# application: polytope

This is the historically first application, and the largest one.

It deals with convex pointed polyhedra. It allows to define a polyhedron either as a convex hull of a point set, an intersection of halfspaces, or as an incidence matrix without any embedding. Then you can ask for a plenty of its (especially combinatorial) properties, construct new polyhedra by modifying it, or study the behavior of the objective functions.

There is a wide range of visualization methods for polyhedra, even for dimensions > 4 and purely combinatorial descriptions, including interfaces to interactive geometry viewers (such as JavaView or geomview), generating PostScript drawings and povray scene files.

**imports from:**common, graph

**uses:**group, ideal, topaz

**Objects**

**Category:**Geometrya lattice that is displaced from the origin, i.e., a set of the form x + L, where x is a non-zero vector and L a (linear) lattice

#### Properties of AffineLattice

A polyhedral cone, not necessarily pointed. Note that in contrast to the vertices of a polytope, the RAYS are given in affine coordinates.

#### Specializations of Cone

#### Properties of Cone

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

**COCIRCUIT_EQUATIONS**: common::SparseMatrix<Rational, NonSymmetric>A matrix whose rows contain the cocircuit equations of P. The columns correspond to the MAX_INTERIOR_SIMPLICES.

Contained in extension`bundled:group`

.**COMBINATORIAL_DIM**: common::IntCombinatorial 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.

**ESSENTIALLY_GENERIC**: common::BoolAll 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

**F2_VECTOR**: common::Matrix<Integer, NonSymmetric>The vector counting the number of incidences between pairs of faces. `f

_{ik}` is the number of incident pairs of `(i+1)`-faces and `(k+1)`-faces. The main diagonal contains the F_VECTOR.**FACETS_THRU_RAYS**: common::IncidenceMatrix<NonSymmetric>Transposed to RAYS_IN_FACETS. Notice that this is a temporary property; it will not be stored in any file.

**FLAG_VECTOR**: common::Vector<Integer>Condensed form of the flag vector, containing all entries indexed by sparse sets in {0, ..., COMBINATORIAL_DIM-1} in the following order: (1, f

_{0}, f_{1}, f_{2}, f_{02}, f_{3}, f_{03}, f_{13}, f_{4}, f_{04}, f_{14}, f_{24}, f_{024}, f_{5}, ...). Use Dehn-Sommerville equations, via user function N_FLAGS, to extend.**FOLDABLE_COCIRCUIT_EQUATIONS**: common::SparseMatrix<Rational, NonSymmetric>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), ...

Contained in extension`bundled:group`

.**F_VECTOR**: common::Vector<Integer>The vector counting the number of faces (`f

_{k}` is the number of `(k+1)`-faces).**GRAPH**: graph::Graph<Undirected>Vertex-edge graph obtained by intersecting the cone with a transversal hyperplane.

**INTERIOR_RIDGE_SIMPLICES**: common::Array<Set<Int>>The (

*d*-1)-dimensional simplices in the interior.Contained in extension`bundled:group`

.**MAX_BOUNDARY_SIMPLICES**: common::Array<Set<Int>>The boundary (

*d*-1)-dimensional simplices of a cone of combinatorial dimension*d*Contained in extension`bundled:group`

.**MAX_INTERIOR_SIMPLICES**: common::Array<Set<Int>>The interior

*d*-dimensional simplices of a cone of combinatorial dimension*d*Contained in extension`bundled:group`

.

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

**CONE_DIM**: common::IntDimension 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.

**EPSILON**: common::Float**Only defined for Cone<Float>**Threshold for zero test for scalar products (e.g. vertex * facet normal)

**FACETS**: common::MatrixFacets 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.

**FACETS_THRU_INPUT_RAYS**: common::IncidenceMatrix<NonSymmetric>Transposed to INPUT_RAYS_IN_FACETS. Notice that this is a temporary property; it will not be stored in any file.

**FULL_DIM**: common::BoolCONE_AMBIENT_DIM and CONE_DIM coincide. Notice that this makes sense also for the derived Polytope class.

**INEQUALITIES_THRU_RAYS**: common::IncidenceMatrix<NonSymmetric>transposed RAYS_IN_INEQUALITIES Notice that this is a temporary property; it will not be stored in any file.

**INPUT_RAYS_IN_FACETS**: common::IncidenceMatrix<NonSymmetric>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.

**LINEALITY_SPACE**: common::MatrixBasis 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.

**LINEAR_SPAN**: common::MatrixDual 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.

**POSITIVE**: common::BoolTrue 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.

**RAYS**: common::MatrixRays 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.

**RAYS_IN_INEQUALITIES**: common::IncidenceMatrix<NonSymmetric>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.

**RAY_SEPARATORS**: common::MatrixThe 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.

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

**EQUATIONS**: common::MatrixEquations 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.

**INEQUALITIES**: common::MatrixInequalities 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.

**INPUT_LINEALITY**: common::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.

**INPUT_RAYS**: common::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 information that depends on the lattice structure of the cone. polymake always works with the integer lattice.

**DEGREE_ONE_GENERATORS**: common::Matrix<Integer, NonSymmetric>**Only defined for Cone<Rational>**Elements of the HILBERT_BASIS for the cone of degree 1 with respect to the MONOID_GRADING.

**GORENSTEIN_CONE**: common::Bool**Only defined for Cone<Rational>**A cone is Gorenstein if it is Q-Gorenstein with index one

**HILBERT_BASIS_GENERATORS**: common::Array<Matrix<Integer, NonSymmetric>>**Only defined for Cone<Rational>**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)

**Depends on:**4ti2 (unbounded) or libnormaliz (bounded)**HILBERT_SERIES**: common::RationalFunction<Rational, Int>**Only defined for Cone<Rational>**Hilbert series of the monoid, given by the intersection of the cone with the lattice Z^d with respect to the MONOID_GRADING

**Depends on:**libnormaliz**HOMOGENEOUS**: common::Bool**Only defined for Cone<Rational>**True if the primitive generators of the rays lie on an affine hyperplane in the span of the rays.

**H_STAR_VECTOR**: common::Vector<Integer>**Only defined for Cone<Rational>**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.

**Depends on:**latte or libnormaliz**MONOID_GRADING**: common::Vector<Integer>**Only defined for Cone<Rational>**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 HOMOGENEOUS cones the affine hyperplane containing the primitive generators will be used.

**N_HILBERT_BASIS**: common::Int**Only defined for Cone<Rational>**The number of elements of the HILBERT_BASIS.

**Q_GORENSTEIN_CONE**: common::Bool**Only defined for Cone<Rational>**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).**Q_GORENSTEIN_CONE_INDEX**: common::Int**Only defined for Cone<Rational>**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.

**SMOOTH_CONE**: common::Bool**Only defined for Cone<Rational>**A cone is

*smooth*if the primitive generators are part of a lattice basis.

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

**TRIANGULATION**: topaz::GeometricSimplicialComplexSome triangulation of the cone using only its RAYS.

#### Properties of TRIANGULATION

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

**BOUNDARY**: topaz::SimplicialComplexAmendment of topaz::SimplicialComplex::BOUNDARY for Cone::TRIANGULATION::BOUNDARY

#### Properties of BOUNDARY

**FACET_TRIANGULATIONS**: common::Array<Set<Int>>For each facet the set of simplex indices of BOUNDARY that triangulate it.

**REFINED_SPLITS**: common::Set<Int>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.

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

**TRIANGULATION_INT**: common::Array<Set<Int>>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.

These properties are for visualization.

**COORDINATE_LABELS**: common::Array<String>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.

**FTR_CYCLIC_NORMAL**: common::Array<Array<Int>>Reordered transposed RAYS_IN_FACETS. Dual to RIF_CYCLIC_NORMAL.

**INEQUALITY_LABELS**: common::Array<String>Unique names assigned to the INEQUALITIES, analogous to RAY_LABELS.

**INPUT_RAY_LABELS**: common::Array<String>Unique names assigned to the INPUT_RAYS, analogous to RAY_LABELS.

**NEIGHBOR_FACETS_CYCLIC_NORMAL**: common::Array<Array<Int>>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.

**NEIGHBOR_RAYS_CYCLIC_NORMAL**: common::Array<Array<Int>>Reordered GRAPH. Dual to NEIGHBOR_FACETS_CYCLIC_NORMAL.

**RAY_LABELS**: common::Array<String>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.**RIF_CYCLIC_NORMAL**: common::Array<Array<Int>>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.

#### User Methods of Cone

These methods are provided for backward compatibility with older versions of polymake only. They should not be used in new code.

**DUAL_DIAMETER**()The diameter of the DUAL_GRAPH

**DUAL_TRIANGLE_FREE**()True if the DUAL_GRAPH contains no triangle

**TRIANGLE_FREE**()True if the GRAPH contains no triangle

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

**CONNECTIVITY**()**DUAL_CONNECTIVITY**()Connectivity of the DUAL_GRAPH this is the minimum number of nodes that have to be removed from the DUAL_GRAPH to make it disconnected

**DUAL_EVEN**()True if the DUAL_GRAPH is bipartite

**face**()For a given set of vertices compute the smallest face F containing them all. Returns a Pair<Set,Set> where the first is the set of vertices of F, while the second is the set of facets containing F. Example: $c=cube(3); print rank($c->VERTICES->minor($c->face([0,1])->first(),All))-1; computes the dimension of the face of the 3-cube which is spanned by the vertices 0 and 1.

**FACET_DEGREES**() → Vector<Int>Facet degrees of the polytope. The

*degree*of a facet is the number of adjacent facets.##### Returns

Vector<Int> - in the same order as FACETS**N_FLAGS**(type ...)Determine the number of flags of a given type.

*type*must belong to {0,...,COMBINATORIAL_DIM-1}. Example: "N_FLAGS(0,3,4)" determines the entry f_{034}of the flag vector.##### Parameters

Int type ... flag type**VERTEX_DEGREES**() → Vector<Int>

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 of the cone

**DIM**()returns the geometric dimension of the cone (including the lineality space) for the dimension of the pointed part ask for COMBINATORIAL_DIM

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

**HILBERT_BASIS**() → Matrix<Integer>**Only defined for Cone<Rational>**for a cone this method returns a Hilbert basis of the cone for a polytope this method returns a Hilbert basis of the homogenization cone of the polytope note: if the cone is not pointed (the polytope is not bounded) then the returned basis is not unique and usually not minimal

##### Returns

Matrix<Integer>

The following methods compute topological invariants.

**DUAL_GRAPH_SIGNATURE**()Difference of the black and white nodes if the DUAL_GRAPH is BIPARTITE. Otherwise -1.

#### Permutations of Cone

**FacetPerm**UNDOCUMENTED

#### Properties of FacetPerm

**VertexPerm**UNDOCUMENTED

#### Properties of VertexPerm

**Category:**Lattice points in conesThe Groebner basis of the homogeneous toric ideal associated to the polytope, the term order is given in matrix form.

#### Properties of GroebnerBasis

**TERM_ORDER_MATRIX**: common::Matrix<Integer, NonSymmetric>The term order in matrix form; if not square, then a tie breaker is used.

**TERM_ORDER_NAME**: common::StringA term order by name; allowed acronyms are

`lex`

,`deglex`

and`degrevlex`

.

**Category:**OptimizationA linear program specified by a linear or abstract objective function

#### Properties of LinearProgram

**ABSTRACT_OBJECTIVE**: common::VectorAbstract objective function. Defines a direction for each edge such that each non-empty face has a unique source and a unique sink. The i-th element is the value of the objective function at vertex number i. Only defined for bounded polytopes.

**Example:**- The following creates a new LinearProgram object and assigns an abstract objective to it:
`> $l = cube(2)->LP(ABSTRACT_OBJECTIVE=>[1,2,3,4]);`

`> print $l->ABSTRACT_OBJECTIVE;`

`1 2 3 4`

**DIRECTED_BOUNDED_GRAPH**: graph::Graph<Directed>Subgraph of BOUNDED_GRAPH. Consists only of directed arcs along which the value of the objective function increases.

**DIRECTED_GRAPH**: graph::Graph<Directed>Subgraph of Polytope::GRAPH. Consists only of directed arcs along which the value of the objective function increases.

**Example:**- The following defines a LinearProgram together with a linear objective for the centered square with side length 2. The directed graph according to the linear objective is stored in a new variable and the corresponding edges are printend.
`> $c = new Vector([0, 1, 0]);`

`> $p = cube(2);`

`> $p->LP(LINEAR_OBJECTIVE=>$c);`

`> $g = $p->LP->DIRECTED_GRAPH;`

`> print $g->EDGES;`

`{0 1}`

`{2 3}`

**LINEAR_OBJECTIVE**: common::VectorLinear objective function. In d-space a linear objective function is given by a (d+1)-vector. The first coordinate specifies a constant that is added to the resulting value.

**Example:**- The following creates a new LinearProgram object and assigns a linear objective to it:
`> $l = cube(2)->LP(LINEAR_OBJECTIVE=>[0,1,1]);`

`> print $l->LINEAR_OBJECTIVE;`

`0 1 1`

**MAXIMAL_FACE**: common::Set<Int>Indices of vertices at which the maximum of the objective function is attained.

**Example:**- The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for the maximal face:
`> $c = new Vector([0, 1, 0]);`

`> $p = cube(2);`

`> $p->LP(LINEAR_OBJECTIVE=>$c);`

`> print $p->LP->MAXIMAL_FACE;`

`{1 3}`

**MAXIMAL_VALUE**: LinearProgram::ScalarMaximum value of the objective function. Negated if linear problem is unbounded.

**Examples:**- The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for the maximal value:
`> $c = new Vector([0, 1, 0]);`

`> $p = cube(2);`

`> $p->LP(LINEAR_OBJECTIVE=>$c);`

`> print $p->LP->MAXIMAL_VALUE;`

`1`

- The following defines a LinearProgram together with a linear objective with bias 3 for the centered square with side length 4 and asks for the maximal value:
`> $c = new Vector([3, 1, 0]);`

`> $p = cube(2,2);`

`> $p->LP(LINEAR_OBJECTIVE=>$c);`

`> print $p->LP->MAXIMAL_VALUE;`

`5`

- The following defines a LinearProgram together with a linear objective for the positive quadrant (unbounded) and asks for the maximal value:
`> $c = new Vector([0, 1, 1]);`

`> $p = facet_to_infinity(simplex(2),0);`

`> $p->LP(LINEAR_OBJECTIVE=>$c);`

`> print $p->LP->MAXIMAL_VALUE;`

`inf`

**MAXIMAL_VERTEX**: common::VectorCoordinates of a (possibly not unique) affine vertex at which the maximum of the objective function is attained.

**Example:**- The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for a maximal vertex:
`> $c = new Vector([0, 1, 0]);`

`> $p = cube(2);`

`> $p->LP(LINEAR_OBJECTIVE=>$c);`

`> print $p->LP->MAXIMAL_VERTEX;`

`1 1 -1`

**MINIMAL_FACE**: common::Set<Int>Similar to MAXIMAL_FACE.

**Example:**- The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for the minimal face:
`> $c = new Vector([0, 1, 0]);`

`> $p = cube(2);`

`> $p->LP(LINEAR_OBJECTIVE=>$c);`

`> print $p->LP->MINIMAL_FACE;`

`{0 2}`

**MINIMAL_VALUE**: LinearProgram::ScalarSimilar to MAXIMAL_VALUE.

**Examples:**- The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for the minimal value:
`> $c = new Vector([0, 1, 0]);`

`> $p = cube(2);`

`> $p->LP(LINEAR_OBJECTIVE=>$c);`

`> print $p->LP->MINIMAL_VALUE;`

`-1`

- The following defines a LinearProgram together with a linear objective with bias 3 for the centered square with side length 4 and asks for the minimal value:
`> $c = new Vector([3, 1, 0]);`

`> $p = cube(2,2);`

`> $p->LP(LINEAR_OBJECTIVE=>$c);`

`> print $p->LP->MINIMAL_VALUE;`

`1`

**MINIMAL_VERTEX**: common::VectorSimilar to MAXIMAL_VERTEX.

**Example:**- The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for a minimal vertex:
`> $c = new Vector([0, 1, 0]);`

`> $p = cube(2);`

`> $p->LP(LINEAR_OBJECTIVE=>$c);`

`> print $p->LP->MINIMAL_VERTEX;`

`1 -1 -1`

**RANDOM_EDGE_EPL**: common::Vector<Rational>Expected average path length for a simplex algorithm employing "random edge" pivoting strategy.

#### User Methods of LinearProgram

**VERTEX_IN_DEGREES**()Array of in-degrees for all nodes of DIRECTED_GRAPH or numbers of objective decreasing edges at each vertex

**VERTEX_OUT_DEGREES**()Array of out-degrees for all nodes of DIRECTED_GRAPH or numbers of objective increasing edges at each vertex

**Category:**SymmetryA symmetric polytope defined as the convex hull of the orbit of a single point under a permutation group acting on coordinates.

**derived from:**SymmetricPolytope##### Type Parameters

Scalar default: Rational#### Properties of OrbitPolytope

**CP_INDICES**: common::Set<Int>The row indices of all core points among the REPRESENTATIVE_CERTIFIERS.

**NOP_GRAPH**: graph::Graph<Directed>The NOP-graph of GEN_POINT with respect to the GENERATING_GROUP. The nodes of the NOP-graph correspond to the REPRESENTATIVE_CERTIFIERS, which represent the different orbit polytopes contained in the given orbit polytope.

**REPRESENTATIVE_CERTIFIERS**: common::MatrixA matrix of representatives of all certifiers for GEN_POINT with respect to the GENERATING_GROUP. A certifier is an integer point in the given orbit polytope. Note that the representative certifiers must be in the same order as the corresponding nodes in the NOP_GRAPH. Further, the CP_INDICES refer to row indices of this property.

**REPRESENTATIVE_CORE_POINTS**: common::MatrixA matrix of representatives of all core points in the given orbit polytope. A core point is an integer point whose orbit polytope is lattice-free (i.e. does not contain integer points besides its vertices).

The POINTS of an object of type PointConfiguration encode a not necessarily convex finite point set. The difference to a parent VectorConfiguration is that the points have homogeneous coordinates, i.e. they will be normalized to have first coordinate 1 without warning.

**derived from:**VectorConfiguration##### Type Parameters

Scalar default: Rational#### Specializations of PointConfiguration

#### Properties of PointConfiguration

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

**COCIRCUIT_EQUATIONS**: common::SparseMatrix<Rational, NonSymmetric>Tells the cocircuit equations that hold for the configuration, one for each interior ridge

Contained in extension`bundled:group`

.**GRAPH**: graph::Graph<Undirected>Graph of the point configuration. Two points are adjacent if they lie in a common edge of the CONVEX_HULL.

**INTERIOR_RIDGE_SIMPLICES**: common::Array<Set<Int>>Tells the number of codimension 1 simplices that are not on the boundary

Contained in extension`bundled:group`

.**MAX_BOUNDARY_SIMPLICES**: common::Array<Set<Int>>Tells the full-dimensional simplices on the boundary that contain no points except for the vertices.

Contained in extension`bundled:group`

.**MAX_INTERIOR_SIMPLICES**: common::Array<Set<Int>>Tells the full-dimensional simplices that contain no points except for the vertices.

Contained in extension`bundled:group`

.**N_MAX_BOUNDARY_SIMPLICES**: common::IntTells the number of MAX_BOUNDARY_SIMPLICES

Contained in extension`bundled:group`

.**N_MAX_INTERIOR_SIMPLICES**: common::IntTells the number of MAX_INTERIOR_SIMPLICES

Contained in extension`bundled:group`

.**SIMPLEXITY_LOWER_BOUND**: common::IntA lower bound for the minimal number of simplices in a triangulation

Contained in extension`bundled:group`

.**SPLITS**: common::MatrixThe splits of the point configuration, i.e., hyperplanes cutting the configuration in two parts such that we have a regular subdivision.

**SPLIT_COMPATIBILITY_GRAPH**: graph::Graph<Undirected>Two SPLITS are compatible if the defining hyperplanes do not intersect in the interior of the point configuration. This defines a graph.

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

**CONVEX_HULL**: PolytopeThe polytope being the convex hull of the point configuration.

#### Properties of CONVEX_HULL

**MULTIPLE_POINTS**: common::BoolTells if multiple points exist. Alias for property VectorConfiguration::MULTIPLE_VECTORS.

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

**POINTS**: common::MatrixThe points of the configuration. Multiples allowed. Alias for property VectorConfiguration::VECTORS.

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

**TRIANGULATION**: topaz::GeometricSimplicialComplexSome triangulation of the point configuration.

#### Properties of TRIANGULATION

**BOUNDARY**: topaz::SimplicialComplexAmendment of topaz::SimplicialComplex::BOUNDARY for PointConfiguration::TRIANGULATION::BOUNDARY

#### Properties of BOUNDARY

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

**FACET_TRIANGULATIONS**: common::Array<Set<Int>>DOC_FIXME: Incomprehensible description! For each facet the set of simplex indices of BOUNDARY that triangulate it.

**GKZ_VECTOR**: common::VectorGKZ-vector

See Chapter 7 in Gelfand, Kapranov, and Zelevinsky:Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994**REFINED_SPLITS**: common::Set<Int>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.

These properties are for visualization.

**PIF_CYCLIC_NORMAL**: common::Array<Array<Int>>Polytope::VIF_CYCLIC_NORMAL of the CONVEX_HULL, but with the indices form POINTS instead of Polytope::VERTICES

**POINT_LABELS**: common::Array<String>Unique names assigned to the POINTS. Alias for property VectorConfiguration::LABELS.

#### User Methods of PointConfiguration

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**()Ambient dimension of the point configuration (without the homogenization coordinate). Similar to Polytope::AMBIENT_DIM.

**DIM**()Affine dimension of the point configuration. Similar to Polytope::DIM.

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

**TRIANGULATION_SIGNS**() → Array<Int>For each simplex in the TRIANGULATION, this contains the sign of the determinant of its coordinate matrix, which is the orientation of the simplex.

##### Returns

Array<Int>

These methods are for visualization.

**VISUAL**() → Visual::PointConfigurationVisualize a point configuration.

**VISUAL_POINTS**() → Visual::ObjectVisualize the POINTS of a point configuration.

Not necessarily bounded or unbounded polyhedron. Nonetheless, the name "Polytope" is used for two reasons: Firstly, combinatorially we always deal with polytopes; see the description of VERTICES_IN_FACETS for details. The second reason is historical. We use homogeneous coordinates, which is why Polytope is derived from Cone. Note that a pointed polyhedron is projectively equivalent to a polytope.

*Scalar*is the numeric data type used for the coordinates.**derived from:**Cone#### Specializations of Polytope

**Polytope<Float>**A pointed polyhedron with float coordinates realized in R

^{d}.It mainly exists for visualization.

Convex hull and related algorithms use floating-point arithmetics. Due to numerical errors inherent to this kind of computations, the resulting combinatorial description can be arbitrarily far away from the truth, or even not correspond to any valid polytope. You have been warned.

None of the standard construction clients produces objects of this type. If you want to get one, create it with the explicit constructor or convert_to.

#### Properties of Polytope

**ALTSHULER_DET**: common::IntegerLet M be the vertex-facet incidence matrix, then the Altshuler determinant is defined as max{det(M ∗ M

^{T}), det(M^{T}∗ M)}.**Example:**- This prints the Altshuler determinant of the built-in pentagonal pyramid (Johnson solid 2):
`> print johnson_solid("pentagonal_pyramid")->ALTSHULER_DET;`

`25`

**BALANCE**: common::IntMaximal dimension in which all facets are balanced.

**Example:**- The following full dimensional polytope given by 10 specific vertices on the 8-dimensional sphere is 3-neighborly. Hence the dual polytope is 3-balanced, where we first center and then polarize it.
`> $p = rand_sphere(8,10,seed=>8866463);`

`> $q = polarize(center($p));`

`> print $q->BALANCE;`

`3`

**BALANCED**: common::BoolDual to NEIGHBORLY.

**Example:**- Since cyclic polytopes generated by vertices on the moment curve are neighborly, their dual polytopes are balanced. The following checks this for the 4-dimensional case by centering the cyclic polytope and then polarizing it:
`> $p = cyclic(4,6);`

`> $q = polarize(center($p));`

`> print $q->BALANCED;`

`1`

**COCUBICAL**: common::BoolDual to CUBICAL.

**Example:**- Since the cross-polytope is dual to a cube of same dimension, it is cocubical. The following checks this for the 3-dimensional case:
`> print cross(3)->COCUBICAL;`

`1`

**COCUBICALITY**: common::IntDual to CUBICALITY.

**Example:**- After stacking a facet of the 3-dimensional cube, its cubicality is lowered to 2. Hence its dual polytope has cocubicality 2 as well. The following produces such a stacked cube and asks for its cocubicality after polarization:
`> $p = stack(cube(3),5);`

`> print polarize($p)->COCUBICALITY;`

`2`

**CUBICAL**: common::BoolTrue if all facets are cubes.

**Examples:**- A k-dimensional cube has k-1-dimensional cubes as facets and is therefore cubical. The following checks if this holds for the 3-dimensional case:
`> print cube(3)->CUBICAL;`

`1`

- This checks if a zonotope generated by 4 random points on the 3-dimensional sphere is cubical, which is always the case.
`> print zonotope(rand_sphere(3,4)->VERTICES)->CUBICAL;`

`1`

**CUBICALITY**: common::IntMaximal dimension in which all facets are cubes.

**Example:**- We will modify the 3-dimensional cube in two different ways. While stacking some facets (in this case facets 4 and 5) preserves the cubicality up to dimension 2, truncating an arbitrary vertex reduces the cubicality to 1.
`> print stack(cube(3),[4,5])->CUBICALITY;`

`2`

`> print truncation(cube(3),5)->CUBICALITY;`

`1`

**DUAL_BOUNDED_H_VECTOR**: common::Vector<Integer>h-vector of the bounded subcomplex, defined for not necessarily bounded polyhedra which are simple (as polyhedra, i.e., VERTEX_DEGREES on the FAR_FACE do not matter). Coincides with the reverse h-vector of the dual simplicial ball. Note that this vector will usually start with a number of zero entries.

**DUAL_H_VECTOR**: common::Vector<Integer>dual h-vector, defined via recursion on the face lattice of a polytope. Coincides for simple polytopes with the combinatorial definition of the h-vector via abstract objective functions.

**EDGE_ORIENTABLE**: common::BoolTrue if there exists an edge-orientation (see EDGE_ORIENTATION for a definition). The polytope is required to be 2-cubical.

**Examples:**- The following checks a 3-dimensional cube for edge orientability:
`> $p = cube(3);`

`> print $p->EDGE_ORIENTABLE;`

`1`

- A 3-dimensinal cube with one stacked facet is still 2-cubical. Therefore we can check for edge orientability:
`> $p = stack(cube(3),5);`

`> print $p->EDGE_ORIENTABLE;`

`1`

**EDGE_ORIENTATION**: common::Matrix<Int, NonSymmetric>List of all edges with orientation, such that for each 2-face the opposite edges point in the same direction. Each line is of the form (u v), which indicates that the edge {u,v} is oriented from u to v. The polytope is required to be 2-cubical.

**Example:**- The following prints a list of oriented edges of a 2-dimensional cube such that opposing edges have the same orientation:
`> $p = cube(2);`

`> print $p->EDGE_ORIENTATION;`

`0 2`

`1 3`

`0 1`

`2 3`

**F2_VECTOR**: common::Matrix<Integer, NonSymmetric>f

_{ik}is the number of incident pairs of i-faces and k-faces; the main diagonal contains the F_VECTOR.**Example:**- The following prints the f2-vector of a 3-dimensional cube: print cube(3)->F2_VECTOR;
`8 24 24`

`24 12 24`

`24 24 6`

**FACETS_THRU_VERTICES**: common::IncidenceMatrix<NonSymmetric>transposed VERTICES_IN_FACETS Notice that this is a temporary property; it will not be stored in any file. Alias for property Cone::FACETS_THRU_RAYS.

**FACE_SIMPLICITY**: common::IntMaximal dimension in which all faces are simple polytopes. This checks the 3-dimensional cube for face simplicity. Since the cube is dual to the cross-polytope of equal dimension and it is simplicial, the result is 3. > print cube(3)->SIMPLICITY; | 3

**FOLDABLE_MAX_SIGNATURE_UPPER_BOUND**: common::IntAn upper bound for the maximal signature of a foldable triangulation of a polytope The signature is the absolute difference of the normalized volumes of black minus white maximal simplices, where only odd normalized volumes are taken into account.

Contained in extension`bundled:group`

.**F_VECTOR**: common::Vector<Integer>f

_{k}is the number of k-faces.**Examples:**- This prints the f-vector of a 3-dimensional cube. The first entry represents the vertices.
`> print cube(3)->F_VECTOR;`

`[8, 12, 6]`

- This prints the f-vector of the 3-dimensional cross-polytope. Since the cube and the cross polytope of equal dimension are dual, their f-vectors are the same up to reversion.
`> print cross(3)->F_VECTOR;`

`[6, 12, 8]`

- After truncating the first standard basis vector of the 3-dimensional cross-polytope the f-vector changes. Only segments of the incident edges of the cut off vertex remain and the intersection of these with the new hyperplane generate four new vertices. These also constitute four new edges and a new facet.
`> print truncation(cross(3),4)->F_VECTOR;`

`[9, 16, 9]`

**GRAPH**: graph::Graph<Undirected>Amendment of Cone::GRAPH for Polytope::GRAPH

#### Properties of GRAPH

**EDGE_DIRECTIONS**: common::EdgeMapDifference of the vertices for each edge (only defined up to signs).

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

**LATTICE_ACCUMULATED_EDGE_LENGTHS**: common::Map<Integer, Int>**Only defined for Polytope::Lattice**a map associating to each edge length of the polytope the number of edges with this length the lattice edge length of an edge is one less than the number of lattice points on that edge

**LATTICE_EDGE_LENGTHS**: common::EdgeMap<Undirected, Integer>**Only defined for Polytope::Lattice**the lattice lengths of the edges of the polytope i.e. for each edge one less than the number of lattice points on that edge

**G_VECTOR**: common::Vector<Integer>(Toric) g-vector, defined via the (generalized) h-vector as g

_{i}= h_{i}- h_{i-1}.**H_VECTOR**: common::Vector<Integer>h-vector, defined via recursion on the face lattice of a polytope. Coincides for simplicial polytopes with the combinatorial definition of the h-vector via shellings

**MOEBIUS_STRIP_EDGES**: common::Matrix<Int, NonSymmetric>Ordered list of edges of a Moebius strip with parallel interior edges. Consists of k lines of the form (v

_{i}w_{i}), for i=1, ..., k.The Moebius strip in question is given by the quadrangles (v

_{i}, w_{i}, w_{i+1},v_{i+1}), for i=1, ..., k-1, and the quadrangle (v_{1}, w_{1}, v_{k}, w_{k}).Validity can be verified with the client validate_moebius_strip. The polytope is required to be 2-cubical.

**MOEBIUS_STRIP_QUADS**: common::Matrix<Int, NonSymmetric>Unordered list of quads which forms a Moebius strip with parallel interior edges. Each line lists the vertices of a quadrangle in cyclic order.

Validity can be verified with the client validate_moebius_strip_quads. The polytope is required to be 2-cubical.

**NEIGHBORLINESS**: common::IntMaximal dimension in which all facets are neighborly.

**Example:**- This determines that the full dimensional polytope given by 10 specific vertices on the 8-dimensional sphere is 3-neighborly, i.e. all 3-dimensional faces are tetrahedra. Hence the polytope is not neighborly.
`> print rand_sphere(8,10,seed=>8866463)->NEIGHBORLINESS;`

`3`

**NEIGHBORLY**: common::BoolTrue if the polytope is neighborly.

**Example:**- This checks the 4-dimensional cyclic polytope with 6 points on the moment curve for neighborliness, i.e. if it is ⌊dim/2⌋ neighborly:
`> print cyclic(4,6)->NEIGHBORLINESS;`

`1`

**N_VERTEX_FACET_INC**: common::IntNumber of pairs of incident vertices and facets. Alias for property Cone::N_RAY_FACET_INC.

**N_VERTICES**: common::IntNumber of VERTICES. Alias for property Cone::N_RAYS.

**Examples:**- The following prints the number of vertices of a 3-dimensional cube:
`> print cube(3)->N_VERTICES;`

`8`

- The following prints the number of vertices of the convex hull of 10 specific points lying in the unit square [0,1]^2:
`> print rand_box(2,10,1,seed=>4583572)->N_VERTICES;`

`4`

**SELF_DUAL**: common::BoolTrue if the polytope is self-dual.

**Examples:**- The following checks if the centered square with side length 2 is self dual:
`> print cube(2)->SELF_DUAL;`

`1`

- The elongated square pyramid (Johnson solid 8) is dual to itself, since the apex of the square pyramid attachted to the cube and the opposing square of the cube swap roles. The following checks this property and prints the result:
`> print johnson_solid(8)->SELF_DUAL;`

`1`

**SIMPLE**: common::BoolTrue if the polytope is simple. Dual to SIMPLICIAL.

**Example:**- This determines if a 3-dimensional cube is simple or not:
`> print cube(3)->SIMPLE;`

`1`

**SIMPLEXITY_LOWER_BOUND**: common::IntA lower bound for the minimal number of simplices in a triangulation

Contained in extension`bundled:group`

.**SIMPLICIAL**: common::BoolTrue if the polytope is simplicial.

**Example:**- A polytope with random vertices uniformly distributed on the unit sphere is simplicial. The following checks this property and prints the result for 8 points in dimension 3:
`> print rand_sphere(3,8)->SIMPLICIAL;`

`1`

**SIMPLICIALITY**: common::IntMaximal dimension in which all faces are simplices.

**Example:**- The 3-dimensional cross-polytope is simplicial, i.e. its simplicity is 2. After truncating an arbitrary vertex the simplicity is reduced to 1.
`> print cross(3)->SIMPLICIALITY;`

`2`

`> print truncation(cross(3),4)->SIMPLICIALITY;`

`1`

**SIMPLICITY**: common::IntMaximal dimension in which all dual faces are simplices.

**Example:**- This checks the 3-dimensional cube for simplicity. Since the cube is dual to the cross-polytope of equal dimension and all its faces are simplices, the result is 2.
`> print cube(3)->SIMPLICITY;`

`2`

**SUBRIDGE_SIZES**: common::Map<Int, Int>Lists for each occurring size (= number of incident facets or ridges) of a subridge how many there are.

**TWO_FACE_SIZES**: common::Map<Int, Int>Lists for each occurring size (= number of incident vertices or edges) of a 2-face how many there are.

**Example:**- This prints the number of facets spanned by 3,4 or 5 vertices a truncated 3-dimensional cube has.
`> $p = truncation(cube(3),5);`

`> print $p->TWO_FACE_SIZES;`

`{(3 1) (4 3) (5 3)}`

**VERTEX_SIZES**: common::Array<Int>Number of incident facets for each vertex. Alias for property Cone::RAY_SIZES.

**Example:**- The following prints the number of incident facets for each vertex of the elongated pentagonal pyramid (Johnson solid 9)
`> print johnson_solid(9)->VERTEX_SIZES;`

`[5 4 4 4 4 4 3 3 3 3 3]`

**VERTICES_IN_FACETS**: common::IncidenceMatrix<NonSymmetric>Vertex-facet incidence matrix, with rows corresponding to facets and columns to vertices. Vertices and facets are numbered from 0 to N_VERTICES-1 rsp. N_FACETS-1, according to their order in VERTICES rsp. FACETS.

This property is at the core of all combinatorial properties. It has the following semantics: (1) The combinatorics of an unbounded and pointed polyhedron is defined to be the combinatorics of the projective closure. (2) The combiantorics of an unbounded polyhedron which is not pointed is defined to be the combinatorics of the quotient modulo the lineality space. Therefore: VERTICES_IN_FACETS and each other property which is grouped under "Combinatorics" always refers to some polytope. Alias for property Cone::RAYS_IN_FACETS.

**Examples:**- The following prints the vertex-facet incidence matrix of a 5-gon by listing all facets as a set of contained vertices in a cyclic order (each line corresponds to an edge):
`> print n_gon(5)->VERTICES_IN_FACETS;`

`{1 2}`

`{2 3}`

`{3 4}`

`{0 4}`

`{0 1}`

- The following prints the Vertex_facet incidence matrix of the standard 3-simplex together with the facet numbers:
`> print rows_numbered(simplex(3)->VERTICES_IN_FACETS);`

`0:1 2 3`

`1:0 2 3`

`2:0 1 3`

`3:0 1 2`

**AFFINE_HULL**: common::MatrixDual basis of the affine hull of the polyhedron. The property AFFINE_HULL appears only in conjunction with the property FACETS. The specification of the property FACETS requires the specification of AFFINE_HULL, and vice versa. Alias for property Cone::LINEAR_SPAN.

**BOUNDED**: common::BoolTrue if and only if LINEALITY_SPACE trivial and FAR_FACE is trivial.

**Example:**- A pyramid over a square is bounded. Removing the base square yields an unbounded pointed polyhedron (the vertices with first entry equal to zero correspond to rays).
`> $p = pyramid(cube(2));`

`> print $p->BOUNDED;`

`1`

`> $q = facet_to_infinity($p,4);`

`> print $q->BOUNDED;`

**CENTERED**: common::BoolTrue if (1, 0, 0, ...) is in the relative interior. If full-dimensional then polar to BOUNDED.

**Example:**- The cube [0,1]^3 is not centered, since the origin is on the boundary. By a small translation we can make it centered:
`> $p = cube(3,0,0);`

`> print $p->CENTERED;`

`> $t = new Vector([-1/2,-1/2,-1/2]);`

`> print translate($p,$t)->CENTERED;`

`1`

**CENTERED_ZONOTOPE**: common::Boolis the zonotope calculated from ZONOTOPE_INPUT_POINTS or ZONOTOPE_INPUT_VECTORS to be centered at the origin? The zonotope is always calculated as the Minkowski sum of all segments conv {x,v}, where * v ranges over the ZONOTOPE_INPUT_POINTS or ZONOTOPE_INPUT_VECTORS, and * x = -v if CENTERED_ZONOTOPE = 1, * x = 0 if CENTERED_ZONOTOPE = 0. Input section only.

**CENTRALLY_SYMMETRIC**: common::BoolTrue if P = -P.

**Example:**- A centered 3-cube is centrally symmetric. By stacking a single facet (5), this property is lost. We can recover it by stacking the opposing facet (4) as well.
`> $p = cube(2);`

`> print $p->CENTRALLY_SYMMETRIC;`

`1`

`> print stack($p,5)->CENTRALLY_SYMMETRIC;`

`> print stack($p,new Set<Int>(4,5))->CENTRALLY_SYMMETRIC;`

`1`

**CONE_AMBIENT_DIM**: common::IntOne more than the dimension of the space in which the polyhedron lives. = dimension of the space in which the homogenization of the polyhedron lives

**CONE_DIM**: common::IntOne more than the dimension of the affine hull of the polyhedron = one more than the dimension of the polyhedron. = dimension of the homogenization of the polyhedron If the polytope is given purely combinatorially, this is the dimension of a minimal embedding space

**Example:**- This prints the cone dimension of a 3-cube. Since the dimension of its affine closure is 3, the result is 4.
`> print cube(3)->CONE_DIM;`

`4`

**FACETS_THRU_POINTS**: common::IncidenceMatrix<NonSymmetric>similar to FACETS_THRU_VERTICES, but with POINTS instead of VERTICES Notice that this is a temporary property; it will not be stored in any file. Alias for property Cone::FACETS_THRU_INPUT_RAYS.

**INEQUALITIES_THRU_VERTICES**: common::IncidenceMatrix<NonSymmetric>transposed VERTICES_IN_INEQUALITIES Alias for property Cone::INEQUALITIES_THRU_RAYS.

**LATTICE**: common::Bool**Only defined for Polytope<Rational>**A polytope is lattice if each vertex has integer coordinates.

**MINIMAL_VERTEX_ANGLE**: common::FloatThe minimal angle between any two vertices (seen from the VERTEX_BARYCENTER).

**MINKOWSKI_CONE**: Cone<Rational>**Only defined for Polytope<Rational>**The cone of all Minkowski summands of the polytope P. Up to scaling, a polytope S is a Minkowski summand of P if and only if the edge directions of S are a subset of those of P, and the closing condition around any 2-face of P is preserved. Coordinates of the cone correspond to the rescaled lengths of the edges of the graph of P (in the order given by the property EDGES of the GRAPH of P). The Minkowski cone is defined as the intersection of all equations given by the closing condition around 2-faces with the positive orthant. For more information see e.g. Klaus Altmann: The versal deformation of an isolated toric Gorenstein singularity

**N_01POINTS**: common::Int**Only defined for Polytope<Rational>**Number of points with 0/1-coordinates in a polytope.

**Depends on:**azove**ONE_VERTEX**: common::VectorA vertex of a pointed polyhedron. Alias for property Cone::ONE_RAY.

**Example:**- This prints the first vertex of the 3-cube (corresponding to the first row in the vertex matrix).
`> print cube(3)->ONE_VERTEX;`

`1 -1 -1 -1`

**POINTED**: common::BoolTrue if the polyhedron does not contain an affine line.

**Example:**- A square does not contain an affine line and is therefore pointed. Removing one facet doens not change this, although it is no longer bounded. After removing two opposing facets, it contains infinitely many affine lines parrallel to the removed facets.
`> $p = cube(2);`

`> print $p->POINTED;`

`1`

`> print facet_to_infinity($p,0)->POINTED;`

`1`

`> print new Polytope(INEQUALITIES=>$p->FACETS->minor([0,1],All))->POINTED;`

**POINTS_IN_FACETS**: common::IncidenceMatrix<NonSymmetric>Similar to VERTICES_IN_FACETS, but with columns corresponding to POINTS instead of VERTICES. This property is a byproduct of convex hull computation algorithms. It is discarded as soon as VERTICES_IN_FACETS is computed. Alias for property Cone::INPUT_RAYS_IN_FACETS.

**QUOTIENT_SPACE**: QuotientSpaceA topological quotient space obtained from a polytope by identifying faces.

Contained in extension`bundled:group`

.**SPECIAL_FACETS**: common::Set<Int>The following is defined for

*CENTERED*polytopes only: A facet is special if the cone over that facet with the origin as the apex contains the*VERTEX_BARYCENTER*. Motivated by Obro's work on Fano polytopes.**SPLITS**: common::MatrixThe splits of the polytope, i.e., hyperplanes cutting the polytope in two parts such that we have a regular subdivision.

**SPLIT_COMPATIBILITY_GRAPH**: graph::Graph<Undirected>Two SPLITS are compatible if the defining hyperplanes do not intersect in the interior of the polytope. This defines a graph.

**STEINER_POINTS**: common::MatrixA weighted inner point depending on the outer angle called Steiner point for all faces of dimensions 2 to d.

**VALID_POINT**: common::VectorSome point belonging to the polyhedron.

**Example:**- This stores a (homogeneous) point belonging to the 3-cube as a vector and prints its coordinates:
`> $v = cube(3)->VALID_POINT;`

`> print $v;`

`1 1 1 1`

**VERTEX_BARYCENTER**: common::VectorThe center of gravity of the vertices of a bounded polytope.

**Example:**- This prints the vertex barycenter of the standard 3-simplex:
`> print simplex(3)->VERTEX_BARYCENTER;`

`1 1/4 1/4 1/4`

**VERTEX_NORMALS**: common::MatrixThe 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. All vectors in this section must be non-zero. Alias for property Cone::RAY_SEPARATORS.

**Example:**- This prints a matrix in which each row represents a normal vector of a hyperplane seperating one vertex of a centered square with side length 2 from the other ones. The first and the last hyperplanes as well as the second and third hyperplanes are the same up to orientation.
`> print cube(2)->VERTEX_NORMALS;`

`0 1/2 1/2`

`0 -1/2 1/2`

`0 1/2 -1/2`

`0 -1/2 -1/2`

**VERTICES**: common::MatrixVertices of the polyhedron. No redundancies are allowed. All vectors in this section must be non-zero. The coordinates are normalized the same way as POINTS. Dual to FACETS. This section is empty if and only if the polytope is empty. The property VERTICES appears only in conjunction with the property LINEALITY_SPACE. The specification of the property VERTICES requires the specification of LINEALITY_SPACE, and vice versa. Alias for property Cone::RAYS.

**Examples:**- To print the vertices (in homogeneous coordinates) of the standard 2-simplex, i.e. a right-angled isoceles triangle, type this:
`> print simplex(2)->VERTICES;`

`(3) (0 1)`

`1 1 0`

`1 0 1`

- If we know some points to be vertices of their convex hull, we can store them as rows in a Matrix and construct a new polytope with it. The following produces a 3-dimensioanl pyramid over the standard 2-simplex with the specified vertices:
`> $M = new Matrix([[1,0,0,0],[1,1,0,0],[1,0,1,0],[1,0,0,3]]);`

`> $p = new Polytope(VERTICES=>$M);`

- The following adds a (square) pyramid to one facet of a 3-cube. We do this by extracting the vertices of the cube via the built-in method and then attach the apex of the pyramid to the matrix.
`> $v = new Vector([1,0,0,3/2]);`

`> $M = cube(3)->VERTICES / $v;`

`> $p = new Polytope(VERTICES=>$M);`

**VERTICES_IN_INEQUALITIES**: common::IncidenceMatrix<NonSymmetric>Similar to VERTICES_IN_FACETS, but with rows corresponding to INEQUALITIES instead of FACETS. This property is a byproduct of convex hull computation algorithms. It is discarded as soon as VERTICES_IN_FACETS is computed. Alias for property Cone::RAYS_IN_INEQUALITIES.

**WEAKLY_CENTERED**: common::BoolTrue if (1, 0, 0, ...) is contained (possibly in the boundary).

**Example:**- The cube [0,1]^3 is only weakly centered, since the origin is on the boundary.
`> $p = cube(3,0,0);`

`> print $p->WEAKLY_CENTERED;`

`1`

`> print $p->CENTERED;`

**ZONOTOPE_INPUT_POINTS**: common::MatrixThe rows of this matrix contain a configuration of affine points in homogeneous cooordinates. The zonotope is obtained as the Minkowski sum of all rows, normalized to x_0 = 1. Thus, if the input matrix has n columns, the ambient affine dimension of the resulting zonotope is n-1.

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

**EQUATIONS**: common::MatrixEquations that hold for all points of the polyhedron.

A vector (A

_{0}, A_{1}, ..., A_{d}) describes the hyperplane of all points (1, x_{1}, ..., x_{d}) such that A_{0}+ A_{1}x_{1}+ ... + A_{d}x_{d}= 0. All vectors in this section must be non-zero.Input section only. Ask for AFFINE_HULL if you want to see an irredundant description of the affine span.

**INEQUALITIES**: common::MatrixInequalities that describe half-spaces such that the polyhedron is their intersection. Redundancies are allowed. Dual to POINTS.

A vector (A

_{0}, A_{1}, ..., A_{d}) defines the (closed affine) half-space of points (1, x_{1}, ..., x_{d}) such that A_{0}+ A_{1}x_{1}+ ... + A_{d}x_{d}>= 0.Input section only. Ask for FACETS and AFFINE_HULL if you want to compute an H-representation from a V-representation.

**POINTS**: common::MatrixPoints such that the polyhedron is their convex hull. Redundancies are allowed. The vector (x

_{0}, x_{1}, ... x_{d}) represents a point in d-space given in homogeneous coordinates. Affine points are identified by x_{0}> 0. Points with x_{0}= 0 can be interpreted as rays.polymake automatically normalizes each coordinate vector, dividing them by the first non-zero element. The clients and rule subroutines can always assume that x

_{0}is either 0 or 1. All vectors in this section must be non-zero. Dual to INEQUALITIES.Input section only. Ask for VERTICES if you want to compute a V-representation from an H-representation. Alias for property Cone::INPUT_RAYS.

**Example:**- Given some (homogeneous) points in 3-space we first construct a matrix containing them. Assume we don't know wether these are all vertices of their convex hull or not. To safely produce a polytope from these points, we set the input to the matrix representing them. In the following the points under consideration are the vertices of the 3-simplex together with their barycenter, which will be no vertex:
`> $M = new Matrix([[1,0,0,0],[1,1,0,0],[1,0,1,0],[1,0,0,1],[1,1/4,1/4,1/4]]);`

`> $p = new Polytope(POINTS=>$M);`

`> print $p->VERTICES;`

`1 0 0 0`

`1 1 0 0`

`1 0 1 0`

`1 0 0 1`

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

**CANONICAL**: common::Bool**Only defined for Polytope::Lattice**The polytope is

*canonical*if there is exactly one interior lattice point.**EHRHART_POLYNOMIAL_COEFF**: common::Vector<Rational>**Only defined for Polytope::Lattice**The coefficients of the Ehrhart polynomial starting at the constant coefficient.

**Depends on:**latte or libnormaliz**FACET_VERTEX_LATTICE_DISTANCES**: common::Matrix<Integer, NonSymmetric>**Only defined for Polytope::Lattice**The entry (i,j) equals the lattice distance of vertex j from facet i.

**FACET_WIDTH**: common::Integer**Only defined for Polytope::Lattice**The maximal integral width of the polytope with respect to the facet normals.

**FACET_WIDTHS**: common::Vector<Integer>**Only defined for Polytope::Lattice**The integral width of the polytope with respect to each facet normal.

**GORENSTEIN**: common::Bool**Only defined for Polytope::Lattice**The polytope is

*Gorenstein*if a dilation of the polytope is REFLEXIVE up to translation.**GORENSTEIN_INDEX**: common::Integer**Only defined for Polytope::Lattice**If the polytope is GORENSTEIN then this is the multiple such that the polytope is REFLEXIVE.

**GORENSTEIN_VECTOR**: common::Vector<Integer>**Only defined for Polytope::Lattice**If the polytope is GORENSTEIN, then this is the unique interior lattice point in the multiple of the polytope that is REFLEXIVE.

**GROEBNER_BASIS**: GroebnerBasis**Only defined for Polytope::Lattice**The Groebner basis for the toric ideal associated to the lattice points in the polytope using any term order.

**LATTICE_BASIS**: common::Matrix<Rational, NonSymmetric>**Only defined for Polytope::Lattice**VERTICES are interpreted as coefficient vectors for this basis given in affine form assumed to the the standard basis if not explicitely specified.

**LATTICE_CODEGREE**: common::Int**Only defined for Polytope::Lattice**COMBINATORIAL_DIM+1-LATTICE_DEGREE or the smallest integer k such that k*P has an interior lattice point.

**LATTICE_DEGREE**: common::Int**Only defined for Polytope::Lattice**The degree of the h*-polynomial or Ehrhart polynomial.

**LATTICE_EMPTY**: common::Bool**Only defined for Polytope::Lattice**True if the polytope contains no lattice points other than the vertices.

**LATTICE_VOLUME**: common::Integer**Only defined for Polytope::Lattice**The normalized volume of the polytope.

**LATTICE_WIDTH**: common::Integer**Only defined for Polytope::Lattice**The minimal integral width of the polytope.

**LATTICE_WIDTH_DIRECTION**: common::Vector<Integer>**Only defined for Polytope::Lattice**One direction which realizes LATTICE_WIDTH of the polytope.

**NORMAL**: common::Bool**Only defined for Polytope::Lattice**The polytope is

*normal*if the cone spanned by P x {1} is generated in height 1.**Depends on:**4ti2 or libnormaliz**REFLEXIVE**: common::Bool**Only defined for Polytope::Lattice**True if the polytope and its dual have integral vertices.

**SMOOTH**: common::Bool**Only defined for Polytope::Lattice**The polytope is

*smooth*if the associated projective variety is smooth; the determinant of the edge directions is +/-1 at every vertex.**TERMINAL**: common::Bool**Only defined for Polytope::Lattice**The polytope is

*terminal*if there is exactly one interior lattice point and all other lattice points are vertices.**VERY_AMPLE**: common::Bool**Only defined for Polytope::Lattice**The polytope is

*very ample*if the Hilbert Basis of the cone spanned by the edge-directions of any vertex lies inside the polytope.**Depends on:**4ti2 or libnormaliz

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

**BOUNDARY_LATTICE_POINTS**: common::Matrix<Integer, NonSymmetric>**Only defined for Polytope<Rational>**The lattice points on the boundary of the polytope, including the vertices.

**INTERIOR_LATTICE_POINTS**: common::Matrix<Integer, NonSymmetric>**Only defined for Polytope<Rational>**The lattice points strictly in the interior of the polytope

**LATTICE_POINTS_GENERATORS**: common::Array<Matrix<Integer, NonSymmetric>>**Only defined for Polytope<Rational>**The lattice points generators in the polytope. The output consists of three matrices [P,R,L], where P are lattice points which are contained in the polytope R are rays and L is the lineality. Together they form a description of all lattice points. Every lattice point can be described as p + lambda*R + mu*L where p is a row in P and lambda has only non-negative integral coordinates and mu has arbitrary integral coordinates.

**Depends on:**4ti2 for unbounded polytopes**N_BOUNDARY_LATTICE_POINTS**: common::Integer**Only defined for Polytope<Rational>**The number of BOUNDARY_LATTICE_POINTS

**N_INTERIOR_LATTICE_POINTS**: common::Integer**Only defined for Polytope<Rational>**The number of INTERIOR_LATTICE_POINTS

Properties which belong to the corresponding (oriented) matroid

These properties provide tools from linear, integer and dicrete optimization. In particular, linear programs are defined here.

Everything in this group is defined for BOUNDED polytopes only.

**POLYTOPAL_SUBDIVISION**: fan::SubdivisionOfPointsPolytopal Subdivision of the polytope using only its vertices.

**RELATIVE_VOLUME**: common::Map<Rational, Rational>**Only defined for Polytope<Rational>**The

*k*-dimensional Euclidean volume of a*k*-dimensional rational polytope embedded in R^n. This value is obtained by summing the square roots of the entries in SQUARED_RELATIVE_VOLUMES using the function*naive_sum_of_square_roots*. Since this latter function does not try very hard to compute the real value, you may have to resort to a computer algebra package. The value is encoded as a map collecting the coefficients of various roots encountered in the sum. For example, {(3 1/2),(5 7)} represents sqrt{3}/2 + 7 sqrt{5}. If the output is not satisfactory, please use a symbolic algebra package.**Example:**- The following prints the 2-dimensional volume of a centered square with side length 2 embedded in the 3-space (the result is 4):
`> $M = new Matrix([1,-1,1,0],[1,-1,-1,0],[1,1,-1,0],[1,1,1,0]);`

`> $p = new Polytope<Rational>(VERTICES=>$M);`

`> print $p->RELATIVE_VOLUME;`

`{(1 4)}`

**SQUARED_RELATIVE_VOLUMES**: common::ArrayArray of the squared relative

*k*-dimensional volumes of the simplices in a triangulation of a*d*-dimensional polytope.**TRIANGULATION**: topaz::GeometricSimplicialComplexAmendment of Cone::TRIANGULATION for Polytope::TRIANGULATION

#### Properties of TRIANGULATION

**GKZ_VECTOR**: common::VectorGKZ-vector

See Chapter 7 in Gelfand, Kapranov, and Zelevinsky:Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994

**VOLUME**: Polytope::ScalarVolume of the polytope.

**Example:**- The following prints the volume of the centered 3-dimensional cube with side length 2:
`> print cube(3)->VOLUME;`

`8`

These properties collect geometric information of a polytope only relevant if it is unbounded, e. g. the far face or the complex of bounded faces.

