES_FACETS
Alias for property
MAXIMAL_CONES_FACETS
.- Type:
-
N_POINTS
Number of
POINTS
. Alias for propertyN_INPUT_RAYS
.- Type:
-
N_VERTICES
- Type:
-
VERTICES
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:
- Type:
- Properties of GROUP:
-
COORDINATE_ACTION
- Type:
- Properties of COORDINATE_ACTION:
-
N_POINTS_GENERATORS
Alias for property
N_INPUT_RAYS_GENERATORS
.- Type:
-
N_VERTICES_GENERATORS
Alias for property
N_RAYS_GENERATORS
.- Type:
-
POINTS_GENERATORS
Alias for property
INPUT_RAYS_GENERATORS
.- Type:
-
VERTICES_GENERATORS
Alias for property
RAYS_GENERATORS
.- Type:
-
-
INPUT_POLYTOPES_ACTION
Alias for property
INPUT_CONES_ACTION
.- Type:
-
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:
- Type:
- Properties of MAXIMAL_POLYTOPES_ACTION:
-
MAXIMAL_POLYTOPES_GENERATORS
Alias for property
MAXIMAL_CONES_GENERATORS
.- Type:
-
N_POINTS_GENERATORS
Alias for property
N_INPUT_RAYS_GENERATORS
.- Type:
-
N_VERTICES_GENERATORS
Alias for property
N_RAYS_GENERATORS
.- Type:
-
POINTS_GENERATORS
Alias for property
INPUT_RAYS_GENERATORS
.- Type:
-
VERTICES_GENERATORS
Alias for property
RAYS_GENERATORS
.- Type:
-
-
POINTS_ACTION
Alias for property
INPUT_RAYS_ACTION
.- Type:
-
REPRESENTATIVE_VERTICES
Alias for property
REPRESENTATIVE_RAYS
.- Type:
Matrix<Scalar,NonSymmetric>
-
VERTICES_ACTION
Alias for property
RAYS_ACTION
.- Type:
-
-
INPUT_POLYTOPES_REPS
Alias for property
INPUT_CONES_REPS
.- Type:
-
MAXIMAL_POLYTOPES_IN_ORBITS
Alias for property
MAXIMAL_CONES_IN_ORBITS
.- Type:
-
MAXIMAL_POLYTOPES_ORBIT_SIZES
Alias for property
MAXIMAL_CONES_ORBIT_SIZES
.- Type:
-
MAXIMAL_POLYTOPES_REPS
Alias for property
MAXIMAL_CONES_REPS
.- Type:
-
MAXIMAL_POLYTOPES_REPS_AFFINE_SPAN_NORMALS
Alias for property
MAXIMAL_CONES_REPS_LINEAR_SPAN_NORMALS
.- Type:
-
MAXIMAL_POLYTOPES_REPS_DIMS
Alias for property
MAXIMAL_CONES_REPS_DIMS
.- Type:
-
MAXIMAL_POLYTOPES_REPS_FACETS
Alias for property
MAXIMAL_CONES_REPS_FACETS
.- Type:
-
N_MAXIMAL_POLYTOPES_ORBITS
Alias for property
N_MAXIMAL_CONE_ORBITS
.- Type:
-
N_VERTICES_ORBITS
Alias for property
N_RAY_ORBITS
.- Type:
-
POINTS_REPS
Alias for property
INPUT_RAYS_REPS
.- Type:
Matrix<Scalar,NonSymmetric>
-
POLYTOPES_ORBIT_SIZES
Alias for property
CONES_ORBIT_SIZES
.- Type:
-
POLYTOPES_REPS
Alias for property
CONES_REPS
.- Type:
-
REPS_AFFINE_SPAN_NORMALS
Alias for property
REPS_LINEAR_SPAN_NORMALS
.- Type:
Matrix<Scalar,NonSymmetric>
-
VERTICES_IMAGES
Alias for property
RAYS_IMAGES
.- Type:
-
VERTICES_IN_ORBITS
Alias for property
RAYS_IN_ORBITS
.- Type:
-
VERTICES_ORBIT_SIZES
Alias for property
RAYS_ORBIT_SIZES
.- Type:
-
VERTICES_REPS
Alias for property
RAYS_REPS
.- Type:
Matrix<Scalar,NonSymmetric>
-
VERTICES_REPS_LABELS
Alias for property
RAYS_REPS_LABELS
.- Type:
Visualization
These properties are for visualization.
-
POINT_LABELS
Alias for property
INPUT_RAY_LABELS
.- Type:
-
VERTEX_LABELS
Alias for property
RAY_LABELS
.- Type:
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:
-
DIM()
Returns the dimension of the linear space spanned by the complex.
- Returns:
Visualization
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. throughmetric_extended_tight_span
.
-
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:
no category
-
VISUAL()
Visualizes the polyhedral complex.
- Options:
- option list
geometric_options
- Returns: