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

Equations that hold for all INPUT_RAYS of the cone. All vectors in this section must be non-zero. Input section only. Ask for LINEAR_SPAN if you want to see an irredundant description of the linear span.

Type: /common/property_types/Algebraic Types/Matrix
Inequalities giving rise to the cone; redundancies are allowed. All vectors in this section must be non-zero. Dual to INPUT_RAYS.

Input section only. Ask for FACETS if you want to compute an H-representation from a V-representation.

Type: /common/property_types/Algebraic Types/Matrix
(Non-homogenous) vectors whose linear span defines a subset of the lineality space of the cone; redundancies are allowed. All vectors in the input must be non-zero. Dual to EQUATIONS.

Input section only. Ask for LINEALITY_SPACE if you want to compute a V-representation from an H-representation.

Type: /common/property_types/Algebraic Types/Matrix
(Non-homogenous) vectors whose positive span form the cone; redundancies are allowed. Dual to INEQUALITIES. All vectors in the input must be non-zero.

Input section only. Ask for RAYS if you want to compute a V-representation from an H-representation.

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

Type: /common/property_types/Algebraic Types/Matrix
The vector counting the number of incidences between pairs of faces. `f<sub>ik</sub>` is the number of incident pairs of `(i+1)`-faces and `(k+1)`-faces. The main diagonal contains the F_VECTOR.

Type: /common/property_types/Basic Types/Bool
All intermediate polytopes (with respect to the given insertion order) in the beneath-and-beyond algorithm are simplicial. We have the implications: RAYS in general position ⇒ ESSENTIALLY_GENERIC ⇒ SIMPLICIAL

Type: /common/property_types/Basic Types/Int
Measures the deviation of the cone from being simple in terms of the GRAPH.

Type: /common/property_types/Algebraic Types/SparseMatrix
A matrix whose rows contain the cocircuit equations of P. The columns correspond to the MAX_INTERIOR_SIMPLICES.

Type: /common/property_types/Basic Types/Array
Number of incident facets for each ray.

Type: /common/property_types/Algebraic Types/Vector
Condensed form of the flag vector, containing all entries indexed by sparse sets in {0, …, COMBINATORIAL_DIM-1} in the following order:

     (1, f<sub>0</sub>, f<sub>1</sub>, f<sub>2</sub>, f<sub>02</sub>, f<sub>3</sub>, f<sub>03</sub>, f<sub>13</sub>, f<sub>4</sub>, f<sub>04</sub>, f<sub>14</sub>, f<sub>24</sub>, f<sub>024</sub>, f<sub>5</sub>, ...).

Use Dehn-Sommerville equations, via user function N_FLAGS, to extend.

Type: /common/property_types/Basic Types/Array
Number of incident rays for each facet.

Type: /common/property_types/Set Types/IncidenceMatrix
Ray-ridge incidence matrix, with rows corresponding to ridges and columns to rays. Rays and ridges are numbered from 0 to N_RAYS-1 rsp. N_RIDGES-1, according to their order in RAYS rsp. RIDGES.

Type: /common/property_types/Basic Types/Bool
True if the cone is simplicial.

Type: /common/property_types/Basic Types/Bool
True if the facets of the cone are simplicial.

Type: /common/property_types/Algebraic Types/SparseMatrix
A matrix whose rows contain the foldable cocircuit equations of P. The columns correspond to 2 * MAX_INTERIOR_SIMPLICES. col 0 = 0, col 1 = first simplex (black copy), col 2 = first simplex (white copy), col 3 = second simplex (black copy), …

Type: /graph/objects/Combinatorics/Graph
Vertex-edge graph obtained by intersecting the cone with a transversal hyperplane.

Type: /common/property_types/Set Types/IncidenceMatrix
Ray-facet incidence matrix, with rows corresponding to facets and columns to rays. Rays and facets are numbered from 0 to N_RAYS-1 rsp. N_FACETS-1, according to their order in RAYS rsp. FACETS.

Type: /common/property_types/Basic Types/Int
Number of pairs of incident vertices and facets.

Type: /common/property_types/Basic Types/Array
The (d-1)-dimensional simplices in the interior.

Type: /polytope/objects/Cone/properties/Combinatorics/HASSE_DIAGRAM
The face lattice of the cone organized as a directed graph. Top and bottom nodes represent the whole cone and the empty face. Every other node corresponds to some proper face of the cone.

Type: /common/property_types/Set Types/IncidenceMatrix
Transposed to RAYS_IN_FACETS. Notice that this is a temporary property; it will not be stored in any file.

Type: /common/property_types/Basic Types/Bool
True if the cone is self-dual.

Type: /common/property_types/Basic Types/Int
The number of RAYS

Type: /graph/objects/Combinatorics/Graph
Facet-ridge graph. Dual to GRAPH.

Type: /common/property_types/Basic Types/Array
The interior d-dimensional simplices of a cone of combinatorial dimension d

Type: /common/property_types/Basic Types/Int
Measures the deviation of the cone from being simple in terms of the DUAL_GRAPH.

Type: /common/property_types/Basic Types/Bool
True if the facets of the cone are simple. Dual to SIMPLICIAL.

Type: /common/property_types/Basic Types/Array
The boundary (d-1)-dimensional simplices of a cone of combinatorial dimension d

Type: /common/property_types/Basic Types/Int
Combinatorial dimension This is the dimension all combinatorial properties of the cone like e.g. RAYS_IN_FACETS or the HASSE_DIAGRAM refer to.

Geometrically, the combinatorial dimension is the dimension of the intersection of the pointed part of the cone with a hyperplane that creates a bounded intersection.

Type: /common/property_types/Algebraic Types/Vector
The vector counting the number of faces (`f<sub>k</sub>` is the number of `(k+1)`-faces).

These properties capture information that depends on the lattice structure of the cone. polymake always works with the integer lattice.

Type: /common/property_types/Algebraic Types/Vector
The coefficients of the Hilbert polynomial, the h^*-polynomial for lattice polytopes, with respect to the MONOID_GRADING starting at the constant coefficient. For lattice polytopes the length of this vector is CONE_DIM. In general the length is one less than the degree of the denominator of the HILBERT_SERIES.

Type: /common/property_types/Algebraic Types/Vector
A grading for the monoid given by the intersection of the cone with the lattice Z^d, should be positive for all generators.

If this property is not specified by the user there are two defaults:

   For rational polytopes the affine hyperplane defined by (1,0,\ldots,0) will be used.
   For [[/polytope/objects/Cone/properties/Lattice points in cones/HOMOGENEOUS]] cones the affine hyperplane containing the primitive generators 
   will be used.

Type: /common/property_types/Basic Types/Int
The number of elements of the HILBERT_BASIS.

Type: /common/property_types/Algebraic Types/Matrix
Elements of the HILBERT_BASIS for the cone of degree 1 with respect to the MONOID_GRADING.

Type: /common/property_types/Basic Types/Array
Generators for the HILBERT_BASIS of a posiibly non-pointed cone the first matrix is a Hilbert basis of a pointed part of the cone the second matrix is a lattice basis of the lineality space note: the pointed part used in this property need not be the same as the one described by RAYS or INPUT_RAYS

     it will be if the cone is pointed (the polytope is bounded)

Type: /common/property_types/Basic Types/Bool
A cone is Gorenstein if it is Q-Gorenstein with index one

Type: /common/property_types/Basic Types/Bool
A cone is smooth if the primitive generators are part of a lattice basis.

Type: /common/property_types/Basic Types/Bool
True if the primitive generators of the rays lie on an affine hyperplane in the span of the rays.

Type: /common/property_types/Algebraic Types/RationalFunction
Hilbert series of the monoid, given by the intersection of the cone with the lattice Z^d with respect to the MONOID_GRADING

Type: /common/property_types/Basic Types/Int
If a cone is Q-Gorenstein, then its index is the common lattice height of the primitive generators with respect to the origin. Otherwise Q_GORENSTEIN_CONE_INDEX is undefined.

Type: /common/property_types/Basic Types/Bool
A cone is Q-Gorenstein if all primitive generators of the cone lie in an affine hyperplane spanned by a lattice functional in the dual cone (but not in the lineality space of the dual cone).

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

Type: /common/property_types/Basic Types/Int
The number of EQUATIONS.

Type: /common/property_types/Algebraic Types/Matrix
Basis of the linear subspace orthogonal to all INEQUALITIES and EQUATIONS All vectors in this section must be non-zero. The property LINEALITY_SPACE appears only in conjunction with the property RAYS, or VERTICES, respectively. The specification of the property RAYS or VERTICES requires the specification of LINEALITY_SPACE, and vice versa.

Type: /common/property_types/Algebraic Types/Vector
A point in the relative interior of the cone.

Type: /common/property_types/Basic Types/Bool
True if the cone does not contain a non-trivial linear subspace.

Type: /common/property_types/Set Types/IncidenceMatrix
Input ray-facet incidence matrix, with rows corresponding to facet and columns to input rays. Input_rays and facets are numbered from 0 to N_INPUT_RAYS-1 rsp. N_FACETS-1, according to their order in INPUT_RAYS rsp. FACETS.

Type: /common/property_types/Basic Types/Int
The dimension of the space in which the cone lives.

Type: /common/property_types/Basic Types/Bool
True if the only valid point in the cone is the unique non-sensical point (0,…,0)

Type: /common/property_types/Basic Types/Int
Dimension of the LINEALITY_SPACE (>0 in the non-POINTED case)

Type: /common/property_types/Basic Types/Float
Threshold for zero test for scalar products (e.g. vertex * facet normal)

Type: /common/property_types/Algebraic Types/Matrix
Rays of the cone. No redundancies are allowed. All vectors in this section must be non-zero. The property RAYS appears only in conjunction with the property LINEALITY_SPACE. The specification of the property RAYS requires the specification of LINEALITY_SPACE, and vice versa.

Type: /common/property_types/Basic Types/Int
The number of INPUT_LINEALITY.

Type: /common/property_types/Basic Types/Int
Dimension of the linear span of the cone = dimension of the cone. If the cone is given purely combinatorially, this is the dimension of a minimal embedding space deduced from the combinatorial structure.

Type: /common/property_types/Set Types/IncidenceMatrix
Ray-inequality incidence matrix, with rows corresponding to facets and columns to rays. Rays and inequalities are numbered from 0 to N_RAYS-1 rsp. number of INEQUALITIES-1, according to their order in RAYS rsp. INEQUALITIES.

Type: /common/property_types/Basic Types/Bool
True if all RAYS of the cone have non-negative coordinates, that is, if the pointed part of the cone lies entirely in the positive orthant.

Type: /common/property_types/Basic Types/Int
The number of INPUT_RAYS.

Type: /common/property_types/Set Types/IncidenceMatrix
Transposed to INPUT_RAYS_IN_FACETS. Notice that this is a temporary property; it will not be stored in any file.

Type: /common/property_types/Algebraic Types/Matrix
Facets of the cone, encoded as inequalities. All vectors in this section must be non-zero. Dual to RAYS. This section is empty if and only if the cone is trivial (e.g. if it encodes an empty polytope). Notice that a polytope which is a single point defines a one-dimensional cone, the face at infinity is a facet. The property FACETS appears only in conjunction with the property LINEAR_SPAN, or AFFINE_HULL, respectively. The specification of the property FACETS requires the specification of LINEAR_SPAN, or AFFINE_HULL, respectively, and vice versa.

Type: /common/property_types/Set Types/IncidenceMatrix
transposed RAYS_IN_INEQUALITIES Notice that this is a temporary property; it will not be stored in any file.

Type: /common/property_types/Algebraic Types/Vector
A ray of a pointed cone.

Type: /common/property_types/Algebraic Types/Matrix
Dual basis of the linear span of the cone. All vectors in this section must be non-zero. The property LINEAR_SPAN appears only in conjunction with the property FACETS. The specification of the property FACETS requires the specification of LINEAR_SPAN, or AFFINE_HULL, respectively, and vice versa.

Type: /common/property_types/Basic Types/Bool
CONE_AMBIENT_DIM and CONE_DIM coincide. Notice that this makes sense also for the derived Polytope class.

Type: /common/property_types/Algebraic Types/Matrix
The i-th row is the normal vector of a hyperplane separating the i-th vertex from the others. This property is a by-product of redundant point elimination algorithm.

Type: /common/property_types/Basic Types/Int
The number of RIDGES.

Type: /common/property_types/Basic Types/Int
The number of FACETS.

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

Type: /polytope/objects/Cone/properties/Symmetry/GROUP
Properties of GROUP:
Symmetry
These properties capture information of the object that is concerned with the action of permutation groups.