**BOUNDED_COMPLEX**: fan::PolyhedralComplex<Rational>Bounded subcomplex. Defined as the bounded cells of the boundary of the pointed part of the polytope. Therefore it only depends on VERTICES_IN_FACETS and FAR_FACE.

#### Properties of BOUNDED_COMPLEX

**GRAPH**: graph::Graph<Undirected>Amendment of fan::PolyhedralFan::GRAPH for Polytope::BOUNDED_COMPLEX::GRAPH

#### Properties of GRAPH

**EDGE_COLORS**: common::EdgeMap<Undirected, Int>Each edge indicates the maximal dimension of a bounded face containing it. Mainly used for visualization purposes.

**EDGE_DIRECTIONS**: common::EdgeMapDifference of the vertices for each edge (only defined up to signs).

**TOWARDS_FAR_FACE**: common::VectorA linear objective function for which each unbounded edge is increasing; only defined for unbounded polyhedra.

These properties are for visualization.

**FACET_LABELS**: common::Array<String>Unique names assigned to the FACETS, analogous to VERTEX_LABELS.

**FTV_CYCLIC_NORMAL**: common::Array<Array<Int>>Reordered transposed VERTICES_IN_FACETS. Dual to VIF_CYCLIC_NORMAL. Alias for property Cone::FTR_CYCLIC_NORMAL.

**GALE_VERTICES**: common::Matrix<Float, NonSymmetric>Coordinates of points for an affine Gale diagram.

**INEQUALITY_LABELS**: common::Array<String>Unique names assigned to the INEQUALITIES, analogous to VERTEX_LABELS.

**NEIGHBOR_VERTICES_CYCLIC_NORMAL**: common::Array<Array<Int>>Reordered GRAPH. Dual to NEIGHBOR_FACETS_CYCLIC_NORMAL. Alias for property Cone::NEIGHBOR_RAYS_CYCLIC_NORMAL.

**POINT_LABELS**: common::Array<String>Unique names assigned to the POINTS, analogous to VERTEX_LABELS. Alias for property Cone::INPUT_RAY_LABELS.

**SCHLEGEL_DIAGRAM**: SchlegelDiagramHolds one special projection (the Schlegel diagram) of the polytope.

**VERTEX_LABELS**: common::Array<String>Unique names assigned to the VERTICES. If specified, they are shown by visualization tools instead of vertex indices.

For a polytope build from scratch, you should create this property by yourself, either manually in a text editor, or with a client program.

If you build a polytope with a construction function taking some other input polytope(s), you can create the labels automatically if you call the function with a

*relabel*option. The exact format of the labels is dependent on the construction, and is described in the corresponding help topic. Alias for property Cone::RAY_LABELS.**VIF_CYCLIC_NORMAL**: common::Array<Array<Int>>Reordered VERTICES_IN_FACETS for 2d and 3d-polytopes. Vertices are listed in the order of their appearance when traversing the facet border counterclockwise seen from outside of the polytope.

For a 2d-polytope (which is a closed polygon), lists all vertices in the border traversing order. Alias for property Cone::RIF_CYCLIC_NORMAL.

#### User Methods of Polytope

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

**CD_INDEX**()Prettily print the cd-index given in CD_INDEX_COEFFICIENTS

**N_RIDGES**()The number of ridges (faces of codimension 2) of the polytope equals the number of edges of the DUAL_GRAPH

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 of the polytope

**contains**(P, v) → Boolchecks whether a given point is contained in a polytope

**contains_in_interior**(P, v) → Boolchecks whether a given point is contained in the strict interior of a polytope

**DIM**()returns the dimension of the polytope

**INNER_DESCRIPTION**() → Array<Matrix<Scalar> >Returns the inner description of a Polytope: [V,L] where V are the vertices and L is the lineality space

##### Returns

Array<Matrix<Scalar> > **labeled_vertices**(label ...) → Set<Int>**MINKOWSKI_CONE_COEFF**(coeff) → Polytope<Rational>**Only defined for Polytope<Rational>**returns the Minkowski summand of a polytope P given by a coefficient vector to the rays of the MINKOWSKI_CONE.

##### Parameters

Vector<Rational> coeff coefficient vector to the rays of the Minkowski summand cone##### Returns

Polytope<Rational> **MINKOWSKI_CONE_POINT**(point) → Polytope<Rational>**Only defined for Polytope<Rational>**returns the Minkowski summand of a polytope P given by a point in the MINKOWSKI_CONE.

**OUTER_DESCRIPTION**() → Array<Matrix<Scalar> >Returns the outer description of a Polytope: [F,A] where F are the facets and A is the affine hull

##### Returns

Array<Matrix<Scalar> >

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

**FACET_POINT_LATTICE_DISTANCES**(v) → Vector<Integer>**Only defined for Polytope::Lattice**Vector containing the distances of a given point

*v*from all facets**N_LATTICE_POINTS_IN_DILATION**(n)**Only defined for Polytope::Lattice**The number of LATTICE_POINTS in the

*n*-th dilation of the polytope##### Parameters

Int n dilation factor**POLYTOPE_IN_STD_BASIS**(P) → Polytope<Rational>**Only defined for Polytope::Lattice**returns a polytope in the integer lattice basis if a LATTICE_BASIS is given

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

**LATTICE_POINTS**() → Matrix<Integer>**Only defined for Polytope<Rational>**Returns the lattice points in bounded Polytopes.

##### Returns

Matrix<Integer>

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

**TRIANGULATION_INT_SIGNS**() → Array<Int>the orientation of the simplices of TRIANGULATION_INT in the given order of the POINTS

##### Returns

Array<Int> - +1/-1 array specifying the sign of the determinant of each simplex**TRIANGULATION_SIGNS**() → Array<Int>For each simplex in the TRIANGULATION, contains the sign of the determinant of its coordinate matrix, telling about its orientation.

##### Returns

Array<Int>

These methods collect geometric information of a polytope only relevant if it is unbounded, e. g. the far face or the complex of bounded faces.

**BOUNDED_DUAL_GRAPH**()Dual graph of the bounded subcomplex.

**BOUNDED_FACETS**() → Set<Int>**BOUNDED_GRAPH**()Graph of the bounded subcomplex.

**BOUNDED_HASSE_DIAGRAM**()HASSE_DIAGRAM constrained to affine vertices Nodes representing the maximal inclusion-independent faces are connected to the top-node regardless of their dimension

**BOUNDED_VERTICES**() → Set<Int>

These methods are for visualization.

**GALE**() → Visual::Gale**SCHLEGEL**() → Visual::SchlegelDiagramCreate a Schlegel diagram and draw it.

##### Options

Visual::Graph::decorations proj_facet decorations for the edges of the projection faceoption list: schlegel_init option list: Visual::Wire::decorations ##### Returns

Visual::SchlegelDiagram **VISUAL**() → Visual::PolytopeVisualize a polytope as a graph (if 1d), or as a solid object (if 2d or 3d), or as a Schlegel diagram (4d).

##### Options

option list: Visual::Polygons::decorations option list: Visual::Wire::decorations option list: Visual::PointSet::decorations option list: geometric_options ##### Returns

Visual::Polytope **VISUAL_BOUNDED_GRAPH**() → Visual::PolytopeGraphVisualize the BOUNDED_COMPLEX.GRAPH of a polyhedron.

##### Options

Int seed random seed value for the string embedderoption list: Visual::Graph::decorations ##### Returns

Visual::PolytopeGraph **VISUAL_DUAL**() → Visual::Object**VISUAL_DUAL_FACE_LATTICE**() → Visual::PolytopeLatticeVisualize the dual face lattice of a polyhedron as a multi-layer graph.

##### Options

Int seed random seed value for the node placementoption list: Visual::Lattice::decorations ##### Returns

Visual::PolytopeLattice **VISUAL_DUAL_GRAPH**() → Visual::GraphVisualize the DUAL_GRAPH of a polyhedron.

##### Options

Int seed random seed value for the string embedderoption list: Visual::Graph::decorations ##### Returns

Visual::Graph **VISUAL_FACE_LATTICE**() → Visual::PolytopeLatticeVisualize the HASSE_DIAGRAM of a polyhedron as a multi-layer graph.

##### Options

Int seed random seed value for the node placementoption list: Visual::Lattice::decorations ##### Returns

Visual::PolytopeLattice **VISUAL_GRAPH**() → Visual::PolytopeGraphVisualize the GRAPH of a polyhedron.

##### Options

Int seed random seed value for the string embedderoption list: Visual::Graph::decorations ##### Returns

Visual::PolytopeGraph **VISUAL_TRIANGULATION_BOUNDARY**() → Visual::ObjectVisualize the TRIANGULATION_BOUNDARY of the polytope.

**Obsolete:**the preferred procedure is to create a SimplicialComplex using the boundary_complex client of the application topaz and call its VISUAL method. FIXME: There is no boundary_complex in topaz.

Polytope propagation means to define a polytope inductively by assigning vectors to arcs of a directed graph. At each node of such a graph a polytope arises as the joint convex hull of the polytopes at the translated sources of the inward pointing arcs.

For details see Joswig: Polytope Propagation on Graphs. Chapter 6 in Pachter/Sturmfels: Algebraic Statistics for Computational Biology, Cambridge 2005.

**derived from:**Polytope#### Properties of PropagatedPolytope

**SUM_PRODUCT_GRAPH**: graph::Graph<Directed>Directed graph to define the propagated polytope. There is a (translation) vector assigned to each arc. We assume that this graph is acyclic with a unique sink.

#### Properties of SUM_PRODUCT_GRAPH

**Category:**SymmetryA topological quotient space obtained from a Polytope by identifying faces. This object will sit inside the polytope.

Contained in extension`bundled:group`

.#### Properties of QuotientSpace

Properties defining a quotient space.

**IDENTIFICATION_GROUP**: group::GroupThe group encoding the quotient space. The faces of the space are the orbits of the faces of the polytope under the group.

**COCIRCUIT_EQUATIONS**: common::SparseMatrix<Rational, NonSymmetric>a SparseMatrix whose rows are the sum of all cocircuit equations corresponding to a fixed symmetry class of interior ridge

**FACES**: common::Array<Array<Set<Int>>>The faces of the quotient space, ordered by dimension. One representative of each orbit class is kept.

**FACE_ORBITS**: common::Array<Set<Array<Set<Int>>>>The orbits of faces of the quotient space, ordered by dimension.

**N_SIMPLICES**: common::Array<Int>The simplices made from points of the quotient space (also internal simplices, not just faces)

**REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES**: common::Array<boost_dynamic_bitset>The (

*d*-1)-dimensional simplices in the interior.**REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES**: common::Array<boost_dynamic_bitset>The boundary (

*d*-1)-dimensional simplices of a cone of combinatorial dimension*d***REPRESENTATIVE_MAX_INTERIOR_SIMPLICES**: common::Array<boost_dynamic_bitset>The interior

*d*-dimensional simplices of a cone of combinatorial dimension*d***SIMPLEXITY_LOWER_BOUND**: common::IntA lower bound for the number of simplices needed to triangulate the quotient space

**SIMPLICIAL_COMPLEX**: topaz::SimplicialComplexA simplicial complex obtained by two stellar subdivisions of the defining polytope.

**SYMMETRY_GROUP**: group::GroupThe symmetry group induced by the symmetry group of the polytope on the FACES of the quotient space

A Schlegel diagram of a polytope.

##### Type Parameters

Scalar default Rational#### Properties of SchlegelDiagram

**ROTATION**: common::Matrix<Float, NonSymmetric>Rotation matrix making the projection facet coinciding with (0 0 0 -1) We want a negatively oriented coordinate system since the view point lies on the negative side of the facet.

**TRANSFORM**: common::MatrixMatrix of a projective transformation mapping the whole polytope into the FACET The points belonging to this facet stay fixed.

**VERTICES**: common::Matrix<Float, NonSymmetric>Coordinates in affine 3-space of the vertices which correspond to a 3-dimensional (Schlegel-) projection of a 4-polytope.

#### User Methods of SchlegelDiagram

**VISUAL**() → Visual::SchlegelDiagramDraw the Schlegel diagram.

##### Options

Visual::Graph::decorations proj_facet decorations for the edges of the projection faceoption list: Visual::Graph::decorations ##### Returns

Visual::SchlegelDiagram

**Category:**SymmetryA cone which is generated by a group and a generating set of inequalities (+equations) or input rays (+input_lineality). The cone is the intersection or the convex hull of all inequalities or input rays in the orbit of the generating set under the GENERATING_GROUP.

**derived from:**Cone##### Type Parameters

Scalar default: Rational#### Properties of SymmetricCone

**GENERATING_GROUP**: group::GroupOfConeThe group which generates the cone by being applied to some GEN_INPUT_RAYS (and GEN_INPUT_LINEALITY) or some GEN_INEQUALITIES (and GEN_EQUATIONS).

**GEN_EQUATIONS**: common::MatrixSome generating equations for (a subset of) the linear span of the symmetric cone. Redundancies are allowed.

Input section only.

**GEN_INEQUALITIES**: common::MatrixSome generating inequalities for the symmetric cone; redundancies are allowed.

Input section only. Ask for REPRESENTATIVE_FACETS if you want a list of representatives for the orbits of facets of a symmetric cone.

**GEN_INPUT_LINEALITY**: common::MatrixSome generating input rays for (a subset of) the lineality space of the symmetric cone. Redundancies are allowed.

Input section only.

**GEN_INPUT_RAYS**: common::MatrixSome generating input rays for the symmetric cone; redundancies are allowed.

Input section only. Ask for REPRESENTATIVE_RAYS if you want a list of representatives for the orbits of rays of a symmetric cone.

#### User Methods of SymmetricCone

**VISUAL_ORBIT_COLORED_GRAPH**() → Visual::PolytopeGraphVisualizes the graph of a symmetric cone: All nodes belonging to one orbit get the same color.

**Category:**SymmetryA polytope which is generated by a group and a generating set of inequalities (+equations) or points (+input_lineality). The polytope is the intersection or the convex hull of all inequalities or points in the orbit of the generating set under the GENERATING_GROUP.

**derived from:**SymmetricCone##### Type Parameters

Scalar default: Rational#### Properties of SymmetricPolytope

#### User Methods of SymmetricPolytope

**AMBIENT_DIM**()must be copied (from common.rules) since SymmetricPolytope is derived from both objects, SymmetricCone and Polytope

**DIM**()must be copied (from common.rules) since SymmetricPolytope is derived from both objects, SymmetricCone and Polytope

Bounded subcomplex of an unbounded polyhedron, which is associated with a finite metric space. The tight span is 1-dimensional if and only if the metric is tree-like. In this sense, the tight span captures the deviation of the metric from a tree-like one.

**derived from:**Polytope#### Properties of TightSpan

Properties of a TightSpan

**TAXA**: common::Array<String>Labels for the rows and columns of the METRIC space. Default TAXA are just consecutive numbers.

#### User Methods of TightSpan

These methods are for visualization.

**VISUAL_BOUNDED_GRAPH**() → Visual::PolytopeGraphVisualize the BOUNDED_COMPLEX.GRAPH of a tight span.

##### Options

Int seed random seed value for the string embedderoption list: Visual::Graph::decorations ##### Returns

Visual::PolytopeGraph **VISUAL_TIGHT_SPAN**() → Visual::GraphThis is a variation of Polytope::VISUAL_BOUNDED_GRAPH for the special case of a tight span. The vertices are embedded according to the METRIC, the others are hanged in between.

##### Options

Int seed random seed value for the string embedderString norm which norm to use when calculating the distances between metric vectors ("max" or "square")option list: Visual::Graph::decorations ##### Returns

Visual::Graph

An object of type VectorConfiguration deals with properties of row vectors, assembled into an n x d matrix called VECTORS. The entries of these row vectors are interpreted as non-homogeneous coordinates. In particular, the coordinates of a VECTOR will *NOT* be normalized to have a leading 1.

##### Type Parameters

Scalar default: Rational#### Properties of VectorConfiguration

**POSITIVE**: common::BoolTrue if all VECTORS have non-negative coordinates, that is, if they all lie entirely in the positive orthant.

**VECTOR_AMBIENT_DIM**: common::IntDimension of the space in which the vector configuration lives. Similar to Cone::CONE_AMBIENT_DIM.

**VECTOR_DIM**: common::IntDimension of the linear hull of the vector configuration. Similar to Cone::CONE_DIM.

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

Properties which belong to the corresponding (oriented) matroid

These properties are for visualization.

**LABELS**: common::Array<String>Unique names assigned to the VECTORS. If specified, they are shown by visualization tools instead of point indices.

**Category:**VisualizationVisualization of the point configuration.

#### User Methods of Visual::PointConfiguration

**POLYTOPAL_SUBDIVISION**(index) → Visual::PointConfigurationVisualize a POLYTOPAL_SUBDIVISION of a point configuration.

##### Parameters

Int index Index of the subdivision to visualize##### Options

option list: Visual::Polygons::decorations ##### Returns

Visual::PointConfiguration **TRIANGULATION**() → Visual::PointConfigurationVisualize the TRIANGULATION of a point configuration

**TRIANGULATION_BOUNDARY**() → Visual::PointConfigurationDraw the edges of the TRIANGULATION_BOUNDARY. The facets are made transparent.

**Category:**VisualizationVisualization of a polytope as a graph (if 1d), or as a solid object (if 2d or 3d), or as a Schlegel diagram (4d).

#### User Methods of Visual::Polytope

**DIRECTED_GRAPH**(lp) → Visual::PolytopeIllustrate the behavior of a linear objective function on the polytope. Superpose the drawing with the directed graph induced by the objective function.

**Example:**- Attaches a linear program to the 3-dimensional cube and visualizes the directed graph, giving the cube a blue facet color
`> $p = cube(3);`

`> $p->LP = new LinearProgram(LINEAR_OBJECTIVE=>[0,0,0,1]);`

`> $p->VISUAL(FacetColor=>"blue")->DIRECTED_GRAPH;`

**LATTICE**() → Visual::PolytopeVisualize the LATTICE_POINTS of a polytope

**Example:**- Visualizes the lattice points of the threedimensional cube.
`> cube(3)->VISUAL->LATTICE;`

**LATTICE_COLORED**() → Visual::PolytopeVisualize the LATTICE_POINTS of a polytope in different colors (interior / boundary / vertices)

**Example:**- Creates the threedimensional unit cube scaled by 1.5 and displays the colored version of its lattice points
`> cube(3,(3/2),0)->VISUAL->LATTICE_COLORED;`

**MIN_MAX_FACE**(lp) → Visual::PolytopeIllustrate the behavior of a linear objective function on the polytope. Draw the facets contained in MAXIMAL_FACE and MINIMAL_FACE in distinct colors.

##### Parameters

LinearProgram lp a LinearProgram object attached to the polytope.##### Options

Color min minimal face decoration (default: yellow vertices and/or facets)Color max maximal face decoration (default: red vertices and/or facets)##### Returns

Visual::Polytope **Example:**- Attaches a linear program to the threedimensional cube and displays the minimal/maximal faces in a different color, choosing purple instead of the default red for the maximal face
`> $p = cube(3);`

`> $p->LP = new LinearProgram(LINEAR_OBJECTIVE=>[0,1,0,0]);`

`> $p->VISUAL->MIN_MAX_FACE(max=>"purple");`

**STEINER**() → Visual::PolytopeAdd the STEINER_POINTS to the 3-d visualization. The facets become transparent.

**Example:**- Displays the Steiner points of a random threedimensional sphere with 20 vertices. The labels of the vertices are turned off.
`> rand_sphere(3,20)->VISUAL(VertexLabels=>"hidden")->STEINER;`

**TRIANGULATION**(t) → Visual::PolytopeAdd the triangulation to the drawing.

You may specify any triangulation of the current polytope. Per default, the TRIANGULATION property is taken. (Currently there is only one possible alternative triangulation: TRIANGULATION_INT).

**Hint:**Use the method*Method -> Effect -> Explode Group of Geometries*of JavaView for better insight in the internal structure.##### Parameters

Array<Set<Int>> t facets of the triangulation##### Options

option list: Visual::Polygons::decorations ##### Returns

Visual::Polytope **Example:**- Displays a triangulation of the threedimensional cube. Facets are made transparent and vertices are hidden.
`> cube(3)->VISUAL->TRIANGULATION(FacetTransparency=>0.7,VertexStyle=>"hidden");`

**TRIANGULATION_BOUNDARY**() → Visual::PolytopeDraw the edges of the TRIANGULATION_BOUNDARY. The facets are made transparent.

**Example:**- Displays the boundary triangulation of the threedimensional cube.
`> cube(3)->VISUAL->TRIANGULATION_BOUNDARY;`

**VERTEX_COLORS**(lp) → Visual::PolytopeIllustrate the behavior of a linear objective function on the polytope. Color the vertices according to the values of the objective function.

##### Parameters

LinearProgram lp a LinearProgram object attached to the polytope##### Options

Color min minimal vertex color (default: yellow)Color max maximal vertex color (default: red)##### Returns

Visual::Polytope **Example:**- Attaches a linear program to the threedimensional cube and displays the minimal/maximal vertices in a different color, choosing purple instead of the default red for the maximal vertices
`> $p = cube(3);`

`> $p->LP = new LinearProgram(LINEAR_OBJECTIVE=>[0,1,0,0]);`

`> $p->VISUAL->VERTEX_COLORS(max=>"purple");`

**Category:**VisualizationVisualization of the graph of a polyhedron.

#### User Methods of Visual::PolytopeGraph

**DIRECTED_GRAPH**(lp) → Visual::PolytopeGraphShow the growth direction of a linear objective function via arrowed edges.

##### Parameters

LinearProgram lp a LinearProgram object attached to the polytope##### Returns

Visual::PolytopeGraph **EDGE_COLORS**() → Visual::PolytopeGraphProduce an edge coloring of a bounded graph from local data in the Hasse diagram.

##### Returns

Visual::PolytopeGraph **MIN_MAX_FACE**(lp) → Visual::PolytopeGraphIllustrate the behavior of a linear objective function on the polytope. The vertices belonging to MINIMAL_FACE and MAXIMAL_FACE are drawn in distinct colors

The spring embedder applies an additional force, which tries to arrange the nodes in the z-axis direction corresponding to the objective function values.

##### Parameters

LinearProgram lp a LinearProgram object attached to the polytope##### Options

Color min minimal face decoration (default: yellow nodes)Color max maximal face decoration (default: red nodes)##### Returns

Visual::PolytopeGraph **VERTEX_COLORS**(lp) → Visual::PolytopeGraphIllustrate the behavior of a linear objective function on the polytope. Color the nodes according to the value the objective function takes on the vertices.

The spring embedder applies an additional force, which tries to arrange the nodes in the z-axis direction corresponding to the objective function values.

##### Parameters

LinearProgram lp a LinearProgram object attached to the polytope.##### Options

Color min minimal face color (default: yellow)Color max maximal face color (default: red)##### Returns

Visual::PolytopeGraph

**Category:**VisualizationVisualization of the HASSE_DIAGRAM of a polyhedron as a multi-layer graph..

#### User Methods of Visual::PolytopeLattice

**MIN_MAX_FACE**(lp) → Visual::PolytopeLatticeIllustrate the behavior of a linear objective function on the polytope. Draw the filters of the MAXIMAL_FACE and MINIMAL_FACE in distinct colors.

##### Parameters

LinearProgram lp a LinearProgram object attached to the polytope##### Options

Color min minimal face decoration (default: yellow border and ingoing edges)Color max maximal face decoration (default: red border and ingoing edges)##### Returns

Visual::PolytopeLattice

**Category:**VisualizationVisualization of the Schlegel diagram of a polytope.

#### User Methods of Visual::SchlegelDiagram

**CONSTRUCTION**() → Visual::SchlegelDiagramVisualize the construction of a 3D Schlegel diagram, that is, the Viewpoint, the 3-polytope and the projection onto one facet.

**DIRECTED_GRAPH**(lp) → Visual::SchlegelDiagramIllustrate the behavior of a linear objective function on the polytope. Superpose the drawing with the directed graph induced by the objective function.

##### Parameters

LinearProgram lp a LinearProgram object attached to the polytope.##### Returns

Visual::SchlegelDiagram **MIN_MAX_FACE**(lp) → Visual::SchlegelDiagramIllustrate the behavior of a linear objective function on the polytope. The vertices belonging to MINIMAL_FACE and MAXIMAL_FACE are drawn in distinct colors

##### Parameters

LinearProgram lp a LinearProgram object attached to the polytope.##### Options

Color min minimal face decoration (default: yellow vertices and/or facets)Color max maximal face decoration (default: red vertices and/or facets)##### Returns

Visual::SchlegelDiagram **SOLID**() → Visual::SchlegelDiagramDraw the facets of the Schlegel diagram as polytopes.

**STEINER**()UNDOCUMENTED

##### Options

option list: Visual::PointSet::decorations **TRIANGULATION_BOUNDARY**() → Visual::SchlegelDiagramDraw the edges of the TRIANGULATION_BOUNDARY

**TRIANGULATION_BOUNDARY_SOLID**() → Visual::SchlegelDiagramDraw the boundary simplices of the triangulation as solid tetrahedra.

**VERTEX_COLORS**(lp) → Visual::SchlegelDiagramIllustrate the behavior of a linear objective function on the polytope. Color the vertices according to the values of the objective function.

##### Parameters

LinearProgram lp a LinearProgram object attached to the polytope.##### Options

Color min minimal vertex color (default: yellow)Color max maximal vertex color (default: red)##### Returns

Visual::SchlegelDiagram

For a finite set of SITES

*S*the Voronoi region of each site is the set of points closest (with respect to Euclidean distance) to the given site. All Voronoi regions (and their faces) form a polyhedral complex which is a vertical projection of the boundary complex of an unbounded polyhedron P(S). This way VoronoiDiagram becomes a derived class from Polytope<Scalar>.**derived from:**Polytope#### Properties of VoronoiDiagram

**DELAUNAY_TRIANGULATION**: common::Array<Set<Int>>Delaunay triangulation of the sites. (Delaunay subdivision, non-simplices are triangulated.)

**ITERATED_VORONOI_GRAPH**: graph::GeometricGraphGraph of the joint Voronoi diagram of the SITES and the vertices of Vor(SITES). The coordinates (homogeneous, before projection) are stored as node attributes. The graph is truncated according to the VORONOI_GRAPH.BOUNDING_BOX. For the default BOUNDING_BOX it may happen that some of the iterated Voronoi vertices are truncated. Create new objects of type VoronoiDiagram to produce proper iterated Voronoi diagrams.

**NN_CRUST_GRAPH**: graph::Graph<Undirected>Graph of the nearest neighbor crust, as defined in:

T. K. Dey and P. Kumar: A simple provable algorithm for curve reconstruction. Proc. 10th. Annu. ACM-SIAM Sympos. Discrete Alg., 1999, 893-894.

Polygonal reconstruction of a smooth planar curve from a finite set of samples. Sampling rate of <= 1/3 suffices.

**NN_GRAPH**: graph::Graph<Undirected>Graph of the nearest neighbors. This is a subgraph of NN_CRUST_GRAPH.

**SITES**: common::MatrixCoordinates of the sites in case the polyhedron is Voronoi. Sites must be pairwise distinct.

**VORONOI_GRAPH**: graph::GeometricGraphGraph of the Voronoi diagram of the SITES. The homogeneous coordinates after projection are stored as node attributes. The graph is truncated according to the BOUNDING_BOX. All vertices of the Voronoi diagram are visible (and represented in the VORONOI_GRAPH) for the default BOUNDING_BOX.

#### User Methods of VoronoiDiagram

These methods are for visualization.

**VISUAL_CRUST**() → Visual::ContainerDraw a Voronoi diagram, its |dual graph and the crust. Use the interactive features of the viewer to select.

**VISUAL_NN_CRUST**() → Visual::ContainerDraw a Voronoi diagram, its dual graph and the nearest neighbor crust. Use the interactive features of the viewer to select.

**VISUAL_VORONOI**() → Visual::ContainerDraw a Voronoi diagram and its dual. Use the interactive features of the viewer to select.

## User Functions

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

**circuits2matrix**(co) → SparseMatrix<Rational>Convert CIRCUITS or COCIRCUITS to a 0/+1/-1 matrix, with one row for each circuit/cocircuit, and as many columns as there are VECTORs/POINTS.

##### Parameters

Set<Pair<Set<Int>,Set<Int>>> co /circuits a set of circuits or cocircuits##### Returns

SparseMatrix<Rational> **cocircuit_equation**(C, rho, index_of) → SparseVector<Int>The cocircuit equations of a cone C corresponding to some interior ridge rho with respect to a list of interior simplices symmetries of the cone are NOT taken into account

Contained in extension`bundled:group`

.##### Parameters

Cone C Set<Int> rho the interior ridgeMap<Set<Int>, Int> index_of the interior_simplices##### Returns

SparseVector<Int> **cocircuit_equations**(C, interior_ridge_simplices, interior_simplices) → SparseMatrix<Int>A matrix whose rows contain the cocircuit equations of a cone C with respect to a list of interior simplices symmetries of the cone are NOT taken into account

Contained in extension`bundled:group`

.##### Parameters

Cone C Array<Set> interior_ridge_simplices Array<Set> interior_simplices ##### Options

String filename where to write the output (default empty)Bool reduce_rows whether to perform row reduction (default 1)Int log_frequency how often to print log messages##### Returns

SparseMatrix<Int> **contraction**(C, v)Contract a vector configuration

*C*along a specified vector*v*.##### Parameters

VectorConfiguration C Int v index of the vector to contract**deletion**(C, v)Delete a specified vector

*v*from a vector configuration*C*.##### Parameters

VectorConfiguration C Int v index of the vector to delete**projected_cocircuit_equations**(C, ridge_rep, isotypic_components) → SparseMatrix<Rational>A SparseMatrix whose rows contain projections of the cocircuit equations of a cone C corresponding to the orbit of a specified ridge onto a direct sum of specified isotypic components

Contained in extension`bundled:group`

.##### Parameters

Cone C Set<Int> ridge_rep interior ridgeSet<Int> isotypic_components the isotypic components to project to##### Returns

SparseMatrix<Rational> **quotient_of_triangulation**(T, G, R) → SparseVectorIn a triangulation T, find the number of representatives of simplices wrt to G, and return the counts in the order indicated by the array R

Contained in extension`bundled:group`

.##### Parameters

Array<Set> T the input triangulation,Array<Array<Int>> G the generators of the symmetry groupArray<Set> R the canonical lex-min representatives of the simplices##### Options

Bool foldable is the triangulation foldable?##### Returns

SparseVector V the number of times a simplex G-isomorphic to each representative in R occurs in T

Functions based on graph isomorphisms.

**congruent**(P1, P2) → ScalarCheck whether two given polytopes

*P1*and*P2*are congruent, i.e. whether there is an affine isomorphism between them that is induced by a (possibly scaled) orthogonal matrix. Returns the scale factor, or 0 if the polytopes are not congruent.We are using the reduction of the congruence problem (for arbitrary point sets) to the graph isomorphism problem due to:

Akutsu, T.: On determining the congruence of point sets in `d` dimensions.Comput. Geom. Theory Appl. 9, 247--256 (1998), no. 4##### Parameters

Polytope P1 the first polytopePolytope P2 the second polytope##### Returns

Scalar the square of the scale factor or 0 if the polytopes are not congruent**Example:**- Let's first consider an isosceles triangle and its image of the reflection in the origin:
`> $t = simplex(2);`

`> $tr = simplex(2,-1);`

Those two are congruent:`> print congruent($t,$tr);`

`1`

If we scale one of them, we get a factor:`> print congruent(scale($t,2),$tr);`

`4`

But if we instead take a triangle that is not isosceles, we get a negative result.`> $tn = new Polytope(VERTICES => [[1,0,0],[1,2,0],[1,0,1]]);`

`> print congruent($t,$tn);`

`0`

**equal_polyhedra**(P1, P2) → BoolTests if the two polyhedra

*P1*and*P2*are equal.##### Parameters

Polytope P1 the first polytopePolytope P2 the second polytope##### Options

Bool verbose Prints information on the difference between P1 and P2 if they are not equal.##### Returns

Bool true if the two polyhedra are equal, false otherwise**Example:**`> $p = new Polytope(VERTICES => [[1,-1,-1],[1,1,-1],[1,-1,1],[1,1,1]]);`

`> print equal_polyhedra($p,cube(2));`

`1`

To see why two polytopes are unequal, try this:`> print equal_polyhedra($p,cube(3),verbose => 1);`

`Cones/Polytopes do no live in the same ambient space.`

`> print equal_polyhedra($p,simplex(2),verbose => 1);`

`Inequality 0 1 0 not satisfied by point 1 -1 -1.`

**find_facet_vertex_permutations**(P1, P2) → Pair<Array<Int>, Array<Int>>Find the permutations of facets and vertices which maps the cone or polyhedron

*P1*to*P2*. The facet permutation is the first component, the vertex permutation is the second component of the return value.Only the combinatorial isomorphism is considered. If the polytopes are not isomorphic, an exception is thrown.

##### Parameters

Cone P1 the first cone/polytopeCone P2 the second cone/polytope##### Returns

Pair<Array<Int>, Array<Int>> the facet and the vertex permutations**included_polyhedra**(P1, P2) → BoolTests if polyhedron

*P1*is included in polyhedron*P2*.##### Parameters

Polytope P1 the first polytopePolytope P2 the second polytope##### Options

Bool verbose Prints information on the difference between P1 and P2 if none is included in the other.##### Returns

Bool 'true' if*P1*is included in*P2*, 'false' otherwise**Example:**`> print included_polyhedra(simplex(3),cube(3));`

`1`

To see in what way the two polytopes differ, try this:`> print included_polyhedra(cube(2),cube(3),verbose=>1);`

`Cones/Polytopes do no live in the same ambient space.`

**isomorphic**(P1, P2) → BoolCheck whether the face lattices of two cones or polytopes are isomorphic. The problem is reduced to graph isomorphism of the vertex-facet incidence graphs.

##### Parameters

Cone P1 the first cone/polytopeCone P2 the second cone/polytope##### Returns

Bool 'true' if the face lattices are isomorphic, 'false' otherwise**Example:**- The following compares the standard 2-cube with a polygon generated as the convex hull of five points. The return value is true since both polygons are quadrangles.
`> $p = new Polytope(POINTS=>[[1,-1,-1],[1,1,-1],[1,-1,1],[1,1,1],[1,0,0]]);`

`> print isomorphic(cube(2),$p);`

`1`

**lattice_isomorphic_smooth_polytopes**(P1, P2) → BoolTests whether two smooth lattice polytopes are lattice equivalent by comparing lattice distances between vertices and facets.

##### Parameters

Polytope P1 the first lattice polytopePolytope P2 the second lattice polytope##### Returns

Bool 'true' if the polytopes are lattice equivalent, 'false' otherwise**Example:**`> $t = new Vector(2,2);`

`> print lattice_isomorphic_smooth_polytopes(cube(2),translate(cube(2),$t));`

`1`

These functions are for checking the consistency of some properties.

**check_inc**(points, hyperplanes, sign, verbose) → BoolCheck coordinate data. For each pair of vectors from two given matrices their inner product must satisfy the given relation.

##### Parameters

Matrix points Matrix hyperplanes String sign composed of one or two characters from [-+0], representing the allowed domain of the vector inner products.Bool verbose print all products violating the required relation##### Returns

Bool 'true' if all relations are satisfied, 'false' otherwise**Example:**- Let's check which vertices of the square lie in its zeroth facet:
`> $H = cube(2)->FACETS->minor([0],All);`

`> print check_inc(cube(2)->VERTICES,$H,'0',1);`

`<1,0> ( 1 1 -1 ) * [ 1 1 0 ] == 2`

`<3,0> ( 1 1 1 ) * [ 1 1 0 ] == 2`

`number of points==4, number of hyperplanes==1, -:0, 0:2, +:2, total:4`

Thus, the first and third vertex don't lie on the hyperplane defined by the facet but on the positive side of it, and the remaining two lie on the hyperplane.

**check_poly**(VIF) → PolytopeTry to check whether a given vertex-facet incidence matrix

*VIF*defines a polytope. Note that a successful certification by check_poly is**not sufficient**to determine whether an incidence matrix actually defines a polytope. Think of it as a plausibility check.##### Parameters

IncidenceMatrix VIF ##### Options

Bool dual transposes the incidence matrixBool verbose prints information about the check.##### Returns

Polytope the resulting polytope under the assumption that*VIF*actually defines a polytope**validate_moebius_strip**(P) → BoolValidates the output of the client edge_orientable, in particular it checks whether the MOEBIUS_STRIP_EDGES form a Moebius strip with parallel opposite edges. Prints a message to stdout.

**validate_moebius_strip_quads**(P) → Matrix<Int>Checks whether the MOEBIUS_STRIP_QUADS form a Moebius strip with parallel opposite edges. Prints a message to stdout and returns the MOEBIUS_STRIP_EDGES if the answer is affirmative.

##### Parameters

Polytope P the given polytope##### Options

Bool verbose print details##### Returns

Matrix<Int> the Moebius strip edges

The following functions allow for the conversion of the coordinate type of cones and polytopes.

**affine_float_coords**(P) → Matrix<Float>Dehomogenize the vertex coordinates and convert them to Float

**Example:**`> print cube(2,1/2)->VERTICES;`

`1 -1/2 -1/2`

`1 1/2 -1/2`

`1 -1/2 1/2`

`1 1/2 1/2`

`> print affine_float_coords(cube(2,1/2));`

`-0.5 -0.5`

`0.5 -0.5`

`-0.5 0.5`

`0.5 0.5`

**convert_to**<Coord> (c) → Cone<Coord>Creates a new Cone object with different coordinate type target coordinate type

*Coord*must be specified in angle brackets e.g. $new_cone = convert_to<Coord>($cone)##### Type Parameters

Coord target coordinate type##### Parameters

Cone c the input cone##### Returns

Cone<Coord> a new cone object or*C*itself it has the requested type**convert_to**<Coord> (P) → Polytope<Coord>provide a Polytope object with desired coordinate type

##### Type Parameters

Coord target coordinate type##### Parameters

Polytope P source object##### Returns

Polytope<Coord> *P*if it already has the requested type, a new object otherwise**Example:**`> print cube(2)->type->full_name;`

`Polytope<Rational>`

`> $pf = convert_to<Float>(cube(2));`

`> print $pf->type->full_name;`

`Polytope<Float>`

Tight spans and their conections to polyhedral geometry

**max_metric**(n) → MatrixCompute a metric such that the f-vector of its tight span is maximal among all metrics with

*n*points.S. Herrmann and M. Joswig: Bounds on the f-vectors of tight spans.Contrib. Discrete Math., Vol.2, 2007 161-184**Example:**- To compute the max-metric of four points and display the f-vector of its tight span, do this:
`> $M = max_metric(5);`

`> $w = new Vector(1,1,1,2,3);`

`> print tight_span($M,$w)->F_VECTOR;`

`6 15 20 15 6`

**min_metric**(n) → MatrixCompute a metric such that the f-vector of its tight span is minimal among all metrics with

*n*points.S. Herrmann and M. Joswig: Bounds on the f-vectors of tight spans.Contrib. Discrete Math., Vol.2, 2007 161-184**Example:**- To compute the min-metric of four points and display the f-vector of its tight span, do this:
`> $M = min_metric(5);`

`> $w = new Vector(1,1,1,2,3);`

`> print tight_span($M,$w)->F_VECTOR;`

`6 15 20 15 6`

**thrackle_metric**(n) → MatrixCompute a metric such that the f-vector of its tight span is maximal among all metrics with

*n*points. This metric can be interpreted as a lifting function for the thrackle triangulation (see de Loera, Sturmfels and Thomas: Groebner Basis and triangultaions of the second hypersimplex)**tight_span**(points, weight, full) → PolytopeCompute the tight span dual to the regular subdivision obtained by lifting

*points*to*weight*and taking the lower complex of the resulting polytope.##### Parameters

Matrix points Vector weight Bool full true if the polytope is full-dimensional. Default value is 1.##### Returns

Polytope (The polymake object TightSpan is only used for tight spans of finite metric spaces, not for tight spans of subdivisions in general.)**Example:**- This computes the tight span dual to a regular subdivision of the squares vertices.
`> $p = tight_span(cube(2)->VERTICES,new Vector(1,1,1,23));`

`> print $p->VERTICES;`

`0 1 1 0`

`0 1 0 1`

`1 -1 0 0`

`1 -1 -11 -11`

`0 1 0 -1`

`0 1 -1 0`

**tight_span**(P) → PolytopeCompute the tight span dual to the regular subdivision of a polytope

*P*obtained by the WEIGHTS and taking the lower complex of the resulting polytope.##### Parameters

Polytope P ##### Returns

Polytope (The polymake object TightSpan is only used for tight spans of finite metric spaces, not for tight spans of subdivisions in general.)**Example:**- The following assigns a regular subdivision induced by weights to the square and then creates the tight span dual to it.
`> $c = cube(2);`

`> $c->POLYTOPAL_SUBDIVISION(WEIGHTS=>[1,1,1,23]);`

`> $p = tight_span($c);`

`> print $p->VERTICES;`

`0 1 1 0`

`0 1 0 1`

`1 -1 0 0`

`1 -1 -11 -11`

`0 1 0 -1`

`0 1 -1 0`

**ts_max_metric**(n) → TightSpan**ts_min_metric**(n) → TightSpan**ts_thrackle_metric**(n) → TightSpanCompute a tight span of a metric such that its f-vector is maximal among all metrics with

*n*points. This metric can be interpreted as a lifting function for the thrackle triangulation (see de Loera, Sturmfels and Thomas: Groebner Basis and triangultaions of the second hypersimplex)

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

**all_steiner_points**(P) → MatrixCompute the Steiner points of all faces of a polyhedron

*P*using a randomized approximation of the angles.*P*must be BOUNDED.**dihedral_angle**(H1, H2) → FloatCompute the dihedral angle between two (oriented) affine or linear hyperplanes.

##### Parameters

Vector<Scalar> H1 : first hyperplaneVector<Scalar> H2 : second hyperplane##### Options

Bool deg output in degrees rather than radians, default is falseBool cone hyperplanes seen as linear hyperplanes, default is false##### Returns

Float **Example:**`> $H1 = new Vector(1,5,5);`

`> $H2 = new Vector(1,-5,5);`

`> print dihedral_angle($H1,$H2,deg=>1);`

`90`

**induced_lattice_basis**(p) → Matrix<Integer>Returns a basis of the affine lattice spanned by the vertices

**Example:**- The vertices of the 2-simplex span all of Z^2...
`> print induced_lattice_basis(simplex(2));`

`0 1 0`

`0 0 1`

...but if we scale it with 2, we get only every second lattice point.`> print induced_lattice_basis(scale(simplex(2),2));`

`0 2 0`

`0 0 2`

**integer_points_bbox**(P) → Matrix<Integer>Enumerate all integer points in the given polytope by searching a bounding box.

**Example:**`> $p = new Polytope(VERTICES=>[[1,1.3,0.5],[1,0.2,1.2],[1,0.1,-1.5],[1,-1.4,0.2]]);`

`> print integer_points_bbox($p);`

`1 0 -1`

`1 -1 0`

`1 0 0`

`1 1 0`

`1 0 1`

**is_vertex**(q, points) → BoolChecks whether there exists a hyperplane seperating the given point

*q*from the*points*via solving a suitable LP (compare cdd redundancy check). If so,*q*would be a new vertex of the polytope P generated by*points*and*q*that is not a vertex in the convex hull of*points*alone. To get the seperating hyperplane, use*seperating_hyperplane*Works without knowing the facets of P!##### Parameters

Vector q the vertex (candidate) which is to be separated from*points*Matrix points the points from which*q*is to be separated##### Returns

Bool 'true' if*q*is a vertex**Example:**`> $q = cube(2)->VERTICES->row(0);`

`> $points = cube(2)->VERTICES->minor(sequence(1,3),All);`

`> print is_vertex($q,$points);`

`1`

**minimal_vertex_angle**(P) → Float**normaliz_compute**(C) → ListCompute degree one elements, Hilbert basis or Hilbert series of a cone C with libnormaliz Hilbert series and Hilbert h-vector depend on the given grading and will not work unless C is HOMOGENEOUS or a MONOID_GRADING is set

Contained in extension`bundled:libnormaliz`

.##### Parameters

Cone C ##### Options

Bool from_facets supply facets instead of rays to normalizBool degree_one_generators compute the generators of degree one, i.e. lattice points of the polytopeBool hilbert_basis compute Hilbert basis of the cone CBool h_star_vector compute Hilbert h-vector of the cone CBool hilbert_series compute Hilbert series of the monoidBool facets compute support hyperplanes (=FACETS,LINEAR_SPAN)Bool rays compute extreme rays (=RAYS)Bool dual_algorithm use the dual algorithm by PottierBool skip_long do not try to use long coordinates firstBool verbose libnormaliz debug output##### Returns

List (Matrix<Integer> degree one generators, Matrix<Integer> Hilbert basis, Vector<Integer> Hilbert h-vector, RationalFunction Hilbert series, Matrix<Rational> facets, Matrix<Rational> linear_span, Matrix<Rational> rays) (only requested items)**print_face_lattice**(VIF, dual)Write the face lattice of a vertex-facet incidence matrix

*VIF*to stdout. If*dual*is set true the face lattice of the dual is printed.##### Parameters

IncidenceMatrix VIF Bool dual **Example:**- To get a nice representation of the squares face lattice, do this:
`> print_face_lattice(cube(2)->VERTICES_IN_FACETS);`

`FACE_LATTICE`

`[ -1 : 4 ]`

`{{0 1} {0 2} {1 3} {2 3}}`

`[ -2 : 4 ]`

`{{0} {1} {2} {3}}`

**steiner_point**(P) → Vector**zonotope_tiling_lattice**(P) → AffineLatticeCalculates a generating set for a tiling lattice for P, i.e., a lattice L such that P + L tiles the affine span of P.

##### Parameters

Polytope P the zonotope##### Options

Bool lattice_origin_is_vertex true if the origin of the tiling lattice should be a vertex of P; default false, ie, the origin will be the barycenter of P##### Returns

AffineLattice **Example:**- This determines a tiling lattice for a parallelogram with the origin as its vertex barycenter and prints it base vectors:
`> $M = new Matrix([[1,1,0],[1,1,1]]);`

`> $p = zonotope($M);`

`> $A = zonotope_tiling_lattice($p);`

`> print $A->BASIS;`

`0 -1 -1`

`0 0 1`

These functions provide tools from linear, integer and dicrete optimization. In particular, linear programs are defined here.

**ball_lifting_lp**(c, facet_index, conv_eps) → PolytopeComputes the inequalities and the linear objective for an LP to lift a simplicial

*d*-ball embedded starshaped in R^{d}.Contained in extension`bundled:local`

.##### Parameters

topaz::GeometricSimplicialComplex c Int facet_index index of the facet to be liftedRational conv_eps some epsilon > 0##### Returns

Polytope **core_point_algo**(p, optLPvalue) → ListAlgorithm to solve highly symmetric integer linear programs (ILP). It is required that the group of the ILP induces the alternating or symmetric group on the set of coordinate directions. The linear objective function is the vector (0,1,1,..,1).

**core_point_algo_Rote**(p, optLPvalue) → ListVersion of core_point_algo with improved running time (according to a suggestion by G. Rote). The core_point_algo is an algorithm to solve highly symmetric integer linear programs (ILP). It is required that the group of the ILP induces the alternating or symmetric group on the set of coordinate directions. The linear objective function is the vector (0,1,1,..,1).

**find_transitive_lp_sol**(Inequalities) → ListAlgorithm to solve symmetric linear programs (LP) of the form max c

^{t}x , c=(0,1,1,..,1) subject to the inequality system given by*Inequalities*. It is required that the symmetry group of the LP acts transitively on the coordinate directions.##### Parameters

Matrix Inequalities the inequalities describing the feasible region##### Returns

List (Vector<Rational> optimal solution, Rational optimal value, Bool feasible, Bool max_bounded)**Example:**- Consider the LP described by the facets of the 3-cube:
`> print find_transitive_lp_sol(cube(3)->FACETS);`

`1 1 1 1311`

The optimal solution is [1,1,1,1], its value under c is 3, and the LP is feasible and bounded in direction of c.

**inner_point**(points) → Vector**lp2poly**<Scalar> (file, testvec, prefix) → Polytope<Scalar>Read a linear programming problem given in LP-Format (as used by cplex & Co.) and convert it to a Polytope<Scalar> object.

**WARNING**The property FEASIBLE is**NOT**computed upon creation. This is done to avoid possibly long computation times before the object becomes available to the caller. This is**NOT**in keeping with standard practice in polymake, but after, all, these are linear programs and not polytopes.##### Type Parameters

Scalar coordinate type of the resulting polytope; default is Rational.##### Parameters

String file filename of a linear programming problem in LP-FormatVector testvec If present, after reading in each line of the LP it is checked whether testvec fulfills itString prefix If testvec is present, all variables in the LP file are assumed to have the form $prefix$i##### Options

Int nocheck Do not automatically compute FEASIBLE for the polytope (recommended for very large LPs)##### Returns

Polytope<Scalar> **poly2lp**(P, LP, maximize, file)Convert a polymake description of a polyhedron to LP format (as used by CPLEX and other linear problem solvers) and write it to standard output or to a

*file*. If*LP*comes with an attachment 'INTEGER_VARIABLES' (of type Array<Bool>), the output will contain an additional section 'GENERAL', allowing for IP computations in CPLEX. If the polytope is not FEASIBLE, the function will throw a runtime error.##### Parameters

Polytope P LinearProgram LP default value:*P*->LPBool maximize produces a maximization problem; default value: 0 (minimize)String file default value: standard output**porta2poly**(file) → Polytope<Rational>Read an .ieq or .poi file (porta input) or .poi.ieq or .ieq.poi (porta output) and convert it to a Polytope<Rational> object

**print_constraints**(C)Write the FACETS / INEQUALITIES and the LINEAR_SPAN / EQUATIONS (if present) of a polytope

*P*or cone*C*in a readable way. COORDINATE_LABELS are adopted if present.##### Parameters

Cone<Scalar> C the given polytope or cone##### Options

Array<String> ineq_labels changes the labels of the inequality rowsArray<String> eq_labels changes the labels of the equation rows**Example:**- The following prints the facet inequalities of the square, changing the labels.
`> print_constraints(cube(2),ineq_labels=>['zero','one','two','three']);`

`Facets:`

`zero: x1 >= -1`

`one: -x1 >= -1`

`two: x2 >= -1`

`three: -x2 >= -1`

**random_edge_epl**(G) → Vector<Rational>Computes a vector containing the expected path length to the maximum for each vertex of a directed graph

*G*. The random edge pivot rule is applied.**rand_aof**(P, start) → Vector<Rational>Produce a random abstract objective function on a given

*simple*polytope*P*. It is assumed that the boundary complex of the dual polytope is extendibly shellable. If, during the computation, it turns out that a certain partial shelling cannot be extended, then this is given instead of an abstract objective function. It is possible (but not required) to specify the index of the starting vertex*start*.##### Parameters

Polytope P a*simple*polytopeInt start the index of the starting vertex; default value: random##### Options

Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome.##### Returns

Vector<Rational> **separating_hyperplane**(q, points) → ListComputes (the normal vector of) a hyperplane which separates a given point

*q*from*points*via solving a suitable LP. The scalar product of the normal vector of the separating hyperplane and a point in*points*is greater or equal than 0 (same behavior as for facets!). If*q*is not a vertex of P=conv(*points*,*q*), the function returns a zero vector and sets*answer*to 'false'. Works without knowing the facets of P!##### Parameters

Vector q the vertex (candidate) which is to be separated from*points*Matrix points the points from which*q*is to be separated##### Returns

List (Bool answer, Vector sep_hyp)**Example:**- The following stores the result in the List @r and then prints the answer and a description of the hyperplane separating the zeroth vertex of the square from the others.
`> $q = cube(2)->VERTICES->row(0);`

`> $points = cube(2)->VERTICES->minor(sequence(1,3),All);`

`> @r = separating_hyperplane($q,$points);`

`> print $r[0];`

`1`

`> print $r[1];`

`0 1/2 1/2`

**separating_hyperplane**(p1, p2) → Vector**totally_dual_integral**(inequalities) → Bool**vertex_colors**(P, LP) → Array<RGB>Calculate RGB-color-values for each vertex depending on a linear or abstract objective function. Maximal and minimal affine vertices are colored as specified. Far vertices (= rays) orthogonal to the linear function normal vector are white. The colors for other affine vertices are linearly interpolated in the HSV color model.

If the objective function is linear and the corresponding LP problem is unbounded, then the affine vertices that would become optimal after the removal of the rays are painted pale.

##### Parameters

Polytope P LinearProgram LP ##### Options

RGB min the minimal RGB valueRGB max the maximal RGB value##### Returns

Array<RGB> **Example:**- This calculates a vertex coloring with respect to a linear program. For a better visualization, we also set the vertex thickness to 2.
`> $p = cube(3);`

`> $p->LP(LINEAR_OBJECTIVE=>[0,1,2,3]);`

`> $v = vertex_colors($p,$p->LP);`

`> $p->VISUAL(VertexColor=>$v,VertexThickness=>2);`

**write_foldable_max_signature_ilp**(P, outfile_name)**write_simplexity_ilp**(P, outfile_name)**write_symmetrized_simplexity_ilp**(P, outfile_name)

Special purpose functions.