• REPRESENTATIVE_MAX_INTERIOR_SIMPLICES
  Type: /common/property_types/Basic Types/Array
  One representative for each orbit of maximal-dimensional interior simplices
• REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES
  Type: /common/property_types/Basic Types/Array
  One representative for each orbit of interior ridge simplices
• BITSET_ACTION
  Type: /group/objects/ImplicitActionOnSets * REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES
  Type: /common/property_types/Basic Types/Array
  One representative for each orbit of maximal-dimensional boundary simplices
• MATRIX_ACTION
  Type: /polytope/objects/Cone/properties/Symmetry/GROUP/properties/Symmetry/MATRIX_ACTION
  Properties of MATRIX_ACTION:
  Symmetry
  These properties capture information of the object that is concerned with the action of permutation groups.
  • RAYS_ORBITS
    Type: /common/property_types/Basic Types/Array
    Alias for property VECTORS_ORBITS.
• REPRESENTATIVE_RAYS
  Type: /common/property_types/Algebraic Types/Matrix * REPRESENTATIVE_BOUNDARY_RIDGE_SIMPLICES
  Type: /common/property_types/Basic Types/Array
  One representative for each orbit of boundary ridge simplices
• REPRESENTATIVE_FACETS
  Type: /common/property_types/Algebraic Types/Matrix

—-

These properties are for visualization.

Type: /common/property_types/Basic Types/Array
Unique names assigned to the FACETS, analogous to RAY_LABELS.

Type: /common/property_types/Basic Types/Array
Unique names assigned to the INPUT_RAYS, analogous to RAY_LABELS.

Type: /common/property_types/Basic Types/Array
Reordered GRAPH. Dual to NEIGHBOR_FACETS_CYCLIC_NORMAL.

Type: /common/property_types/Basic Types/Array
Unique names assigned to the coordinate directions, analogous to RAY_LABELS. For Polytopes this should contain “inhomog_var” for the homogenization coordinate and this will be added automatically if necessary and CONE_AMBIENT_DIM can be computed.

Type: /common/property_types/Basic Types/Array
Unique names assigned to the RAYS. If specified, they are shown by visualization tools instead of ray indices.

For a cone built from scratch, you should create this property by yourself, either manually in a text editor, or with a client program. If you build a cone with a construction client taking some other input cone(s), you can create the labels automatically if you call the client with a relabel option. The exact format of the labels is dependent on the construction, and is described by the corresponding client.

Type: /common/property_types/Basic Types/Array
Reordered transposed RAYS_IN_FACETS. Dual to RIF_CYCLIC_NORMAL.

Type: /common/property_types/Basic Types/Array
Reordered DUAL_GRAPH for 3d-cones. The neighbor facets are listed in the order corresponding to RIF_CYCLIC_NORMAL, so that the first two vertices in RIF_CYCLIC_NORMAL make up the ridge to the first neighbor facet and so on.

Type: /common/property_types/Basic Types/Array
Unique names assigned to the INEQUALITIES, analogous to RAY_LABELS.

Type: /common/property_types/Basic Types/Array
Reordered RAYS_IN_FACETS for 2d and 3d-cones. Rays are listed in the order of their appearance when traversing the facet border counterclockwise seen from outside of the origin.

These properties collect information about triangulations of the object and properties usually computed from such, as the volume.

Type: /common/property_types/Basic Types/Array
Conceptually, similar to TRIANGULATION, but using INPUT_RAYS. However, here we use a small object type. The main reason for the existence of this property (in this form) is the beneath_beyond algorithm, which automatically produces this data as a by-product of the conversion from INPUT_RAYS to FACETS. And that data is too valuable to throw away. Use big objects of type VectorConfiguration if you want to work with triangulations using redundant points.

Type: /polytope/objects/Cone/properties/Triangulation and volume/TRIANGULATION
Some triangulation of the cone using only its RAYS.
Properties of TRIANGULATION:
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.

• REFINED_SPLITS
  Type: /common/property_types/Set Types/Set
  The splits that are coarsenings of the current TRIANGULATION.

If the triangulation is regular these form the unique split decomposition of the corresponding weight function.

• BOUNDARY
  Type: /polytope/objects/Cone/properties/Triangulation and volume/TRIANGULATION/properties/Combinatorics/BOUNDARY
  Augmented subobject BOUNDARY that triangulate it.

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

• WEIGHTS
  Type: /common/property_types/Algebraic Types/Vector
  Weight vector to construct a regular TRIANGULATION.

Must be generic.