**edge_orientable**(P)Checks whether a 2-cubical polytope

*P*is*edge-orientable*(in the sense of Hetyei), that means that there exits an orientation of the edges such that for each 2-face the opposite edges point in the same direction. It produces the certificates EDGE_ORIENTATION if the polytope is edge-orientable, or MOEBIUS_STRIP_EDGES otherwise. In the latter case, the output can be checked with the client validate_moebius_strip.##### Parameters

Polytope P the given 2-cubical polytope**lawrence_matrix**(M) → Matrix**matroid_indices_of_hypersimplex_vertices**() → Set<Int>For a given matroid return the bases as a subset of the vertices of the hypersimplex

**violations**(P, q) → SetCheck which relations, if any, are violated by a point.

##### Parameters

Polytope P Vector q ##### Options

String section Which section of P to test against qInt violating_criterion has the options: +1 (positive values violate; this is the default), 0 (*non*zero values violate), -1 (negative values violate)##### Returns

Set **Example:**- This calculates and prints the violated equations defining a square with the origin as its center and side length 2 with respect to a certain point:
`> $p = cube(2);`

`> $v = new Vector([1,2,2]);`

`> $S = violations($p,$v);`

`> print $S;`

`{1 3}`

**wronski_center_ideal**(L, lambda)Returns a system of polynomials which is necessary to check if degeneration avoids center of projection: compute eliminant e(s); this must not have a zero in (0,1)

##### Parameters

Matrix<Int> L lattice pointsVector<Int> lambda height function on lattice points**wronski_polynomial**(M, lambda, coeff, s)Returns a Wronski polynomial of a topaz::FOLDABLE triangulation of a lattice polytope

##### Parameters

Matrix<Int> M points (in homogeneous coordinates); affinely span the spaceVector<Int> lambda height function on lattice pointsArray<Rational> coeff coefficientsRational s additional Parameter in the polynomial##### Options

topaz::SimplicialComplex triangulation The triangulation of the pointset corresponding to the lifting functionRing ring the ring in which the polynomial should be**wronski_system**(M, lambda, coeff_array, s)Returns a Wronski system of a topaz::FOLDABLE triangulation of a lattice polytope

##### Parameters

Matrix<Int> M points (in homogeneous coordinates); affinely span the spaceVector<Int> lambda height function on lattice pointsArray<Array<Rational>> coeff_array coefficientsRational s additional Parameter in the polynomial##### Options

topaz::SimplicialComplex triangulation The triangulation of the pointset corresponding to the lifting functionRing ring the ring in which the polynomial should be

Various constructions of cones.

**normal_cone**(p, v, outer) → ConeComputes the normal cone of

*p*at the vertex*v*. By default this is the inner normal cone.##### Parameters

Polytope p Int v vertex number which is not contained in the far faceBool outer asks for outer normal cone. Default value is 0 (= inner)##### Returns

Cone **Example:**- To compute the outer normal cone of the 3-cube, do this:
`> $c = normal_cone(cube(3),0,1);`

`> print $c->RAYS;`

`-1 0 0`

`0 -1 0`

`0 0 -1`

**recession_cone**(P) → Cone<Scalar>retuns the recession cone (tail cone, characteristic cone) of a polytope

**subcone**(C) → ConeMake a subcone from a cone.

Constructing a point configuration, either from scratch or from existing objects.

**minkowski_sum**(P1, P2) → PointConfigurationProduces the Minkowski sum of

*P1*and*P2*.**Example:**`> $P1 = new PointConfiguration(POINTS=>simplex(2)->VERTICES);`

`> $P2 = new PointConfiguration(POINTS=>[[1,1,1],[1,-1,1],[1,1,-1],[1,-1,-1],[1,0,0]]);`

`> $m = minkowski_sum($P1,$P2);`

`> print $m->POINTS;`

`1 1 1`

`1 -1 1`

`1 1 -1`

`1 -1 -1`

`1 0 0`

`1 2 1`

`1 0 1`

`1 2 -1`

`1 0 -1`

`1 1 0`

`1 1 2`

`1 -1 2`

`1 1 0`

`1 -1 0`

`1 0 1`

**minkowski_sum**(lambda, P1, mu, P2) → PointConfigurationProduces the polytope

*lambda***P1*+*mu***P2*, where * and + are scalar multiplication and Minkowski addition, respectively.**Example:**`> $P1 = new PointConfiguration(POINTS=>simplex(2)->VERTICES);`

`> $P2 = new PointConfiguration(POINTS=>[[1,1,1],[1,-1,1],[1,1,-1],[1,-1,-1],[1,0,0]]);`

`> $m = minkowski_sum($P1,$P2);`

`> print $m->POINTS;`

`1 3 3`

`1 -3 3`

`1 3 -3`

`1 -3 -3`

`1 0 0`

`1 4 3`

`1 -2 3`

`1 4 -3`

`1 -2 -3`

`1 1 0`

`1 3 4`

`1 -3 4`

`1 3 -2`

`1 -3 -2`

`1 0 1`

Polytope constructions which take graphs as input.

**flow_polytope**<Scalar> (G, Arc_Bounds, source, sink) → PolytopeProduces the flow polytope of a directed Graph

*G*=(V,E) with a given*source*and*sink*. The flow polytope has the following outer description: forall v in V-{source, sink}: sum_{e in E going into v} x_e - sum_{e in E going out of v} x_e = 0sum_{e in E going into source} x_e - sum_{e in E going out of source} x_e <= 0

sum_{e in E going into sink} x_e - sum_{e in E going out of sink} x_e >= 0

forall e in E: x_e <= given bound on edge e

##### Type Parameters

Scalar ##### Parameters

Graph<Directed> G EdgeMap<Directed, Scalar> Arc_Bounds Int source Int sink ##### Returns

Polytope **flow_polytope**<Scalar> (G, Arc_Bounds, source, sink) → PolytopeProduces the flow polytope of a directed Graph

*G*=(V,E) with a given*source*and*sink*. The flow polytope has the following outer description: forall v in V-{source, sink}: sum_{e in E going into v} x_e - sum_{e in E going out of v} x_e = 0sum_{e in E going into source} x_e - sum_{e in E going out of source} x_e <= 0

sum_{e in E going into sink} x_e - sum_{e in E going out of sink} x_e >= 0

forall e in E: x_e <= given bound on edge e

##### Type Parameters

Scalar ##### Parameters

Graph<Directed> G Array<Scalar> Arc_Bounds Int source Int sink ##### Returns

Polytope **fractional_cut_polytope**(G) → Polytope**fractional_matching_polytope**(G) → Polytope**tutte_lifting**(G) → Polytope**weighted_digraph_polyhedron**(encoding) → polytope::PolytopeWeighted digraph polyhedron of a directed graph with a weight function. The graph and the weight function are combined into a matrix.

Polytope constructions which take other big objects as input.

**matroid_polytope**(m) → Polytope<Rational>Produce the matroid polytope from a matroid

*m*. Each vertex corresponds to a basis of the matroid, the non-bases coordinates get value 0, the bases coordinates get value 1.##### Parameters

matroid::Matroid m ##### Options

Bool inequalities also generate INEQUALITIES, if CONNECTED##### Returns

Polytope<Rational>

An important way of constructing polytopes is to modify an already existing polytope.

Actually, these functions don't alter the input polytope (it is forbidden in polymake), but create a new polytope object.

Many functions can at your choice either calculate the vertex or facet coordinates, or constrain themselves on the purely combinatorial description of the resulting polytope.

**bipyramid**(P, z, z_prime)Make a bipyramid over a pointed polyhedron. The bipyramid is the convex hull of the input polyhedron

*P*and two points (*v*,*z*), (*v*,*z_prime*) on both sides of the affine span of*P*. For bounded polyhedra, the apex projections*v*to the affine span of*P*coincide with the vertex barycenter of*P*.##### Parameters

Polytope P Scalar z distance between the vertex barycenter and the first apex, default value is 1.Scalar z_prime distance between the vertex barycenter and the second apex, default value is -*z*.##### Options

Bool no_coordinates : don't compute the coordinates, purely combinatorial description is produced.Bool relabel copy the vertex labels from the original polytope, label the new vertices with "Apex" and "Apex'".**Example:**- Here's a way to construct the 3-dimensional cross polytope:
`> $p = bipyramid(bipyramid(cube(1)));`

`> print equal_polyhedra($p,cross(3));`

`1`

**blending**(P1, v1, P2, v2) → PolytopeCompute the blending of two polyhedra at simple vertices. This is a slightly less standard construction. A vertex is

*simple*if its vertex figure is a simplex.Moving a vertex

*v*of a bounded polytope to infinity yields an unbounded polyhedron with all edges through*v*becoming mutually parallel rays. Do this to both input polytopes*P1*and*P2*at simple vertices*v1*and*v2*, respectively. Up to an affine transformation one can assume that the orthogonal projections of*P1*and*P2*in direction*v1*and*v2*, respectively, are mutually congruent.Any bijection b from the set of edges through

*v1*to the edges through*v2*uniquely defines a way of glueing the unbounded polyhedra to obtain a new bounded polytope, the*blending*with respect to b. The bijection is specified as a*permutation*of indices 0 1 2 etc. The default permutation is the identity.The number of vertices of the blending is the sum of the numbers of vertices of the input polytopes minus 2. The number of facets is the sum of the numbers of facets of the input polytopes minus the dimension.

The resulting polytope is described only combinatorially.

**cayley_embedding**(P_0, P_1, t_0, t_1) → PolytopeCreate a Cayley embedding of two polytopes (one of them must be pointed). The vertices of the first polytope

*P_0*get embedded to*(t_0,0)*and the vertices of the second polytope*P_1*to*(0,t_1)*.Default values are

*t_0*=*t_1*=1.The option

*relabel*creates an additional section VERTEX_LABELS.**cayley_embedding**(A) → PolytopeCreate a Cayley embedding of an array (P1,...,Pm) of polytopes. All polytopes must have the same dimension, at least one of them must be pointed, and all must be defined over the same number type. Each vertex

*v*of the*i*-th polytope is embedded to*v*|*t_i e_i*, where*t_i*is the*i*-th entry of the optional array*t*.The option

*relabel*creates an additional section VERTEX_LABELS.##### Parameters

Polytope A the input polytopes##### Options

Array<Scalar> factors array of scaling factors for the Cayley embedding; defaults to the all-1 vectorBool relabel ##### Returns

Polytope **cayley_polytope**(P_Array) → PolytopeConstruct the cayley polytope of a set of pointed lattice polytopes contained in

*P_Array*which is the convex hull of P_{1}×e_{1}, ..., P_{k}×e_{k}where e_{1}, ...,e_{k}are the standard unit vectors in R^{k}. In this representation the last k coordinates always add up to 1. The option*proj*projects onto the complement of the last coordinate.##### Parameters

Array<Polytope> P_Array an array containing the lattice polytopes P_{1},...,P_{k}##### Options

Bool proj ##### Returns

Polytope **cells_from_subdivision**(P, cells) → Polytope<Scalar>Extract the given

*cells*of the subdivision of a polyhedron and create a new polyhedron that has as vertices the vertices of the cells.##### Parameters

Polytope<Scalar> P Set<Int> cells ##### Options

Bool relabel copy the vertex labels from the original polytope##### Returns

Polytope<Scalar> **Example:**- First we create a nice subdivision for a small polytope:
`> $p = new Polytope(VERTICES=>[[1,0,0],[1,0,1],[1,1,0],[1,1,1],[1,3/2,1/2]]);`

`> $p->POLYTOPAL_SUBDIVISION(MAXIMAL_CELLS=>[[0,1,3],[1,2,3],[2,3,4]]);`

Then we create the polytope that has as vertices the vertices from cell 1 and 2, while keeping their labels.`> $c = cells_from_subdivision($p,[1,2],relabel=>1);`

`> print $c->VERTICES;`

`1 0 1`

`1 1 0`

`1 1 1`

`1 3/2 1/2`

`> print $c->VERTEX_LABELS;`

`1 2 3 4`

**cell_from_subdivision**(P, cell) → PolytopeExtract the given

*cell*of the subdivision of a polyhedron and write it as a new polyhedron.##### Parameters

Polytope P Int cell ##### Options

Bool relabel copy the vertex labels from the original polytope##### Returns

Polytope **Example:**- First we create a nice subdivision for our favourite 2-polytope, the square:
`> $p = cube(2);`

`> $p->POLYTOPAL_SUBDIVISION(MAXIMAL_CELLS=>[[0,1,3],[1,2,3]]);`

Then we extract the 0-th cell, copying the vertex labels.`> $c = cell_from_subdivision($p,0,relabel=>1);`

`> print $c->VERTICES;`

`1 1 -1`

`1 -1 1`

`1 1 1`

`> print $c->VERTEX_LABELS;`

`1 2 3`

**conv**(P_Array) → PropagatedPolytopeConstruct a new polyhedron as the convex hull of the polyhedra given in

*P_Array*.**dual_linear_program**(P, maximize) → PolytopeProduces the dual linear program for a given linear program of the form min {cx | Ax >= b, Bx = d}. Here (A,b) are given by the FACETS (or the INEQAULITIES), and (B,d) are given by the AFFINE_HULL (or by the EQUATIONS) of the polytope P, while the objective function c comes from an LP subobject.

**edge_middle**(P) → Polytope**facet**(P, facet) → ConeExtract the given

*facet*of a polyhedron and write it as a new polyhedron.##### Parameters

Cone P Int facet ##### Options

Bool no_coordinates don't copy the coordinates, produce purely combinatorial description.Bool relabel copy the vertex labels from the original polytope.##### Returns

Cone **Example:**- To create a cone from the vertices of the zeroth facet of the 3-cube, type this:
`> $p = facet(cube(3),0);`

**facet_to_infinity**(P, i) → PolytopeMake an affine transformation such that the i-th facet is transformed to infinity

**Example:**- This generates the polytope that is the positive quadrant in 2-space:
`> $p = new Polytope(VERTICES=>[[1,-1,-1],[1,0,1],[1,1,0]]);`

`> $pf = facet_to_infinity($q,2);`

`> print $pf->VERTICES;`

`1 0 0`

`0 0 1`

`0 1 0`

**free_sum**(P1, P2) → PolytopeConstruct a new polyhedron as the free sum of two given bounded ones.

##### Parameters

Polytope P1 Polytope P2 ##### Options

Bool force_centered if the input polytopes must be centered. Defaults to true.Bool no_coordinates produces a pure combinatorial description. Defaluts to false.##### Returns

Polytope **Example:**`> $p = free_sum(cube(2),cube(2));`

`> print $p->VERTICES;`

`1 -1 -1 0 0`

`1 1 -1 0 0`

`1 -1 1 0 0`

`1 1 1 0 0`

`1 0 0 -1 -1`

`1 0 0 1 -1`

`1 0 0 -1 1`

`1 0 0 1 1`

**free_sum_decomposition**(P) → Array<Polytope>Decompose a given polytope into the free sum of smaller ones

**free_sum_decomposition_indices**(P) → Array<Set>Decompose a given polytope into the free sum of smaller ones, and return the vertex indices of the summands

**Example:**`> $p = free_sum(cube(1),cube(1));`

`> print $p->VERTICES;`

`1 -1 0`

`1 1 0`

`1 0 -1`

`1 0 1`

`> print free_sum_decomposition_indices($p);`

`{0 1}`

`{2 3}`

**gc_closure**(P) → Polytope**integer_hull**(P) → Polytope**intersection**(C ...) → ConeConstruct a new polyhedron or cone as the intersection of given polyhedra and/or cones. Works only if all CONE_AMBIENT_DIM values are equal. If the input contains both cones and polytopes, the output will be a polytope.

**Example:**`> $p = intersection(cube(2),cross(2,3/2));`

`> print $p->VERTICES;`

`1 1 1/2 -1`

`1 1 1/2`

`1 1/2 1`

`1 1 -1/2`

`1 -1/2 1`

`1 -1 1/2`

`1 -1 -1/2`

`1 -1/2 -1`

**join_polytopes**(P1, P2) → PolytopeConstruct a new polyhedron as the join of two given bounded ones.

##### Parameters

Polytope P1 Polytope P2 ##### Options

Bool no_coordinates produces a pure combinatorial description.##### Returns

Polytope **Example:**- To join two squares, use this:
`> $p = join_polytopes(cube(2),cube(2));`

`> print $p->VERTICES;`

`1 -1 -1 -1 0 0`

`1 1 -1 -1 0 0`

`1 -1 1 -1 0 0`

`1 1 1 -1 0 0`

`1 0 0 1 -1 -1`

`1 0 0 1 1 -1`

`1 0 0 1 -1 1`

`1 0 0 1 1 1`

**lattice_bipyramid**(P, v, v_prime, z, z_prime) → PolytopeMake a lattice bipyramid over a polyhedron. The bipyramid is the convex hull of the input polyhedron

*P*and two points (*v*,*z*), (*v_prime*,*z_prime*) on both sides of the affine span of*P*.##### Parameters

Polytope P Vector v basis point for the first apexVector v_prime basis for the second apex If*v_prime*is omitted,*v*will be used for both apices. If both*v*and*v_prime*are omitted, it tries to find two vertices which don't lie in a common facet. If no such vertices can be found or*P*is a simplex, it uses an interior lattice point as both*v*and*v_prime*.Rational z height for the first apex, default value is 1Rational z_prime height for the second apex, default value is -*z*##### Options

Bool relabel copy the vertex labels from the original polytope, label the new vertices with "Apex" and "Apex'".##### Returns

Polytope **Example:**- To create the bipyramid over a square and keep the vertex labels, do this:
`> $p = lattice_bipyramid(cube(2),new Vector(1,0,0),relabel=>1);`

`> print $p->VERTICES;`

`1 -1 -1 0`

`1 1 -1 0`

`1 -1 1 0`

`1 1 1 0`

`1 0 0 1`

`1 0 0 -1`

`> print $p->VERTEX_LABELS;`

`0 1 2 3 Apex Apex'`

**lattice_pyramid**(P, z, v) → PolytopeMake a lattice pyramid over a polyhedron. The pyramid is the convex hull of the input polyhedron

*P*and a point*v*outside the affine span of*P*.##### Parameters

Polytope P Rational z the height for the apex (*v*,*z*), default value is 1.Vector v the lattice point to use as apex, default is the first vertex of*P*.##### Options

Bool relabel copy the original vertex labels, label the new top vertex with "Apex".##### Returns

Polytope **Example:**- To create the pyramid of height 5 over a square and keep the vertex labels, do this:
`> $p = lattice_pyramid(cube(2),5,new Vector(1,0,0),relabel=>1);`

`> print $p->VERTICES;`

`1 -1 -1 0`

`1 1 -1 0`

`1 -1 1 0`

`1 1 1 0`

`1 0 0 5`

`> print $p->VERTEX_LABELS;`

`0 1 2 3 Apex`

**make_totally_dual_integral**(P) → Polytope**mapping_polytope**(P1, P2) → PolytopeConstruct a new polytope as the

*mapping polytope*of two polytopes*P1*and*P2*. The mapping polytope is the set of all affine maps from R^{p}to R^{q}, that map*P1*into*P2*.The label of a new facet corresponding to v

_{1}and h_{1}will have the form "v_{1}*h_{1}".**minkowski_sum**(P1, P2) → PolytopeProduces the Minkowski sum of

*P1*and*P2*.**Example:**- The following stores the minkowski sum of a square and a triangle in the variable $p and then prints its vertices.
`> $p = minkowski_sum(cube(2),simplex(2));`

`> print $p->VERTICES;`

`1 -1 -1`

`1 2 -1`

`1 -1 2`

`1 2 1`

`1 1 2`

**minkowski_sum**(lambda, P1, mu, P2) → PolytopeProduces the polytope

*lambda***P1*+*mu***P2*, where * and + are scalar multiplication and Minkowski addition, respectively.**Example:**- The following stores the minkowski sum of a scaled square and a triangle in the variable $p and then prints its vertices.
`> $p = minkowski_sum(2,cube(2),1,simplex(2));`

`> print $p->VERTICES;`

`1 -2 -2`

`1 3 -2`

`1 -2 3`

`1 3 2`

`1 2 3`

**minkowski_sum_fukuda**(summands) → Polytope<Scalar>Computes the (VERTICES of the)

*Minkowski sum*of a list of polytopes using the algorithm by Fukuda described inKomei Fukuda, From the zonotope construction to the Minkowski addition of convex polytopes, J. Symbolic Comput., 38(4):1261-1272, 2004.**Example:**`> $p = minkowski_sum_fukuda([cube(2),simplex(2),cross(2)]);`

`> print $p->VERTICES;`

`1 -2 -1`

`1 -1 -2`

`1 3 -1`

`1 3 1`

`1 2 -2`

`1 -2 2`

`1 -1 3`

`1 1 3`

**mixed_integer_hull**(P, int_coords) → PolytopeProduces the mixed integer hull of a polyhedron

**pointed_part**(P) → Polytope**prism**(P, z1, z2) → PolytopeMake a prism over a pointed polyhedron. The prism is the product of the input polytope

*P*and the interval [*z1*,*z2*].##### Parameters

Polytope P the input polytopeScalar z1 the left endpoint of the interval; default value: -1Scalar z2 the right endpoint of the interval; default value: -*z1*##### Options

Bool no_coordinates only combinatorial information is handledBool relabel creates an additional section VERTEX_LABELS; the bottom facet vertices get the labels from the original polytope; the labels of their clones in the top facet get a tick (') appended.##### Returns

Polytope **Example:**- The following saves the prism over the square and the interval [-2,2] to the variable $p while relabeling, and then prints a nice representation of its vertices.
`> $p = prism(cube(2),-2,relabel=>1);`

`> print labeled($p->VERTICES,$p->VERTEX_LABELS);`

`0:1 -1 -1 -2 1:1 1 -1 -2 2:1 -1 1 -2 3:1 1 1 -2 0':1 -1 -1 2 1':1 1 -1 2 2':1 -1 1 2 3':1 1 1 2`

**product**(P1, P2) → PolytopeConstruct a new polytope as the product of two given polytopes

*P1*and*P2*.##### Parameters

Polytope P1 Polytope P2 ##### Options

Bool no_coordinates only combinatorial information is handledBool relabel creates an additional section VERTEX_LABELS; the label of a new vertex corresponding to v_{1}⊕ v_{2}will have the form LABEL_1*LABEL_2.##### Returns

Polytope **Example:**- The following builds the product of a square and an interval while relabeling, and then prints a nice representation of its vertices.
`> $p = product(cube(2),cube(1),relabel=>1);`

`> print labeled($p->VERTICES,$p->VERTEX_LABELS);`

`0*0:1 -1 -1 -1 0*1:1 -1 -1 1 1*0:1 1 -1 -1 1*1:1 1 -1 1 2*0:1 -1 1 -1 2*1:1 -1 1 1 3*0:1 1 1 -1 3*1:1 1 1 1`

**projection**(P, indices) → ConeOrthogonally project a pointed polyhedron to a coordinate subspace.

The subspace the polyhedron

*P*is projected on is given by indices in the set*indices*. The option*revert*inverts the coordinate list. The client scans for all coordinate sections and produces proper output from each. If a description in terms of inequalities is found, the client performs Fourier-Motzkin elimination unless the*nofm*option is set. Setting the*nofm*option is useful if the corank of the projection is large; in this case the number of inequalities produced grows quickly.##### Parameters

Cone P Array<Int> indices ##### Options

Bool revert inverts the coordinate listBool nofm suppresses Fourier-Motzkin elimination##### Returns

Cone **Example:**- project the 3-cube along the first coordinate, i.e. to the subspace spanned by the second and third coordinate:
`> $p = projection(cube(3),[1],revert=>1);`

`> print $p->VERTICES;`

`1 1 -1`

`1 1 1`

`1 -1 1`

`1 -1 -1`

**projection_full**(P) → ConeOrthogonally project a polyhedron to a coordinate subspace such that redundant columns are omitted, i.e., the projection becomes full-dimensional without changing the combinatorial type. The client scans for all coordinate sections and produces proper output from each. If a description in terms of inequalities is found, the client performs Fourier-Motzkin elimination unless the

*nofm*option is set. Setting the*nofm*option is useful if the corank of the projection is large; in this case the number of inequalities produced grows quickly.**pyramid**(P, z) → PolytopeMake a pyramid over a polyhedron. The pyramid is the convex hull of the input polyhedron

*P*and a point*v*outside the affine span of*P*. For bounded polyhedra, the projection of*v*to the affine span of*P*coincides with the vertex barycenter of*P*.##### Parameters

Polytope P Scalar z is the distance between the vertex barycenter and*v*, default value is 1.##### Options

Bool no_coordinates don't compute new coordinates, produce purely combinatorial description.Bool relabel copy vertex labels from the original polytope, label the new top vertex with "Apex".##### Returns

Polytope **Example:**- The following saves the pyramid of height 2 over the square to the variable $p. The vertices are relabeled.
`> $p = pyramid(cube(2),2,relabel=>1);`

To print the vertices and vertex labels of the newly generated pyramid, do this:`> print $p->VERTICES;`

`1 -1 -1 0`

`1 1 -1 0`

`1 -1 1 0`

`1 1 1 0`

`1 0 0 2`

`> print $p->VERTEX_LABELS;`

`0 1 2 3 Apex`

**rand_inner_points**(P, n) → PolytopeProduce a polytope with

*n*random points from the input polytope*P*. Each generated point is a convex linear combination of the input vertices with uniformly distributed random coefficients. Thus, the output points can't in general be expected to be distributed uniformly within the input polytope; cf. unirand for this. The polytope must be BOUNDED.**rand_vert**(V, n) → MatrixSelects

*n*random vertices from the set of vertices*V*. This can be used to produce random polytopes which are neither simple nor simplicial as follows: First produce a simple polytope (for instance at random, by using rand_sphere, rand, or unirand). Then use this client to choose among the vertices at random. Generalizes random 0/1-polytopes.**spherize**(P) → Polytope**Example:**- The following scales the 2-dimensional cross polytope by 23 and then projects it back onto the unit circle.
`> $p = scale(cross(2),23);`

`> $s = spherize($p);`

`> print $s->VERTICES;`

`1 1 0`

`1 -1 0`

`1 0 1`

`1 0 -1`

**stack**(P, stack_facets) → PolytopeStack a (simplicial or cubical) polytope over one or more of its facets.

For each facet of the polytope

*P*specified in*stack_facets*, the barycenter of its vertices is lifted along the normal vector of the facet. In the simplicial case, this point is directly added to the vertex set, thus building a pyramid over the facet. In the cubical case, this pyramid is truncated by a hyperplane parallel to the original facet at its half height. This way, the property of being simplicial or cubical is preserved in both cases.The option

*lift*controls the exact coordinates of the new vertices. It should be a rational number between 0 and 1, which expresses the ratio of the distance between the new vertex and the stacked facet, to the maximal possible distance. When*lift*=0, the new vertex would lie on the original facet.*lift*=1 corresponds to the opposite extremal case, where the new vertex hit the hyperplane of some neighbor facet. As an additional restriction, the new vertex can't lie further from the facet as the vertex barycenter of the whole polytope. Alternatively, the option*noc*(no coordinates) can be specified to produce a pure combinatorial description of the resulting polytope.##### Parameters

Polytope P Set<Int> stack_facets the facets to be stacked; A single facet to be stacked is specified by its number. Several facets can be passed in a Set or in an anonymous array of indices: [n1,n2,...] Special keyword*All*means that all factes are to be stacked.##### Options

Rational lift controls the exact coordinates of the new vertices; rational number between 0 and 1; default value: 1/2Bool no_coordinates produces a pure combinatorial description (in contrast to*lift*)Bool relabel creates an additional section VERTEX_LABELS; New vertices get labels 'f(FACET_LABEL)' in the simplicial case, and 'f(FACET_LABEL)-NEIGHBOR_VERTEX_LABEL' in the cubical case.##### Returns

Polytope **Example:**- To generate a cubical polytope by stacking all facets of the 3-cube to height 1/4, do this:
`> $p = stack(cube(3),All,lift=>1/4);`

**stellar_all_faces**(P, d) → PolytopePerform a stellar subdivision of all proper faces, starting with the facets.

Parameter

*d*specifies the lowest dimension of the faces to be divided. It can also be negative, then treated as the co-dimension. Default is 1, that is, the edges of the polytope.**stellar_indep_faces**(P, in_faces) → PolytopePerform a stellar subdivision of the faces

*in_faces*of a polyhedron*P*.The faces must have the following property: The open vertex stars of any pair of faces must be disjoint.

**tensor**(P1, P2) → PolytopeConstruct a new polytope as the convex hull of the tensor products of the vertices of two polytopes

*P1*and*P2*. Unbounded polyhedra are not allowed. Does depend on the vertex coordinates of the input.**Example:**- The following creates the tensor product polytope of two squares and then prints its vertices.
`> $p = tensor(cube(2),cube(2));`

`> print $p->VERTICES;`

`1 1 1 1 1`

`1 -1 1 -1 1`

`1 1 -1 1 -1`

`1 -1 1 1 -1`

`1 1 1 -1 -1`

`1 1 -1 -1 1`

`1 -1 -1 1 1`

`1 -1 -1 -1 -1`

**truncation**(P, trunc_vertices) → PolytopeCut off one or more vertices of a polyhedron.

The exact location of the cutting hyperplane(s) can be controlled by the option

*cutoff*, a rational number between 0 and 1. When*cutoff*=0, the hyperplane would go through the chosen vertex, thus cutting off nothing. When*cutoff*=1, the hyperplane touches the nearest neighbor vertex of a polyhedron.Alternatively, the option

*no_coordinates*can be specified to produce a pure combinatorial description of the resulting polytope, which corresponds to the cutoff factor 1/2.##### Parameters

Polytope P Set<Int> trunc_vertices the vertex/vertices to be cut off; A single vertex to be cut off is specified by its number. Several vertices can be passed in a Set or in an anonymous array of indices: [n1,n2,...] Special keyword*All*means that all vertices are to be cut off.##### Options

Scalar cutoff controls the exact location of the cutting hyperplane(s); rational number between 0 and 1; default value: 1/2Bool no_coordinates produces a pure combinatorial description (in contrast to*cutoff*)Bool relabel creates an additional section VERTEX_LABELS; New vertices get labels of the form 'LABEL1-LABEL2', where LABEL1 is the original label of the truncated vertex, and LABEL2 is the original label of its neighbor.##### Returns

Polytope **Example:**- To truncate the second vertex of the square at 1/4, try this:
`> $p = truncation(cube(2),2,cutoff=>1/4);`

`> print $p->VERTICES;`

`1 -1 -1`

`1 1 -1`

`1 1 1`

`1 -1 1/2`

`1 -1/2 1`

**unirand**(P, n) → PolytopeProduce a polytope with

*n*random points that are uniformly distributed within the given polytope*P*.*P*must be bounded and full-dimensional.##### Parameters

Polytope P Int n the number of random points##### Options

Bool boundary forces the points to lie on the boundary of the given polytopeInt seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome.##### Returns

Polytope **Examples:**- This produces a polytope as the convex hull of 5 random points in the square with the origin as its center and side length 2:
`> $p = unirand(cube(2),5);`

- This produces a polytope as the convex hull of 5 random points on the boundary of the square with the origin as its center and side length 2:
`> $p = unirand(cube(2),5,boundary=>1);`

**vertex_figure**(p, n) → PolytopeConstruct the vertex figure of the vertex

*n*of a polyhedron. The vertex figure is dual to a facet of the dual polytope.##### Parameters

Polytope p Int n number of the chosen vertex##### Options

Scalar cutoff controls the exact location of the cutting hyperplane. It should lie between 0 and 1. Value 0 would let the hyperplane go through the chosen vertex, thus degenerating the vertex figure to a single point. Value 1 would let the hyperplane touch the nearest neighbor vertex of a polyhedron. Default value is 1/2.Bool no_coordinates skip the coordinates computation, producing a pure combinatorial description.Bool relabel inherit vertex labels from the corresponding neighbor vertices of the original polytope.##### Returns

Polytope **Example:**- This produces a vertex figure of one vertex of a 3-dimensional cube with the origin as its center and side length 2. The result is a 2-simplex.
`> $p = vertex_figure(cube(3),5);`

`> print $p->VERTICES;`

`1 1 -1 0`

`1 1 0 1`

**wedge**(P, facet, z, z_prime) → PolytopeMake a wedge from a polytope over the given

*facet*. The polytope must be bounded. The inclination of the bottom and top side facet is controlled by*z*and*z_prime*, which are heights of the projection of the old vertex barycenter on the bottom and top side facet respectively.##### Parameters

Polytope P , must be boundedInt facet the `cutting edge'.Rational z default value is 0.Rational z_prime default value is -*z*, or 1 if*z*==0.##### Options

Bool no_coordinates don't compute coordinates, pure combinatorial description is produced.Bool relabel create vertex labels: The bottom facet vertices obtain the labels from the original polytope; the labels of their clones in the top facet get a tick (') appended.##### Returns

Polytope **Example:**- This produces the wedge from a square (over the facet 0), which yields a prism over a triangle:
`> $p = wedge(cube(2),0);`

`> print $p->VERTICES;`

`1 -1 -1 0`

`1 1 -1 0`

`1 -1 1 0`

`1 1 1 0`

`1 1 -1 2`

`1 1 1 2`

With these clients you can create polytopes belonging to various parameterized families which occur frequently in polytope theory, as well as several kinds of random polytopes. Regular polytopes and their friends are listed separately.

**associahedron**(d) → Polytope**binary_markov_graph**(observation) → PropagatedPolytopeDefines a very simple graph for a polytope propagation related to a Hidden Markov Model. The propagated polytope is always a polygon. For a detailed description see

M. Joswig: Polytope propagation, in: Algebraic statistics and computational biologyby L. Pachter and B. Sturmfels (eds.), Cambridge, 2005.**binary_markov_graph**(observation)##### Parameters

String observation encoded as a string of "0" and "1".**birkhoff**(n, even) → PolytopeConstructs the Birkhoff polytope of dimension

*n*^{2}. It is the polytope of*n*x*n*stochastic matrices (encoded as*n*^{2}row vectors), thus matrices with non-negative entries whose row and column entries sum up to one. Its vertices are the permutation matrices.**cyclic**(d, n) → PolytopeProduce a

*d*-dimensional cyclic polytope with*n*points. Prototypical example of a neighborly polytope. Combinatorics completely known due to Gale's evenness criterion. Coordinates are chosen on the (spherical) moment curve at integer steps from*start*, or 0 if unspecified. If*spherical*is true the vertices lie on the sphere with center (1/2,0,...,0) and radius 1/2. In this case (the necessarily positive) parameter*start*defaults to 1.##### Parameters

Int d the dimensionInt n the number of points##### Options

Int start defaults to 0 (or to 1 if spherical)Bool spherical defaults to false##### Returns

Polytope **Example:**- To create the 2-dimensional cyclic polytope with 6 points on the sphere, starting at 3:
`> $p = cyclic(2,6,start=>3,spherical=>1);`

`> print $p->VERTICES;`

`1 1/10 3/10`

`1 1/26 5/26`

`1 1/37 6/37`

`1 1/50 7/50`

`1 1/65 8/65`

**cyclic_caratheodory**(d, n) → PolytopeProduce a

*d*-dimensional cyclic polytope with*n*points. Prototypical example of a neighborly polytope. Combinatorics completely known due to Gale's evenness criterion. Coordinates are chosen on the trigonometric moment curve. For cyclic polytopes from other curves, see polytope::cyclic.**delpezzo**(d, scale) → Polytope<Scalar>Produce a

*d*-dimensional del-Pezzo polytope, which is the convex hull of the cross polytope together with the all-ones and minus all-ones vector.All coordinates are +/-

*scale*or 0.##### Parameters

Int d the dimensionScalar scale the absolute value of each non-zero vertex coordinate. Needs to be positive. The default value is 1.##### Returns

Polytope<Scalar> **dwarfed_cube**(d) → Polytope**dwarfed_product_polygons**(d, s) → Polytope**explicit_zonotope**(zones) → PolytopeProduce the POINTS of a zonotope as the iterated Minkowski sum of all intervals [-x,x], where x ranges over the rows of the input matrix

*zones*.##### Parameters

Matrix zones the input vectors##### Options

Bool rows_are_points the rows of the input matrix represent affine points(true, default) or linear vectors(false)##### Returns

Polytope **Example:**`> $M = new Matrix([1,1],[1,-1]);`

`> $p = explicit_zonotope($M,rows_are_points=>0);`

`> print $p->VERTICES;`

`1 2 0`

`1 0 -2`

`1 0 2`

`1 -2 0`

**fano_simplex**(d) → Polytope**fractional_knapsack**(b) → PolytopeProduce a knapsack polytope defined by one linear inequality (and non-negativity constraints).

**goldfarb**(d, e, g) → PolytopeProduces a

*d*-dimensional Goldfarb cube if*e*<1/2 and*g*<=e/4. The Goldfarb cube is a combinatorial cube and yields a bad example for the Simplex Algorithm using the Shadow Vertex Pivoting Strategy. Here we use the description as a deformed product due to Amenta and Ziegler. For*e*<1/2 and*g*=0 we obtain the Klee-Minty cubes.**goldfarb_sit**(d, eps, delta) → PolytopeProduces a

*d*-dimensional variation of the Klee-Minty cube if*eps*<1/2 and*delta*<=1/2. This Klee-Minty cube is scaled in direction x_(d-i) by (eps*delta)^i. This cube is a combinatorial cube and yields a bad example for the Simplex Algorithm using the 'steepest edge' Pivoting Strategy. Here we use a scaled description of the construction of Goldfarb and Sit.**hypersimplex**(k, d) → PolytopeProduce the hypersimplex Δ(

*k*,*d*), that is the the convex hull of all 0/1-vector in R^{d}with exactly*k*1s. Note that the output is never full-dimensional.##### Parameters

Int k number of 1sInt d ambient dimension##### Options

Bool group Bool no_vertices do not compute verticesBool no_facets do not compute facetsBool no_vif do not compute vertices in facets##### Returns

Polytope **Example:**- This creates the hypersimplex in dimension 4 with vertices with exactly two 1-entries and computes its symmetry group:
`> $h = hypersimplex(2,4,group=>1);`

**hypertruncated_cube**<Scalar> (d, k, lambda) → Polytope<Scalar>Produce a

*d*-dimensional hypertruncated cube. With symmetric linear objective function (0,1,1,...,1).##### Type Parameters

Scalar Coordinate type of the resulting polytope. Unless specified explicitly, deduced from the type of bound values, defaults to Rational.##### Parameters

Int d the dimensionScalar k cutoff parameterScalar lambda scaling of extra vertex##### Returns

Polytope<Scalar> **klee_minty_cube**(d, e) → Polytope**k_cyclic**(n, s) → PolytopeProduce a (rounded) 2*k-dimensional k-cyclic polytope with

*n*points, where k is the length of the input vector*s*. Special cases are the bicyclic (k=2) and tricyclic (k=3) polytopes. Only possible in even dimensions.The parameters s_i can be specified as integer, floating-point, or rational numbers. The coordinates of the i-th point are taken as follows:

cos(s_1 * 2πi/n),sin(s_1 * 2πi/n),...cos(s_k * 2πi/n),sin(s_k * 2πi/n)Warning: Some of the k-cyclic polytopes are not simplicial. Since the components are rounded, this function might output a polytope which is not a k-cyclic polytope!

More information can be found in the following references:

P. Schuchert: "Matroid-Polytope und Einbettungen kombinatorischer Mannigfaltigkeiten",PhD thesis, TU Darmstadt, 1995.Z. Smilansky: "Bi-cyclic 4-polytopes",Isr. J. Math. 70, 1990, 82-92**Example:**- To produce a (not exactly) regular pentagon, type this:
`> $p = k_cyclic(5,[1]);`

**lecture_hall_simplex**(d) → Polytope**long_and_winding**(r) → Polytope<PuiseuxFraction<Max, Rational, Rational> >Produce polytope in dimension 2r+2 with 3r+4 facets such that the total curvature of the central path is at least Omega(2^r). This establishes a counter-example to a continuous analog of the Hirsch conjecture by Deza, Terlaky and Zinchenko, Adv. Mech. Math. 17 (2009). The construction and its analysis can be found in Allamigeon, Benchimol, Gaubert and Joswig, arXiv: 1405.4161

##### Parameters

Int r defining parameter##### Options

Rational eval_ratio parameter for evaluating the puiseux rational functionsInt eval_exp to evaluate at eval_ratio^eval_exp, default: 1Float eval_float parameter for evaluating the puiseux rational functions##### Returns

Polytope<PuiseuxFraction<Max, Rational, Rational> > **max_GC_rank**(d) → Polytope**multiplex**(d, n) → PolytopeProduce a combinatorial description of a multiplex with parameters

*d*and*n*. This yields a self-dual*d*-dimensional polytope with*n*+1 vertices.They are introduced by

T. Bisztriczky,On a class of generalized simplices, Mathematika 43:27-285, 1996,see also

M.M. Bayer, A.M. Bruening, and J.D. Stewart,A combinatorial study of multiplexes and ordinary polytopes,Discrete Comput. Geom. 27(1):49--63, 2002.**neighborly_cubical**(d, n) → PolytopeProduce the combinatorial description of a neighborly cubical polytope. The facets are labelled in oriented matroid notation as in the cubical Gale evenness criterion.

See Joswig and Ziegler, Discr. Comput. Geom. 24:315-344 (2000).**newton**(p) → Polytope<Rational>Produce the Newton polytope of a polynomial

*p*.**Example:**- Create the newton polytope of 1+x^2+y like so:
`> $r=new Ring(qw(x y));`

`> ($x,$y)=$r->variables;`

`> $p=1+($x^2)+$y;`

`> $n = newton($p);`

`> print $n->VERTICES;`

`1 0 0`

`1 0 1`

`1 2 0`

**n_gon**(n, r) → PolytopeProduce a regular

*n*-gon. All vertices lie on a circle of radius*r*. The radius defaults to 1.**Example:**- To store the regular pentagon in the variable $p and calculate its symmetry group, do this:
`> $p = n_gon(5,group=>1);`

`> print $p->GROUP->GENERATORS;`

`0 4 3 2 1`

`1 2 3 4 0`

**perles_irrational_8_polytope**() → Polytope**permutahedron**(d) → PolytopeProduce a

*d*-dimensional permutahedron. The vertices correspond to the elements of the symmetric group of degree*d*+1.**Example:**- To create the 3-permutahedron and also compute its symmetry group, do this:
`> $p = permutahedron(3,group=>1);`

`> print $p->GROUP->GENERATORS;`

`1 0 2 3`

`3 0 1 2`

**pile**(sizes) → PolytopeProduce a (

*d*+1)-dimensional polytope from a pile of cubes. Start with a*d*-dimensional pile of cubes. Take a generic convex function to lift this polytopal complex to the boundary of a (*d*+1)-polytope.##### Parameters

Vector<Int> sizes a vector (s_{1},...,s_{d}, where s_{i}specifies the number of boxes in the i-th dimension.##### Returns

Polytope **pseudo_delpezzo**(d, scale) → Polytope<Scalar>Produce a

*d*-dimensional del-Pezzo polytope, which is the convex hull of the cross polytope together with the all-ones vector.All coordinates are +/-

*scale*or 0.##### Parameters

Int d the dimensionScalar scale the absolute value of each non-zero vertex coordinate. Needs to be positive. The default value is 1.##### Returns

Polytope<Scalar> **rand_box**(d, n, b) → PolytopeComputes the convex hull of

*n*points sampled uniformly at random from the integer points in the cube [0,*b*]^{d}.**rand_cyclic**(d, n) → PolytopeComputes a random instance of a cyclic polytope of dimension

*d*on*n*vertices by randomly generating a Gale diagram whose cocircuits have alternating signs.**rand_metric**<Scalar> (n) → Matrix**rand_metric_int**<Scalar> (n) → Matrix**rand_sphere**(d, n) → Polytope**rss_associahedron**(l) → PolytopeProduce a polytope of constrained expansions in dimension

*l*according toRote, Santos, and Streinu: Expansive motions and the polytope of pointed pseudo-triangulations.Discrete and computational geometry, 699--736, Algorithms Combin., 25, Springer, Berlin, 2003.**signed_permutahedron**(d) → Polytope**simplex**(d, scale) → PolytopeProduce the standard

*d*-simplex. Combinatorially equivalent to a regular polytope corresponding to the Coxeter group of type A_{d-1}. Optionally, the simplex can be scaled by the parameter*scale*.**Examples:**- To print the vertices (in homogeneous coordinates) of the standard 2-simplex, i.e. a right-angled isoceles triangle, type this:
`> print simplex(2)->VERTICES;`

`(3) (0 1)`

`1 1 0`

`1 0 1`

The first row vector is sparse and encodes the origin. - To create a 3-simplex and also calculate its symmetry group, type this:
`> simplex(3,group=>1);`

**stable_set**(G) → Polytope**transportation**(r, c) → Polytope**upper_bound_theorem**(d, n) → Polytope**Example:**- This produces the combinatorial data as mentioned above for 5 points in dimension 3 and prints the h-vector:
`> $p = upper_bound_theorem(3,5);`

`> print $p->H_VECTOR;`

`1 2 2 1`

**zonotope**(M) → Polytope<Scalar>Create a zonotope from a matrix whose rows are input points or vectors.

This method merely defines a Polytope object with the property ZONOTOPE_INPUT_POINTS.

##### Parameters

Matrix<Scalar> M input points or vectors##### Options

Bool rows_are_points true if M are points instead of vectors; default trueBool centered true if output should be centered; default true##### Returns

Polytope<Scalar> the zonotope generated by the input points or vectors**Examples:**- The following produces a parallelogram with the origin as its vertex barycenter:
`> $M = new Matrix([[1,1,0],[1,1,1]]);`

`> $p = zonotope($M);`

`> print $p->VERTICES;`

`1 0 -1/2`

`1 0 1/2`

`1 -1 -1/2`

`1 1 1/2`

- The following produces a parallelogram with the origin being a vertex (not centered case):
`> $M = new Matrix([[1,1,0],[1,1,1]]);`

`> $p = zonotope($M,centered=>0);`

`> print $p->VERTICES;`

`1 0 0`

`1 1 1`

`1 1 0`

`1 2 1`

**zonotope_vertices_fukuda**(M) → Matrix<E>Create the vertices of a zonotope from a matrix whose rows are input points or vectors.

**Example:**- The following stores the vertices of a parallelogram with the origin as its vertex barycenter and prints them:
`> $M = new Matrix([[1,1,0],[1,1,1]]);`

`> print zonotope_vertices_fukuda($M);`

`1 0 -1/2`

`1 0 1/2`

`1 -1 -1/2`

`1 1 1/2`

A way of constructing vector configurations is to modify an already existing vector configuration.

**projection**(P, indices) → VectorConfigurationOrthogonally project a vector configuration to a coordinate subspace.

The subspace the VectorConfiguration

*P*is projected on is given by indices in the set*indices*. The option*revert*inverts the coordinate list.##### Parameters

VectorConfiguration P Array<Int> indices ##### Options

Bool revert inverts the coordinate list##### Returns

VectorConfiguration **projection_full**(P) → VectorConfigurationOrthogonally project a vector configuration to a coordinate subspace such that redundant columns are omitted, i.e., the projection becomes full-dimensional without changing the combinatorial type.

Functions producing big objects which are not contained in application polytope.

**coxeter_group**(type) → group::GroupOfPolytopeProduces the Coxeter group of type

*type*. The Dynkin diagrams of the different types can be found in the description of the clients simple_roots_type_*.##### Parameters

String type the type of the Coxeter group##### Returns

group::GroupOfPolytope the Coxeter group of type*type***crosscut_complex**(p) → topaz::SimplicialComplexProduce the

*crosscut complex*of the boundary of a polytope.##### Parameters

Polytope p ##### Options

Bool geometric_realization create a topaz::GeometricSimplicialComplex; default is true##### Returns

topaz::SimplicialComplex

This includes the Platonic solids and their generalizations into two directions. In dimension 3 there are the Archimedean, Catalan and Johnson solids. In higher dimensions there are the simplices, the cubes, the cross polytopes and three other regular 4-polytopes.

**archimedean_solid**(s) → PolytopeCreate Archimedean solid of the given name. Some polytopes are realized with floating point numbers and thus not exact; Vertex-facet-incidences are correct in all cases.

##### Parameters

String s the name of the desired Archimedean solidPossible values:- 'truncated_tetrahedron'
- Truncated tetrahedron. Regular polytope with four triangular and four hexagonal facets.
- 'cuboctahedron'
- Cuboctahedron. Regular polytope with eight triangular and six square facets.
- 'truncated_cube'
- Truncated cube. Regular polytope with eight triangular and six octagonal facets.
- 'truncated_octahedron'
- Truncated Octahedron. Regular polytope with six square and eight hexagonal facets.
- 'rhombicuboctahedron'
- Rhombicuboctahedron. Regular polytope with eight triangular and 18 square facets.
- 'truncated_cuboctahedron'
- Truncated Cuboctahedron. Regular polytope with 12 square, eight hexagonal and six octagonal facets.
- 'snub_cube'
- Snub Cube. Regular polytope with 32 triangular and six square facets. The vertices are realized as floating point numbers. This is a chiral polytope.
- 'icosidodecahedron'
- Icosidodecahedon. Regular polytope with 20 triangular and 12 pentagonal facets.
- 'truncated_dodecahedron'
- Truncated Dodecahedron. Regular polytope with 20 triangular and 12 decagonal facets.
- 'truncated_icosahedron'
- Truncated Icosahedron. Regular polytope with 12 pentagonal and 20 hexagonal facets.
- 'rhombicosidodecahedron'
- Rhombicosidodecahedron. Regular polytope with 20 triangular, 30 square and 12 pentagonal facets.
- 'truncated_icosidodecahedron'
- Truncated Icosidodecahedron. Regular polytope with 30 square, 20 hexagonal and 12 decagonal facets.
- 'snub_dodecahedron'
- Snub Dodecahedron. Regular polytope with 80 triangular and 12 pentagonal facets. The vertices are realized as floating point numbers. This is a chiral polytope.

##### Returns

Polytope **Example:**- To show the mirror image of the snub cube use:
`> scale(archimedean_solid('snub_cube'),-1)->VISUAL;`

**catalan_solid**(s) → PolytopeCreate Catalan solid of the given name. Some polytopes are realized with floating point numbers and thus not exact; Vertex-facet-incidences are correct in all cases.

##### Parameters

String s the name of the desired Catalan solidPossible values:- 'triakis_tetrahedron'
- Triakis Tetrahedron. Dual polytope to the Truncated Tetrahedron, made of 12 isosceles triangular facets.
- 'triakis_octahedron'
- Triakis Octahedron. Dual polytope to the Truncated Cube, made of 24 isosceles triangular facets.
- 'rhombic_dodecahedron'
- Rhombic dodecahedron. Dual polytope to the cuboctahedron, made of 12 rhombic facets.
- 'tetrakis_hexahedron'
- Tetrakis hexahedron. Dual polytope to the truncated octahedron, made of 24 isosceles triangluar facets.
- 'disdyakis_dodecahedron'
- Disdyakis dodecahedron. Dual polytope to the truncated cuboctahedron, made of 48 scalene triangular facets.
- 'pentagonal_icositetrahedron'
- Pentagonal Icositetrahedron. Dual polytope to the snub cube, made of 24 irregular pentagonal facets. The vertices are realized as floating point numbers.
- 'pentagonal_hexecontahedron'
- Pentagonal Hexecontahedron. Dual polytope to the snub dodecahedron, made of 60 irregular pentagonal facets. The vertices are realized as floating point numbers.
- 'rhombic_triacontahedron'
- Rhombic triacontahedron. Dual polytope to the icosidodecahedron, made of 30 rhombic facets.
- 'triakis_icosahedron'
- Triakis icosahedron. Dual polytope to the icosidodecahedron, made of 30 rhombic facets.
- 'deltoidal_icositetrahedron'
- Deltoidal Icositetrahedron. Dual polytope to the rhombicubaoctahedron, made of 24 kite facets.
- 'pentakis_dodecahedron'
- Pentakis dodecahedron. Dual polytope to the truncated icosahedron, made of 60 isosceles triangular facets.
- 'deltoidal_hexecontahedron'
- Deltoidal hexecontahedron. Dual polytope to the rhombicosidodecahedron, made of 60 kite facets.
- 'disdyakis_triacontahedron'
- Disdyakis triacontahedron. Dual polytope to the truncated icosidodecahedron, made of 120 scalene triangular facets.

##### Returns

Polytope **cross**<Scalar> (d, scale) → Polytope<Scalar>Produce a

*d*-dimensional cross polytope. Regular polytope corresponding to the Coxeter group of type B_{d-1}= C_{d-1}.All coordinates are +/-

*scale*or 0.##### Type Parameters

Scalar Coordinate type of the resulting polytope. Unless specified explicitly, deduced from the type of bound values, defaults to Rational.##### Parameters

Int d the dimensionScalar scale the absolute value of each non-zero vertex coordinate. Needs to be positive. The default value is 1.##### Options

Bool group add a symmetry group description to the resulting polytope##### Returns

Polytope<Scalar> **Example:**- To create the 3-dimensional cross polytope, type
`> $p = cross(3);`

Check out it's vertices and volume:`> print $p->VERTICES;`

`1 1 0 0`

`1 -1 0 0`

`1 0 1 0`

`1 0 -1 0`

`1 0 0 1`

`1 0 0 -1`

`> print cross(3)->VOLUME;`

`4/3`

If you rather had a bigger one, type`> $p_scaled = cross(3,2);`

`> print $p_scaled->VOLUME;`

`32/3`

To also calculate the symmetry group, do this:`> $p = cross(3,group=>1);`

You can then print the generators of this group like this:`> print $p->GROUP->GENERATORS;`

`1 0 2 3 4 5`

`2 3 0 1 4 5`

`0 1 4 5 2 3`

**cube**<Scalar> (d, x_up, x_low) → Polytope<Scalar>Produce a

*d*-dimensional cube. Regular polytope corresponding to the Coxeter group of type B_{d-1}= C_{d-1}.The bounding hyperplanes are x

_{i}<=*x_up*and x_{i}>=*x_low*.##### Type Parameters

Scalar Coordinate type of the resulting polytope. Unless specified explicitly, deduced from the type of bound values, defaults to Rational.##### Parameters

Int d the dimensionScalar x_up upper bound in each dimensionScalar x_low lower bound in each dimension##### Options

Bool group add a symmetry group description to the resulting polytope##### Returns

Polytope<Scalar> **Examples:**- This yields a +/-1 cube of dimension 3 and stores it in the variable $c.
`> $c = cube(3);`

- This stores a standard unit cube of dimension 3 in the variable $c.
`> $c = cube(3,0);`

- This prints the area of a square with side length 4 translated to have its vertex barycenter at [5,5]:
`> print cube(2,7,3)->VOLUME;`

`16`

**cuboctahedron**() → Polytope**dodecahedron**() → Polytope**icosahedron**() → Polytope**icosidodecahedron**() → Polytope**johnson_solid**(n) → Polytope**johnson_solid**(s) → PolytopeCreate Johnson solid with the given name. Some polytopes are realized with floating point numbers and thus not exact; Vertex-facet-incidences are correct in all cases.

##### Parameters

String s the name of the desired Johnson polytopePossible values:- 'square_pyramid'
- Square pyramid with regular facets. Johnson solid J1.
- 'pentagonal_pyramid'
- Pentagonal pyramid with regular facets. Johnson solid J2.
- 'triangular_cupola'
- Triangular cupola with regular facets. Johnson solid J3.
- 'square_cupola'
- Square cupola with regular facets. Johnson solid J4.
- 'pentagonal_cupola'
- Pentagonal cupola with regular facets. Johnson solid J5.
- 'pentagonal_rotunda'
- Pentagonal rotunda with regular facets. Johnson solid J6.
- 'elongated_triangular_pyramid'
- Elongated triangular pyramid with regular facets. Johnson solid J7.
- 'elongated_square_pyramid'
- Elongated square pyramid with regular facets. Johnson solid J8.
- 'elongated_pentagonal_pyramid'
- Elongated pentagonal pyramid with regular facets. Johnson solid J9. The vertices are realized as floating point numbers.
- 'gyroelongated_square_pyramid'
- Gyroelongated square pyramid with regular facets. Johnson solid J10. The vertices are realized as floating point numbers.
- 'gyroelongated_pentagonal_pyramid'
- Gyroelongated pentagonal pyramid with regular facets. Johnson solid J11.
- 'triangular_bipyramid'
- Triangular bipyramid with regular facets. Johnson solid J12.
- 'pentagonal_bipyramid'
- Pentagonal bipyramid with regular facets. Johnson solid J13. The vertices are realized as floating point numbers.
- 'elongated_triangular_bipyramid'
- Elongated triangular bipyramid with regular facets. Johnson solid J14.
- 'elongated_square_bipyramid'
- Elongated square bipyramid with regular facets. Johnson solid J15.
- 'elongated_pentagonal_bipyramid'
- Elongated pentagonal bipyramid with regular facets. Johnson solid J16. The vertices are realized as floating point numbers.
- 'gyroelongated_square_bipyramid'
- Gyroelongted square bipyramid with regular facets. Johnson solid J17. The vertices are realized as floating point numbers.
- 'elongated_triangular_cupola'
- Elongted triangular cupola with regular facets. Johnson solid J18. The vertices are realized as floating point numbers.
- 'elongated_square_cupola'
- Elongted square cupola with regular facets. Johnson solid J19.
- 'elongated_pentagonal_cupola'
- Elongted pentagonal cupola with regular facets. Johnson solid J20 The vertices are realized as floating point numbers.
- 'elongated_pentagonal_rotunda'
- Elongated pentagonal rotunda with regular facets. Johnson solid J21. The vertices are realized as floating point numbers.
- 'gyroelongated_triangular_cupola'
- Gyroelongted triangular cupola with regular facets. Johnson solid J22. The vertices are realized as floating point numbers.
- 'gyroelongated_square_cupola'
- Gyroelongted square cupola with regular facets. Johnson solid J23. The vertices are realized as floating point numbers.
- 'gyroelongated_pentagonal_cupola'
- Gyroelongted pentagonal cupola with regular facets. Johnson solid J24. The vertices are realized as floating point numbers.
- 'gyroelongated_pentagonal_rotunda'
- Gyroelongted pentagonal rotunda with regular facets. Johnson solid J25. The vertices are realized as floating point numbers.
- 'gyrobifastigium'
- Gyrobifastigium with regular facets. Johnson solid J26.
- 'triangular_orthobicupola'
- Triangular orthobicupola with regular facets. Johnson solid J27.
- 'square_orthobicupola'
- Square orthobicupola with regular facets. Johnson solid J28.
- 'square_gyrobicupola'
- Square gyrobicupola with regular facets. Johnson solid J29.
- 'pentagonal_orthobicupola'
- Pentagonal orthobicupola with regular facets. Johnson solid J30. The vertices are realized as floating point numbers.
- 'pentagonal_gyrobicupola'
- Pentagonal gyrobicupola with regular facets. Johnson solid J31. The vertices are realized as floating point numbers.
- 'pentagonal_orthocupolarotunda'
- Pentagonal orthocupolarotunda with regular facets. Johnson solid J32. The vertices are realized as floating point numbers.
- 'pentagonal_gyrocupolarotunda'
- Pentagonal gyrocupolarotunda with regular facets. Johnson solid J33. The vertices are realized as floating point numbers.
- 'pentagonal_orthobirotunda'
- Pentagonal orthobirotunda with regular facets. Johnson solid J32. The vertices are realized as floating point numbers.
- 'elongated_triangular_orthbicupola'
- Elongated triangular orthobicupola with regular facets. Johnson solid J35. The vertices are realized as floating point numbers.
- 'elongated_triangular_gyrobicupola'
- Elongated triangular gyrobicupola with regular facets. Johnson solid J36. The vertices are realized as floating point numbers.
- 'elongated_square_gyrobicupola'
- Elongated square gyrobicupola with regular facets. Johnson solid J37.
- 'elongated_pentagonal_orthobicupola'
- Elongated pentagonal orthobicupola with regular facets. Johnson solid J38. The vertices are realized as floating point numbers.
- 'elongated_pentagonal_gyrobicupola'
- Elongated pentagonal gyrobicupola with regular facets. Johnson solid J39. The vertices are realized as floating point numbers.
- 'elongated_pentagonal_orthocupolarotunda'
- Elongated pentagonal orthocupolarotunda with regular facets. Johnson solid J40. The vertices are realized as floating point numbers.
- 'elongated_pentagonal_gyrocupolarotunda'
- Elongated pentagonal gyrocupolarotunda with regular facets. Johnson solid J41. The vertices are realized as floating point numbers.
- 'elongated_pentagonal_orthobirotunda'
- Elongated pentagonal orthobirotunda with regular facets. Johnson solid J42. The vertices are realized as floating point numbers.
- 'elongated_pentagonal_gyrobirotunda'
- Elongated pentagonal gyrobirotunda with regular facets. Johnson solid J43. The vertices are realized as floating point numbers.
- 'gyroelongated_triangular_bicupola'
- Gyroelongated triangular bicupola with regular facets. Johnson solid J44. The vertices are realized as floating point numbers.
- 'elongated_square_bicupola'
- Elongated square bicupola with regular facets. Johnson solid J45. The vertices are realized as floating point numbers.
- 'gyroelongated_pentagonal_bicupola'
- Gyroelongated pentagonal bicupola with regular facets. Johnson solid J46. The vertices are realized as floating point numbers.
- 'gyroelongated_pentagonal_cupolarotunda'
- Gyroelongated pentagonal cupolarotunda with regular facets. Johnson solid J47. The vertices are realized as floating point numbers.
- 'gyroelongated_pentagonal_birotunda'
- Gyroelongated pentagonal birotunda with regular facets. Johnson solid J48. The vertices are realized as floating point numbers.
- 'augmented_triangular_prism'
- Augmented triangular prism with regular facets. Johnson solid J49. The vertices are realized as floating point numbers.
- 'biaugmented_triangular_prism'
- Biaugmented triangular prism with regular facets. Johnson solid J50. The vertices are realized as floating point numbers.
- 'triaugmented_triangular_prism'
- Triaugmented triangular prism with regular facets. Johnson solid J51. The vertices are realized as floating point numbers.
- 'augmented_pentagonal_prism'
- Augmented prantagonal prism with regular facets. Johnson solid J52. The vertices are realized as floating point numbers.
- 'biaugmented_pentagonal_prism'
- Augmented pentagonal prism with regular facets. Johnson solid J53. The vertices are realized as floating point numbers.
- 'augmented_hexagonal_prism'
- Augmented hexagonal prism with regular facets. Johnson solid J54. The vertices are realized as floating point numbers.
- 'parabiaugmented_hexagonal_prism'
- Parabiaugmented hexagonal prism with regular facets. Johnson solid J55. The vertices are realized as floating point numbers.
- 'metabiaugmented_hexagonal_prism'
- Metabiaugmented hexagonal prism with regular facets. Johnson solid J56. The vertices are realized as floating point numbers.
- 'triaugmented_hexagonal_prism'
- triaugmented hexagonal prism with regular facets. Johnson solid J57. The vertices are realized as floating point numbers.
- 'augmented_dodecahedron'
- Augmented dodecahedron with regular facets. Johnson solid J58. The vertices are realized as floating point numbers.
- 'parabiaugmented_dodecahedron'
- Parabiaugmented dodecahedron with regular facets. Johnson solid J59. The vertices are realized as floating point numbers.
- 'metabiaugmented_dodecahedron'
- Metabiaugmented dodecahedron with regular facets. Johnson solid J60. The vertices are realized as floating point numbers.
- 'triaugmented_dodecahedron'
- Triaugmented dodecahedron with regular facets. Johnson solid J61. The vertices are realized as floating point numbers.
- 'metabidiminished_icosahedron'
- Metabidiminished icosahedron with regular facets. Johnson solid J62.
- 'tridiminished_icosahedron'
- Tridiminished icosahedron with regular facets. Johnson solid J63.
- 'augmented_tridiminished_icosahedron'
- Augmented tridiminished icosahedron with regular facets. Johnson solid J64. The vertices are realized as floating point numbers.
- 'augmented_truncated_tetrahedron'
- Augmented truncated tetrahedron with regular facets. Johnson solid J65.
- 'augmented_truncated_cube'
- Augmented truncated cube with regular facets. Johnson solid J66.
- 'biaugmented_truncated_cube'
- Biaugmented truncated cube with regular facets. Johnson solid J67.
- 'augmented_truncated_dodecahedron'
- Augmented truncated dodecahedron with regular facets. Johnson solid J68. The vertices are realized as floating point numbers.
- 'parabiaugmented_truncated_dodecahedron'
- Parabiaugmented truncated dodecahedron with regular facets. Johnson solid J69. The vertices are realized as floating point numbers.
- 'metabiaugmented_truncated_dodecahedron'
- Metabiaugmented truncated dodecahedron with regular facets. Johnson solid J70. The vertices are realized as floating point numbers.
- 'triaugmented_truncated_dodecahedron'
- Triaugmented truncated dodecahedron with regular facets. Johnson solid J71. The vertices are realized as floating point numbers.
- 'gyrate_rhombicosidodecahedron'
- Gyrate rhombicosidodecahedron with regular facets. Johnson solid J72. The vertices are realized as floating point numbers.
- 'parabigyrate_rhombicosidodecahedron'
- Parabigyrate rhombicosidodecahedron with regular facets. Johnson solid J73. The vertices are realized as floating point numbers.
- 'metabigyrate_rhombicosidodecahedron'
- Metabigyrate rhombicosidodecahedron with regular facets. Johnson solid J74. The vertices are realized as floating point numbers.
- 'trigyrate_rhombicosidodecahedron'
- Trigyrate rhombicosidodecahedron with regular facets. Johnson solid J75. The vertices are realized as floating point numbers.
- 'diminished_rhombicosidodecahedron'
- Diminished rhombicosidodecahedron with regular facets. Johnson solid J76.
- 'paragyrate_diminished_rhombicosidodecahedron'
- Paragyrate diminished rhombicosidodecahedron with regular facets. Johnson solid J77. The vertices are realized as floating point numbers.
- 'metagyrate_diminished_rhombicosidodecahedron'
- Metagyrate diminished rhombicosidodecahedron with regular facets. Johnson solid J78. The vertices are realized as floating point numbers.
- 'bigyrate_diminished_rhombicosidodecahedron'
- Bigyrate diminished rhombicosidodecahedron with regular facets. Johnson solid J79. The vertices are realized as floating point numbers.
- 'parabidiminished_rhombicosidodecahedron'
- Parabidiminished rhombicosidodecahedron with regular facets. Johnson solid J80.
- 'metabidiminished_rhombicosidodecahedron'
- Metabidiminished rhombicosidodecahedron with regular facets. Johnson solid J81.
- 'gyrate_bidiminished_rhombicosidodecahedron'
- Gyrate bidiminished rhombicosidodecahedron with regular facets. Johnson solid J82. The vertices are realized as floating point numbers.
- 'triminished_rhombicosidodecahedron'
- Tridiminished rhombicosidodecahedron with regular facets. Johnson solid J83.
- 'snub_disphenoid'
- Snub disphenoid with regular facets. Johnson solid J84. The vertices are realized as floating point numbers.
- 'snub_square_antisprim'
- Snub square antiprism with regular facets. Johnson solid J85. The vertices are realized as floating point numbers.
- 'sphenocorona'
- Sphenocorona with regular facets. Johnson solid J86. The vertices are realized as floating point numbers.
- 'augmented_sphenocorona'
- Augmented sphenocorona with regular facets. Johnson solid J87. The vertices are realized as floating point numbers.
- 'sphenomegacorona'
- Sphenomegacorona with regular facets. Johnson solid J88. The vertices are realized as floating point numbers.
- 'hebesphenomegacorona'
- Hebesphenomegacorona with regular facets. Johnson solid J89. The vertices are realized as floating point numbers.
- 'disphenocingulum'
- Disphenocingulum with regular facets. Johnson solid J90. The vertices are realized as floating point numbers.
- 'bilunabirotunda'
- Bilunabirotunda with regular facets. Johnson solid J91.
- 'triangular_hebesphenorotunda'
- Triangular hebesphenorotunda with regular facets. Johnson solid J92.

##### Returns

Polytope **platonic_solid**(s) → PolytopeCreate Platonic solid of the given name.

##### Parameters

String s the name of the desired Platonic solidPossible values:- 'tetrahedron'
- Tetrahedron. Regular polytope with four triangular facets.
- 'cube'
- Cube. Regular polytope with six square facets.
- 'octahedron'
- Octahedron. Regular polytope with eight triangular facets.
- 'dodecahedron'
- Dodecahedron. Regular polytope with 12 pentagonal facets.
- 'icosahedron'
- Icosahedron. Regular polytope with 20 triangular facets.

##### Returns

Polytope **regular_120_cell**() → Polytope**regular_24_cell**() → Polytope**regular_600_cell**() → Polytope**regular_simplex**(d) → PolytopeProduce a regular

*d*-simplex embedded in R^d with edge length sqrt(2).**Examples:**- To print the vertices (in homogeneous coordinates) of the regular 2-simplex, i.e. an equilateral triangle, type this:
`> print regular_simplex(2)->VERTICES;`

`1 1 0`

`1 0 1`

`1 1/2-1/2r3 1/2-1/2r3`

The polytopes cordinate type is QuadraticExtension<Rational>, thus numbers that can be represented as a + b*sqrt(c) with Rational numbers a, b and c. The last row vectors entrys thus represent the number 1/2*(1-sqrt(3)). - To store a regular 3-simplex in the variable $s and also calculate its symmetry group, type this:
`> $s = regular_simplex(3,group=>1);`

You can then print the groups generators like so:`> print $s->GROUP->GENERATORS;`

`1 0 2`

`2 0 1`

**rhombicosidodecahedron**() → Polytope**rhombicuboctahedron**() → Polytope**simple_roots_type_A**(index) → SparseMatrixProduce the simple roots of the Coxeter arrangement of type A Indices are taken w.r.t. the Dynkin diagram 0 ---- 1 ---- ... ---- n-1 Note that the roots lie at infinity to facilitate reflecting in them, and are normalized to length sqrt{2}.

**simple_roots_type_B**(index) → SparseMatrixProduce the simple roots of the Coxeter arrangement of type B Indices are taken w.r.t. the Dynkin diagram 0 ---- 1 ---- ... ---- n-2 --(4)--> n-1 Note that the roots lie at infinity to facilitate reflecting in them, and are normalized to length sqrt{2}.

**simple_roots_type_C**(index) → SparseMatrixProduce the simple roots of the Coxeter arrangement of type C Indices are taken w.r.t. the Dynkin diagram 0 ---- 1 ---- ... ---- n-2 <--(4)-- n-1 Note that the roots lie at infinity to facilitate reflecting in them, and are normalized to length sqrt{2}.

**simple_roots_type_D**(index) → SparseMatrixProduce the simple roots of the Coxeter arrangement of type D Indices are taken w.r.t. the Dynkin diagram n-2 / 0 - 1 - ... - n-3

n-1 Note that the roots lie at infinity to facilitate reflecting in them, and are normalized to length sqrt{2}.

**simple_roots_type_E6**() → SparseMatrixProduce the simple roots of the Coxeter arrangement of type E6 Indices are taken w.r.t. the Dynkin diagram 3 | | 0 ---- 1 ---- 2 ---- 4 ---- 5 Note that the roots lie at infinity to facilitate reflecting in them.

##### Returns

SparseMatrix **simple_roots_type_E7**() → SparseMatrixProduce the simple roots of the Coxeter arrangement of type E7 Indices are taken w.r.t. the Dynkin diagram 4 | | 0 ---- 1 ---- 2 ---- 3 ---- 5 ---- 6 Note that the roots lie at infinity to facilitate reflecting in them.

##### Returns

SparseMatrix **simple_roots_type_E8**() → SparseMatrixProduce the simple roots of the Coxeter arrangement of type E8 Indices are taken w.r.t. the Dynkin diagram 5 | | 0 ---- 1 ---- 2 ---- 3 ---- 4 ---- 6 ---- 7 Note that the roots lie at infinity to facilitate reflecting in them.

##### Returns

SparseMatrix **simple_roots_type_F4**() → SparseMatrixProduce the simple roots of the Coxeter arrangement of type F4 Indices are taken w.r.t. the Dynkin diagram 0 ---- 1 --(4)--> 2 ---- 3

##### Returns

SparseMatrix **simple_roots_type_G2**() → SparseMatrixProduce the simple roots of the Coxeter arrangement of type G2 Indices are taken w.r.t. the Dynkin diagram 0 <--(6)-- 1

##### Returns

SparseMatrix **simple_roots_type_H3**() → SparseMatrixProduce the simple roots of the Coxeter arrangement of type H3 Indices are taken w.r.t. the Dynkin diagram 0 --(5)-- 1 ---- 2 Note that the roots lie at infinity to facilitate reflecting in them, and are normalized to length 2

##### Returns

SparseMatrix **simple_roots_type_H4**() → SparseMatrixProduce the simple roots of the Coxeter arrangement of type H4 Indices are taken w.r.t. the Dynkin diagram 0 --(5)-- 1 ---- 2 ---- 3 Note that the roots lie at infinity to facilitate reflecting in them, and are normalized to length sqrt{2}

##### Returns

SparseMatrix **tetrahedron**() → Polytope**truncated_cube**() → Polytope**truncated_cuboctahedron**() → PolytopeCreate truncated cuboctahedron. An Archimedean solid. This is actually a misnomer. The actual truncation of a cuboctahedron is normally equivalent to this construction, but has two different edge lengths. This construction has regular 2-faces.

##### Returns

Polytope **truncated_dodecahedron**() → Polytope**truncated_icosahedron**() → PolytopeCreate exact truncated icosahedron in Q(sqrt{5}). An Archimedean solid. Also known as the soccer ball.

##### Returns

Polytope **truncated_icosidodecahedron**() → Polytope**truncated_octahedron**() → PolytopeCreate truncated octahedron. An Archimedean solid. Also known as the 3-permutahedron.

##### Returns

Polytope **wythoff**(type, rings) → PolytopeProduce the orbit polytope of a point under a Coxeter arrangement with exact coordinates, possibly in a qudratic extension field of the rationals

Topologic cell complexes defined as quotients over polytopes modulo a discrete group.

**cs_quotient**(P)For a centrally symmetric polytope, divide out the central symmetry, i.e, identify diametrically opposite faces.

Contained in extension`bundled:group`

.##### Parameters

Polytope P , centrally symmetric**cylinder_2**() → PolytopeReturn a 2-dimensional

*cylinder*obtained by identifying two opposite sides of a square.Contained in extension`bundled:group`

.##### Returns

Polytope **quarter_turn_manifold**() → PolytopeReturn the 3-dimensional Euclidean manifold obtained by identifying opposite faces of a 3-dimensional cube by a quarter turn. After identification, two classes of vertices remain.

Contained in extension`bundled:group`

.##### Returns

Polytope **write_quotient_space_simplexity_ilp**()outputs a linear program whose optimal value is a lower bound for the number of simplices necessary to triangulate the polytope in such a way that its symmetries respect the triangulation of the boundary.

Contained in extension`bundled:group`

.

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

**alternating_group**(degree, domain) → group::GroupOfPolytopeConstructs an

*alternating group*of given*degree*. (See also group::alternating_group.)Contained in extension`bundled:group`

.**combinatorial_symmetries**(poly, on_vertices) → group::GroupOfPolytopeCompute the combinatorial symmetries (i.e., automorphisms of the face lattice) of a given polytope

*poly*. If*on_vertices*is set to 1, the function returns a GroupOfPolytope which acts on the vertices. If*on_vertices*is set to any other number, the function returns a GroupOfPolytope which acts on the facets of the polytope. If*on_vertices*is unspecified, both groups are returned.##### Parameters

Polytope poly Int on_vertices specifies whether the returned group should act on vertices (1) or on facets (2)##### Returns

group::GroupOfPolytope the combinatorial symmetry group acting on the vertices or the facets or (group::GroupOfPolytope, group::GroupOfPolytope) = (group on vertices, group on facets) if*on_vertices*is undefined**Example:**- To get the vertex and facet symmetry groups of the square and print their generators, type the following:
`> ($gv,$gf) = combinatorial_symmetries(cube(2));`

`> print $gv->GENERATORS;`

`2 3 0 1`

`1 0 2 3`

`> print $gf->GENERATORS;`

`0 2 1 3`

`1 0 3 2`

**convert_coord_action**(group, mat, dom_out) → group::GroupConverts the generators of a group acting on coordinates to generators of the corresponding group which acts on the rows of the given matrix

*mat*. The parameter*dom_out*specifies whether*mat*describes vertices or facets.Contained in extension`bundled:group`

.##### Parameters

group::Group group input group acting on coordinatesMatrix mat vertices or facets of a polytopeInt dom_out OnRays(1) or OnFacets(2)##### Options

String name an optional name for the output group##### Returns

group::Group a new group object with the generators induced on the new domain**convert_group_domain**(group, VIF) → group::GroupConverts the generators of the input group from the domain onRays to generators on the domain onFacets, and vice versa.

Contained in extension`bundled:group`

.##### Parameters

group::Group group IncidenceMatrix VIF the vertex-facet incidence matrix of the cone or polytope##### Options

String name an optional name for the output group##### Returns

group::Group a new group object with the generators induced on the new domain**cyclic_group**(degree, domain) → group::GroupOfPolytopeConstructs a

*cyclic group*of given*degree*. (See also group::cyclic_group.)Contained in extension`bundled:group`

.**group_from_cyclic_notation0**(group, domain) → group::GroupOfPolytopeConstructs a group from a string with generators in cyclic notation. All numbers in the string are 0-based, meaning that 0 is the smallest number allowed.

Contained in extension`bundled:group`

.##### Parameters

String group generators in cyclic notationInt domain of the polytope symmetry group. 1 for action on vertex indices, 2 for action" on facet indices, 3 for action on coordinates##### Returns

group::GroupOfPolytope **Example:**`> $g = group_from_cyclic_notation0("(0,2)(1,3)",0);`

`> print $g->GENERATORS;`

`2 3 0 1`

**group_from_cyclic_notation1**(group, domain) → group::GroupOfPolytopeConstructs a group from a string with generators in cyclic notation. All numbers in the string are 1-based, meaning that 1 is the smallest number allowed. Example: "(1,3)(2,4)"

Contained in extension`bundled:group`

.##### Parameters

String group generators in cyclic notationInt domain of the polytope symmetry group. 1 for action on vertex indices, 2 for action" on facet indices, 3 for action on coordinates##### Returns

group::GroupOfPolytope # @example > $g = group_from_cyclic_notation1("(1,3)(2,4)",0); > print $g->GENERATORS; | 2 3 0 1**lattice_automorphisms_smooth_polytope**(P) → Array<Array<Int>>Returns a generating set for the lattice automorphism group of a smooth polytope

*P*by comparing lattice distances between vertices and facets.##### Parameters

Polytope P the given polytope##### Returns

Array<Array<Int>> the generating set for the lattice automorphism group**Example:**`> print lattice_automorphisms_smooth_polytope(cube(2));`

`2 3 0 1`

`1 0 3 2`

`0 2 1 3`

**linear_symmetries**(m) → group::GroupComputes the linear symmetries of a matrix

*m*whose rows describe a point configuration via 'sympol'.Contained in extension`bundled:group`

.##### Parameters

Matrix m holds the points as rows whose linear symmetry group is to be computed##### Returns

group::Group the linear symmetry group of*m***Example:**`> $ls = linear_symmetries(cube(2)->VERTICES);`

`> print $ls->GENERATORS;`

`0 2 1 3`

`3 1 2 0`

`2 3 0 1`

**linear_symmetries**(c, dual) → group::GroupOfConeComputes the linear symmetries of a given polytope

*p*via 'sympol'. If the input is a cone, it may compute only a subgroup of the linear symmetry group.Contained in extension`bundled:group`

.##### Parameters

Cone c the cone (or polytope) whose linear symmetry group is to be computedBool dual true if group action on vertices, false if action on facets##### Returns

group::GroupOfCone the linear symmetry group of*p*(or a subgroup if*p*is a cone)**nestedOPGraph**(gen_point, points, lattice_points, group, verbose) → ARRAYConstructs the NOP-graph of an orbit polytope. It is used by the rule for the NOP_GRAPH.

##### Parameters

Vector gen_point the generating pointMatrix points the vertices of the orbit polytopeMatrix lattice_points the lattice points of the orbit polytopegroup::GroupOfPolytope group the generating groupBool verbose print out additional information##### Returns

ARRAY ($Graph, $lp_reps, $minInStartOrbit, \@core_point_reps, $CPindices)**orbit_polytope**(gen_point, group) → OrbitPolytopeConstructs the orbit polytope of a given point

*gen_point*with respect to a given permutation group*group*.##### Parameters

Vector<Rational> gen_point the basis point of the orbit polytopegroup::GroupOfPolytope group a group acting on coordinates##### Returns

OrbitPolytope the orbit polytope of*gen_point*w.r.t.*group***ortho_project**(p) → Polytope**representation_conversion_up_to_symmetry**(c, a, dual, rayCompMethod) → ListComputes the dual description of a polytope up to its linear symmetry group.

Contained in extension`bundled:group`

.##### Parameters

Cone c the cone (or polytope) whose dual description is to be computedgroup::Group a Bool dual true if V to H, false if H to VInt rayCompMethod specifies sympol's method of ray computation via lrs(0), cdd(1), beneath_and_beyond(2), ppl(3)##### Returns

List (Bool success indicator, Matrix<Rational> vertices/inequalities, Matrix<Rational> lineality/equations)**symmetric_group**(degree, domain) → group::GroupOfPolytopeConstructs a

*symmetric group*of given*degree*. (See also group::symmetric_group.)Contained in extension`bundled:group`

.##### Parameters

Int degree Int domain of the polytope's symmetry group. 1 for action on vertex indices, 2 for action" on facet indices, 3 for action on coordinates##### Returns

group::GroupOfPolytope **Example:**`> $g = symmetric_group(5,0);`

`> print $g->GENERATORS;`

`1 0 2 3 4`

`0 2 1 3 4`

`0 1 3 2 4`

`0 1 2 4 3`

**truncated_orbit_polytope**(v, group, eps) → SymmetricPolytopeConstructs an orbit polytope of a given point

*v*with respect to a given group*group*, in which all vertices are cut off by hyperplanes in distance*eps*Contained in extension`bundled:group`

.##### Parameters

Vector v point of which orbit polytope is to be constructedgroup::GroupOfPolytope group group for which orbit polytope is to be constructedRational eps scaled distance by which the vertices of the orbit polytope are to be cut off##### Returns

SymmetricPolytope the truncated orbit polytope**visualizeNOP**(orb, colors_ref, trans_ref)Visualizes all (nested) orbit polytopes contained in

*orb*in one picture.##### Parameters

OrbitPolytope orb the orbit polytopeARRAY colors_ref the reference to an array of colorsARRAY trans_ref the reference to an array of transparency values**visualizeNOPGraph**(orb, filename)Visualizes the NOP-graph of an orbit polytope. Requires 'graphviz' and a Postscript viewer. Produces a file which is to be processed with the program 'dot' from the graphviz package. If 'dot' is installed, the NOP-graph is visualized by the Postscript viewer.

##### Parameters

OrbitPolytope orb the orbit polytopeString filename the filename for the 'dot' file

These functions take a realized polytope and produce a new one by applying a suitable affine or projective transformation in order to obtain some special coordinate description but preserve the combinatorial type.

For example, before you can polarize an arbitrary polyhedron, it must be transformed to a combinatorially equivalent bounded polytope with the origin as a relatively interior point. It is achieved with the sequence orthantify - bound - center - polarize.

**ambient_lattice_normalization**(p) → PolytopeTransform to a full-dimensional polytope while preserving the ambient lattice Z^n

##### Parameters

Polytope p the input polytope,##### Options

Bool store_transform store the reverse transformation as an attachement##### Returns

Polytope - the transformed polytope defined by its vertices. Facets are only written if available in*p*.**Examples:**- Consider a line segment embedded in 2-space containing three lattice points:
`> $p = new Polytope(VERTICES=>[[1,0,0],[1,2,2]]);`

`> print ambient_lattice_normalization($p)->VERTICES;`

`1 0`

`1 2`

The ambient lattice of the projection equals the intersection of the affine hull of $p with Z^2. - Another line segment containing only two lattice points:
`> $p = new Polytope(VERTICES=>[[1,0,0],[1,1,2]]);`

`> $P = ambient_lattice_normalization($p,store_transform=>1);`

`> print $P->VERTICES;`

`1 0`

`1 1`

To get the transformation, do the following:`> print $M = $P->get_attachment(REVERSE_LATTICE_PROJECTION);`

`1 0 0`

`0 1 2`

`> print $P->VERTICES * $M;`

`1 0 0`

`1 1 2`

**bound**(P) → PolytopeMake a positive polyhedron bounded. Apply a projective linear transformation to a polyhedron mapping the far hyperplane to the hyperplane spanned by the unit vectors. The origin (1,0,...,0) is fixed.

The input polyhedron should be POSITIVE; i.e. no negative coordinates.

**Example:**- Observe the transformation of a simple unbounded 2-polyhedron:
`> $P = new Polytope(VERTICES=>[[1,0,0],[0,1,1],[0,0,1]]);`

`> print bound($P)->VERTICES;`

`1 0 0`

`1 1/2 1/2`

`1 0 1`

As you can see, the far points are mapped to the hyperplane spanned by (1,1,0) and (1,0,1).

**center**(P) → PolytopeMake a polyhedron centered. Apply a linear transformation to a polyhedron

*P*such that a relatively interior point (preferably the vertex barycenter) is moved to the origin (1,0,...,0).**Example:**- Consider this triangle not containing the origin:
`> $P = new Polytope(VERTICES => [[1,1,1],[1,2,1],[1,1,2]]);`

`> $origin = new Vector([1,0,0]);`

`> print $PC->contains_in_interior($origin);`

`> $PC = center($P);`

`> print $PC->contains_in_interior($origin);`

`1`

This is what happened to the vertices:`> print $PC->VERTICES;`

`1 -1/3 -1/3`

`1 2/3 -1/3`

`1 -1/3 2/3`

There also exists a property to check whether a polytope is centered:`> print $PC->CENTERED;`

`1`

**orthantify**(P, v) → PolytopeMake a polyhedron POSITIVE. Apply an affine transformation to a polyhedron such that the vertex

*v*is mapped to the origin (1,0,...,0) and as many facets through this vertex as possible are mapped to the bounding facets of the first orthant.##### Parameters

Polytope P Int v vertex to be moved to the origin. By default it is the first affine vertex of the polyhedron.##### Returns

Polytope **Example:**- To orthantify the square, moving its first vertex to the origin, do this:
`> $p = orthantify(cube(2),1);`

`> print $p->VERTICES;`

`1 2 0`

`1 0 0`

`1 2 2`

`1 0 2`

**polarize**(C) → ConeGiven a bounded, centered, not necessarily full-dimensional polytope

*P*, produce its polar with respect to the standard Euclidean scalar product. Note that the definition of the polar has changed after version 2.10: the polar is reflected in the origin to conform with cone polarization If*P*is not full-dimensional, the output will contain lineality orthogonal to the affine span of*P*. In particular, polarize() of a pointed polytope will always produce a full-dimensional polytope. If you want to compute the polar inside the affine hull you may use the pointed_part client afterwards.**Example:**- To save the polar of the square in the variable $p and then print its vertices, do this:
`> $p = polarize(cube(2));`

`> print $p->VERTICES;`

`1 1 0`

`1 -1 0`

`1 0 1`

`1 0 -1`

**revert**(P) → PolytopeApply a reverse transformation to a given polyhedron

*P*. All transformation clients keep track of the polytope's history. They write or update the attachment REVERSE_TRANSFORMATION.Applying revert to the transformed polytope reconstructs the original polytope.

**Example:**- The following translates the square and then reverts the transformation:
`> $v = new Vector(1,2);`

`> $p = translate(cube(2),$v);`

`> print $p->VERTICES;`

`1 0 1`

`1 2 1`

`1 0 3`

`1 2 3`

`> $q = revert($p);`

`> print $q->VERTICES;`

`1 -1 -1`

`1 1 -1`

`1 -1 1`

`1 1 1`

**scale**(P, factor, store) → PolytopeScale a polyhedron

*P*by a given scaling parameter*factor*.##### Parameters

Polytope P the polyhedron to be scaledScalar factor the scaling factorBool store stores the reverse transformation as an attachment (REVERSE_TRANSFORMATION); default value: 1.##### Returns

Polytope **Example:**- To sacle the square by 23, do this:
`> $p = scale(cube(2),23);`

`> print $p->VERTICES;`

`1 -23 -23`

`1 23 -23`

`1 -23 23`

`1 23 23`

The transformation matrix is stored in an attachment:`> print $p->get_attachment('REVERSE_TRANSFORMATION');`

`1 0 0`

`0 1/23 0`

`0 0 1/23`

To reverse the transformation, you can use the*revert*function.`> $q = revert($p);`

`> print $q->VERTICES;`

`1 -1 -1`

`1 1 -1`

`1 -1 1`

`1 1 1`

**transform**(P, trans, store) → PolytopeTransform a polyhedron

*P*according to the linear transformation*trans*.##### Parameters

Polytope P the polyhedron to be transformedMatrix trans the transformation matrixBool store stores the reverse transformation as an attachment (REVERSE_TRANSFORMATION); default value: 1.##### Returns

Polytope **Example:**- This translates the square by (23,23) and stores the transformation:
`> $M = new Matrix([1,23,23],[0,1,0],[0,0,1]);`

`> $p = transform(cube(2),$M,1);`

`> print $p->VERTICES;`

`1 22 22`

`1 24 22`

`1 22 24`

`1 24 24`

To retrieve the attached transformation, use this:`> print $p->get_attachment('REVERSE_TRANSFORMATION');`

`1 -23 -23`

`0 1 0`

`0 0 1`

Check out the*revert*function to learn how to undo the transformation. It might have been more comfortable to use the*translate*function to achieve the above result.

**translate**(P, trans, store) → PolytopeTranslate a polyhedron

*P*by a given translation vector*trans*.##### Parameters

Polytope P the polyhedron to be translatedVector trans the translation vectorBool store stores the reverse transformation as an attachment (REVERSE_TRANSFORMATION); default value: 1.##### Returns

Polytope **Example:**- This translates the square by (23,23) and stores the transformation:
`> $t = new Vector(23,23);`

`> $p = translate(cube(2),$t);`

`> print $p->VERTICES;`

`1 22 22`

`1 24 22`

`1 22 24`

`1 24 24`

To retrieve the attached transformation, use this:`> print $p->get_attachment('REVERSE_TRANSFORMATION');`

`1 -23 -23`

`0 1 0`

`0 0 1`

Check out the*revert*function to learn how to undo the transformation.

**vertex_lattice_normalization**(p) → PolytopeTransform to a full-dimensional polytope while preserving the lattice spanned by vertices induced lattice of new vertices = Z^dim

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

**barycentric_subdivision**(c) → topaz::SimplicialComplexCreate a simplicial complex as a barycentric subdivision of a given cone or polytope. Each vertex in the new complex corresponds to a face in the old complex.

##### Parameters

Cone c input cone or polytope##### Options

Bool relabel generate vertex labels from the faces of the original complex; default trueBool geometric_realization create a topaz::GeometricSimplicialComplex; default is true##### Returns

topaz::SimplicialComplex **barycentric_subdivision**(pc) → PointConfigurationCreate a simplicial complex as the barycentric subdivision of a given point configuration. Each vertex in the new complex corresponds to a face in the old complex.

##### Parameters

PointConfiguration pc input point configuration##### Options

Bool relabel generate vertex labels from the faces of the original complex; default trueBool geometric_realization ##### Returns

PointConfiguration **coherency_index**(points, w1, w2)**coherency_index**(p1, p2)**common_refinement**(points, sub1, sub2, dim) → IncidenceMatrixComputes the common refinement of two subdivisions of

*points*. It is assumed that there exists a common refinement of the two subdivisions.##### Parameters

Matrix points IncidenceMatrix sub1 first subdivisionIncidenceMatrix sub2 second subdivisionInt dim dimension of the point configuration##### Returns

IncidenceMatrix the common refinement**Example:**- A simple 2-dimensional set of points:
`> $points = new Matrix<Rational>([[1,0,0],[1,1,0],[1,0,1],[1,1,1],[1,2,1]]);`

Two different subdivisions...`> $sub1 = new IncidenceMatrix([[0,1,2],[1,2,3,4]]);`

`> $sub2 = new IncidenceMatrix([[1,3,4],[0,1,2,3]]);`

...and their common refinement:`> print common_refinement($points,$sub1,$sub2,2);`

`{0 1 2}`

`{1 3 4}`

`{1 2 3}`

**common_refinement**(p1, p2) → Polytope**delaunay_triangulation**(V) → Array<Set<Int>>Compute the Delaunay triangulation of the given SITES of a VoronoiDiagram

*V*. If the sites are not in general position, the non-triangular facets of the Delaunay subdivision are triangulated (by applying the beneath-beyond algorithm).**Example:**`> $VD = new VoronoiDiagram(SITES=>[[1,1,1],[1,0,1],[1,-1,1],[1,1,-1],[1,0,-1],[1,-1,-1]]);`

`> $D = delaunay_triangulation($VD);`

`> print $D;`

`{1 2 4}`

`{2 4 5}`

`{0 1 3}`

`{1 3 4}`

**foldable_max_signature_ilp**(d, points, volume, cocircuit_equations) → LinearProgram<Rational>Set up an ILP whose MAXIMAL_VALUE is the maximal signature of a foldable triangulation of a polytope, point configuration or quotient manifold

Contained in extension`bundled:group`

.##### Parameters

Int d the dimension of the input polytope, point configuration or quotient manifoldMatrix points the input points or verticesRational volume the volume of the convex hullSparseMatrix cocircuit_equations the matrix of cocircuit equations##### Options

String filename a name for a file in .lp format to store the linear program##### Returns

LinearProgram<Rational> an ILP that provides the result**foldable_max_signature_upper_bound**(d, points, volume, cocircuit_equations) → IntegerCalculate the LP relaxation upper bound to the maximal signature of a foldable triangulation of polytope, point configuration or quotient manifold

Contained in extension`bundled:group`

.##### Parameters

Int d the dimension of the input polytope, point configuration or quotient manifoldMatrix points the input points or verticesRational volume the volume of the convex hullSparseMatrix cocircuit_equations the matrix of cocircuit equations##### Returns

Integer the optimal value of an LP that provides a bound**interior_and_boundary_ridges**(P) → Pair<Array<Set>,Array<Set>>Find the (

*d*-1)-dimensional simplices in the interior and in the boundary of a*d*-dimensional polytope or coneContained in extension`bundled:group`

.**Example:**`> print interior_and_boundary_ridges(cube(2));`

`<{0 3}`

`{1 2}`

`>`

`<{0 1}`

`{0 2}`

`{1 3}`

`{2 3}`

`>`

**is_regular**(points, subdiv) → Pair<Bool,Vector>For a given subdivision

*subdiv*of*points*tests if the subdivision is regular and if yes computes a weight vector inducing this subdivsion. The output is a pair of Bool and the weight vector. Options can be used to ensure properties of the resulting vector. The default is having 0 on all vertices of the first face of*subdiv*.##### Parameters

Matrix points in homogeneous coordinatesArray<Set<Int> > subdiv ##### Options

Matrix<Scalar> equations system of linear equation the cone is cut with.Set<Int> lift_to_zero gives only lifting functions lifting the designated vertices to 0Int lift_face_to_zero gives only lifting functions lifting all vertices of the designated face to 0##### Returns

Pair<Bool,Vector> **Example:**- A regular subdivision of the square, with the first cell lifted to zero:
`> $points = cube(2)->VERTICES;`

`> print is_regular($points,[[0,1,3],[1,2,3]],lift_to_zero=>[0,1,3]);`

`1 <0 0 1 0>`

**is_subdivision**(points, faces)Checks whether

*faces*forms a valid subdivision of*points*, where*points*is a set of points, and*faces*is a collection of subsets of (indices of)*points*. If the set of interior points of*points*is known, this set can be passed by assigning it to the option*interior_points*. If*points*are in convex position (i.e., if they are vertices of a polytope), the option*interior_points*should be set to [ ] (the empty set).**Example:**- Two potential subdivisions of the square without innter points:
`> $points = cube(2)->VERTICES;`

`> print is_subdivision($points,[[0,1,3],[1,2,3]],interior_points=>[ ]);`

`1`

`> print is_subdivision($points,[[0,1,2],[1,2]],interior_points=>[ ]);`

**iterated_barycentric_subdivision**(c, n) → topaz::SimplicialComplexCreate a simplicial complex as an iterated barycentric subdivision of a given cone or polytope.

##### Parameters

Cone c input cone or polytopeInt n how many times to subdivide##### Options

Bool relabel write labels of new points; default is falseBool geometric_realization create a topaz::GeometricSimplicialComplex; default is false##### Returns

topaz::SimplicialComplex **max_interior_simplices**(P) → Array<Set>Find the maximal interior simplices of a polytope P. Symmetries of P are NOT taken into account.

Contained in extension`bundled:group`

.**Example:**`> print max_interior_simplices(cube(2));`

`{0 1 2}`

`{0 1 3}`

`{0 2 3}`

`{1 2 3}`

**max_interior_simplices**(P)find the maximal interior simplices of a point configuration that DO NOT contain any point in their closure, except for the vertices. Symmetries of the configuration are NOT taken into account.

Contained in extension`bundled:group`

.##### Parameters

PointConfiguration P the input point configuration**metric2hyp_triang**(FMS) → Polytope**metric2splits**(D) → Array<Pair<Set>>Computes all non-trivial splits of a metric space

*D*(encoded as a symmetric distance matrix).**mixed_volume**(P1, P2, Pn) → ScalarProduces the mixed volume of polytopes P

_{1},P_{2},...,P_{n}.##### Parameters

Polytope<Scalar> P1 first polytopePolytope<Scalar> P2 second polytopePolytope<Scalar> Pn last polytope##### Returns

Scalar mixed volume**Example:**`> print mixed_volume(cube(2),simplex(2));`

`4`

**n_triangulations**(M, optimization) → IntegerCalculates the number of triangulations of the input points given as rows of a matrix. This can be space intensive.

##### Parameters

Matrix M points in the projective planeBool optimization defaults to 1, where 1 includes optimization and 0 excludes it##### Returns

Integer number of triangulations**Example:**- To print the number of possible triangulations of a square, do this:
`> print n_triangulations(cube(2)->VERTICES);`

`2`

**placing_triangulation**(Points) → Array<Set<Int>>Compute the placing triangulation of the given point set using the beneath-beyond algorithm.

##### Parameters

Matrix Points the given point set##### Options

Bool non_redundant whether it's already known that*Points*are non-redundantArray<Int> permutation placing order of*Points*, must be a valid permutation of (0..Points.rows()-1)##### Returns

Array<Set<Int>> **Example:**- To compute the placing triangulation of the square (of whose vertices we know that they're non-redundant), do this:
`> $t = placing_triangulation(cube(2)->VERTICES,non_redundant=>1);`

`> print $t;`

`{0 1 2}`

`{1 2 3}`

**points2metric**(points) → MatrixDefine a metric by restricting the Euclidean distance function to a given set of

*points*. Due to floating point computations (sqrt is used) the metric defined may not be exact. If the option*max*or*l1*is set to true the max-norm or l1-norm is used instead (with exact computation).##### Parameters

Matrix points ##### Options

Bool max triggers the usage of the max-norm (exact computation)Bool l1 triggers the usage of the l1-norm (exact computation)##### Returns

Matrix **Example:**`> print points2metric(cube(2),max=>1);`

`0 2 2 2`

`2 0 2 2`

`2 2 0 2`

`2 2 2 0`

**poly2metric**(P) → MatrixDefine a metric by restricting the Euclidean distance function to the vertex set of a given polytope

*P*. Due to floating point computations (sqrt is used) the metric defined may not be exact. If the option*max*or*l1*is set to true the max-norm or l1-norm is used instead (with exact computation).##### Parameters

Polytope P ##### Options

Bool max triggers the usage of the max-norm (exact computation)##### Returns

Matrix **Example:**`> print points2metric(cube(2)->VERTICES,max=>1);`

`0 2 2 2`

`2 0 2 2`

`2 2 0 2`

`2 2 2 0`

**quotient_space_simplexity_ilp**(d, V, volume, cocircuit_equations) → LinearProgramSet up an LP whose MINIMAL_VALUE is a lower bound for the minimal number of simplices needed to triangulate a polytope, point configuration or quotient manifold

Contained in extension`bundled:group`

.##### Parameters

Int d the dimension of the input polytope, point configuration or quotient manifoldMatrix V the input points or verticesScalar volume the volume of the convex hullSparseMatrix cocircuit_equations the matrix of cocircuit equations##### Options

String filename a name for a file in .lp format to store the linear program##### Returns

LinearProgram an LP that provides a lower bound**quotient_space_simplexity_lower_bound**(d, V, volume, cocircuit_equations) → IntegerCalculate a lower bound for the minimal number of simplices needed to triangulate a polytope, point configuration or quotient manifold

Contained in extension`bundled:group`

.##### Parameters

Int d the dimension of the input polytope, point configuration or quotient manifoldMatrix V the input points or verticesScalar volume the volume of the convex hullSparseMatrix cocircuit_equations the matrix of cocircuit equations##### Returns

Integer the optimal value of an LP that provides a lower bound**regularity_lp**(points, subdiv) → Polytope<Scalar>For a given subdivision

*subdiv*of*points*determines a*LinearProgram*to decide whether the subdivision is regular. The output a Polytope with an attached LP. Options can be used to ensure properties of the resulting LP. The default is having 0 on all vertices of the first face of*subdiv*.##### Parameters

Matrix points in homogeneous coordinatesArray<Set<Int> > subdiv ##### Options

Matrix<Scalar> equations system of linear equation the cone is cut with.Set<Int> lift_to_zero gives only lifting functions lifting the designated vertices to 0Int lift_face_to_zero gives only lifting functions lifting all vertices of the designated face to 0Scalar epsilon minimum distance from all inequalities##### Returns

Polytope<Scalar> **regular_subdivision**(points, weights) → Array<Set<Int>>Compute a regular subdivision of the polytope obtained by lifting

*points*to*weights*and taking the lower complex of the resulting polytope. If the weight is generic the output is a triangulation.**Example:**- The following generates a regular subdivision of the square.
`> $w = new Vector(2,23,2,2);`

`> $r = regular_subdivision(cube(2)->VERTICES,$w);`

`> print $r;`

`{0 1 3}`

`{0 2 3}`

**secondary_cone**(points, subdiv) → ConeFor a given subdivision

*subdiv*of*points*tests computes the corresponding secondary cone. If the subdivision is not regular, the cone will be the secondary cone of the finest regular coarsening of*subdiv*. (See option*test_regularity*) Options can be used to make the Cone POINTED.##### Parameters

Matrix points in homogeneous coordinatesArray<Set<Int> > subdiv ##### Options

Matrix<Scalar> equations system of linear equation the cone is cut with.Set<Int> lift_to_zero gives only lifting functions lifting the designated vertices to 0Int lift_face_to_zero gives only lifting functions lifting all vertices of the designated face to 0Bool test_regularity throws an exception if the subdivision is not regular##### Returns

Cone **simplexity_ilp**(d, points, the, volume, cocircuit_equations) → LinearProgramSet up an ILP whose MINIMAL_VALUE is the minimal number of simplices needed to triangulate a polytope, point configuration or quotient manifold

Contained in extension`bundled:group`

.##### Parameters

Int d the dimension of the input polytope, point configuration or quotient manifoldMatrix points the input points or verticesArray<Set> the representatives of maximal interior simplicesScalar volume the volume of the convex hullSparseMatrix cocircuit_equations the matrix of cocircuit equations##### Options

String filename a name for a file in .lp format to store the linear program##### Returns

LinearProgram an LP that provides a lower bound**simplexity_ilp_with_angles**(d, points, the, volume, cocircuit_equations) → LinearProgramSet up an ILP whose MINIMAL_VALUE is the minimal number of simplices needed to triangulate a polytope, point configuration or quotient manifold

Contained in extension`bundled:group`

.##### Parameters

Int d the dimension of the input polytope, point configuration or quotient manifoldMatrix points the input points or verticesArray<Set> the (representative) maximal interior simplicesScalar volume the volume of the convex hullSparseMatrix cocircuit_equations the matrix of cocircuit equations##### Options

String filename a name for a file in .lp format to store the linear program##### Returns

LinearProgram an LP that provides a lower bound**simplexity_lower_bound**(d, points, volume, cocircuit_equations) → IntegerCalculate the LP relaxation lower bound for the minimal number of simplices needed to triangulate a polytope, point configuration or quotient manifold

Contained in extension`bundled:group`

.##### Parameters

Int d the dimension of the input polytope, point configuration or quotient manifoldMatrix points the input points or verticesScalar volume the volume of the convex hullSparseMatrix cocircuit_equations the matrix of cocircuit equations##### Returns

Integer the optimal value of an LP that provides a lower bound**splits_in_subdivision**(vertices, subdivision, splits) → Set<Int>Tests which of the

*splits*of a polyhedron are coarsenings of the given*subdivision*.##### Parameters

Matrix vertices the vertices of the polyhedronArray<Set<Int>> subdivision a subdivision of the polyhedronMatrix splits the splits of the polyhedron##### Returns

Set<Int> **split_compatibility_graph**(splits, P) → Graph**split_polyhedron**(P) → Polytope**staircase_weight**(k, l) → Vector<Rational>Gives a weight vector for the staircase triangulation of the product of a

*k-1*- and an*l-1*-dimensional simplex.##### Parameters

Int k the number of vertices of the first simplexInt l the number of vertices of the second simplex##### Returns

Vector<Rational> **Example:**- The following creates the staircase triangulation of the product of the 2- and the 1-simplex.
`> $w = staircase_weight(3,2);`

`> $p = product(simplex(2),simplex(1));`

`> $p->POLYTOPAL_SUBDIVISION(WEIGHTS=>$w);`

`> print $p->POLYTOPAL_SUBDIVISION->MAXIMAL_CELLS;`

`{0 2 4 5}`

`{0 1 3 5}`

`{0 2 3 5}`

**stellar_subdivision**(pc, faces) → PointConfigurationComputes the complex obtained by stellar subdivision of all

*faces*of the TRIANGULATION of the PointConfiguration.##### Parameters

PointConfiguration pc input point configurationArray<Set<Int>> faces list of faces to subdivide##### Options

Bool no_labels : do not write any labels##### Returns

PointConfiguration **symmetrized_foldable_max_signature_ilp**(d, points, volume, generators, symmetrized_foldable_cocircuit_equations) → LinearProgram<Rational>Set up an ILP whose MAXIMAL_VALUE is the maximal signature of a foldable triangulation of a polytope, point configuration or quotient manifold

Contained in extension`bundled:group`

.##### Parameters

Int d the dimension of the input polytope, point configuration or quotient manifoldMatrix points the input points or verticesRational volume the volume of the convex hullArray<Array<Int>> generators the generators of the symmetry groupSparseMatrix symmetrized_foldable_cocircuit_equations the matrix of symmetrized cocircuit equations##### Options

String filename a name for a file in .lp format to store the linear program##### Returns

LinearProgram<Rational> an ILP that provides the result**symmetrized_foldable_max_signature_upper_bound**(d, points, volume, cocircuit_equations) → IntegerCalculate the LP relaxation upper bound to the maximal signature of a foldable triangulation of polytope, point configuration or quotient manifold

Contained in extension`bundled:group`

.##### Parameters

Int d the dimension of the input polytope, point configuration or quotient manifoldMatrix points the input points or verticesRational volume the volume of the convex hullSparseMatrix cocircuit_equations the matrix of cocircuit equations##### Returns

Integer the optimal value of an LP that provides a bound**universal_polytope**(P) → Polytope**universal_polytope**(P, reps, cocircuit_equations) → PolytopeCalculate the universal polytope of a polytope, point configuration or quotient manifold

Contained in extension`bundled:group`

.##### Parameters

Polytope P the input polytopeArray<Set> reps the representatives of maximal interior simplicesSparseMatrix cocircuit_equations the matrix of cocircuit equations##### Returns

Polytope **universal_polytope**(PC) → PolytopeCalculate the universal polytope of a point configuration

Contained in extension`bundled:group`

.

These functions are for visualization.

**bounding_box**(V, surplus_k, voronoi) → MatrixIntroduce artificial boundary facets (which are always vertical, i.e., the last coordinate is zero) to allow for bounded images of unbounded polyhedra (e.g. Voronoi polyhedra). If the

*voronoi*flag is set, the last direction is left unbounded.**vlabels**(vertices, wo_zero) → ARRAYCreates vertex labels for visualization from the

*vertices*of the polytope. The parameter*wo_zero*decides whether the entry at position 0 (homogenizing coordinate) is omitted (1) or included (0) in the label string."##### Parameters

Matrix vertices the vertices of the polytopeBool wo_zero includes (0) or omits (1) the entry at position 0##### Returns

ARRAY a reference to an array of vertex label strings**Example:**- This prints the vertex labels for the square with the origin as its center and side length 2, where we omit the leading 1:
`> $l = vlabels(cube(2)->VERTICES,1);`

`> print join(', ', @{$v});`

`(-1,-1), (1,-1), (-1,1), (1,1)`

## Common Option Lists

These options are for visualization.

**schlegel_init**Initial properties of the Schlegel diagram to be displayed.

##### Options

Int FACET index of the projection facet, see Visual::SchlegelDiagram::FACETRational ZOOM zoom factor, see Visual::SchlegelDiagram::ZOOMVector FACET_POINT Vector INNER_POINT