# 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

•

### AffineLattice

Category: Geometry

a 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

•

### Cone

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

•
Cone<Rational>

An affine rational cone realized in Rd.

•
Cone<Float>

An affine cone with float coordinates realized in Rd.

•
Cone::ExactCoord

An affine cone with an exact coordinate type, like Rational.

•

### Backward compatibility

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

•
DIAMETER () → Int

The diameter of the GRAPH of the cone

##### Returns
 Int
•
DUAL_DIAMETER () → Int

The diameter of the DUAL_GRAPH

##### Returns
 Int
•
DUAL_TRIANGLE_FREE () → Bool

True if the DUAL_GRAPH contains no triangle

##### Returns
 Bool
•
TRIANGLE_FREE () → Bool

True if the GRAPH contains no triangle

##### Returns
 Bool
•

### Combinatorics

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 () → Int

Connectivity of the GRAPH this is the minimum number of nodes that have to be removed from the GRAPH to make it disconnected

##### Returns
 Int
•
DUAL_CONNECTIVITY () → Int

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

##### Returns
 Int
•
DUAL_EVEN () → Bool

True if the DUAL_GRAPH is bipartite

##### Returns
 Bool
•
EVEN () → Bool

True if the GRAPH is bipartite

##### Returns
 Bool
•
faces_of_dim (c) → Array<Set<Int>>

Output the faces of a given dimension

##### Parameters
 Cone c the input cone
##### Returns
 Array>
•
FACET_DEGREES () → Vector<Int>

Facet degrees of the polytope. The degree of a facet is the number of adjacent facets.

##### Returns
 Vector - in the same order as FACETS
•
N_FLAGS (type ...) → Int

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 f034 of the flag vector.

##### Parameters
 Int type ... flag type
##### Returns
 Int
•
VERTEX_DEGREES () → Vector<Int>

Ray degrees of the cone

##### Returns
 Vector - in the same order as RAYS
•

### Geometry

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

•
AMBIENT_DIM () → Int

returns the dimension of the ambient space of the cone

##### Returns
 Int
•
contains (v) → Bool

checks whether a given point is contained in a cone

##### Parameters
 Vector v point
##### Returns
 Bool
•
contains_in_interior (v) → Bool

checks whether a given point is contained in the strict interior of a cone

##### Parameters
 Vector v point
##### Returns
 Bool
•
DIM () → Int

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

##### Returns
 Int
•

### Lattice points in cones

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
•

### Topology

The following methods compute topological invariants.

•

### Visualization

These methods are for visualization.

•
VISUAL () → Visual::Cone

Visualizes the cone, intersected with the unit ball.

##### Options
 option list: Visual::Polygons::decorations option list: geometric_options_linear
##### Returns
 Visual::Cone

•

### GroebnerBasis

Category: Lattice points in cones

The Groebner basis of the homogeneous toric ideal associated to the polytope, the term order is given in matrix form.

•

### LinearProgram

Category: Optimization

A linear program specified by a linear or abstract objective function

#### Properties of LinearProgram

•
ABSTRACT_OBJECTIVE: common::Vector

Abstract 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 $g->EDGES; {0 1} {2 3} • LINEAR_OBJECTIVE: common::Vector Linear objective function. In d-space a linear objective function is given by a (d+1)-vector. The first coordinate specifies a constant that is added to the resulting value. Example: • The following creates a new LinearProgram object and assigns a linear objective to it:>$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::Scalar

Maximum 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 $p->LP->MAXIMAL_VALUE; inf • MAXIMAL_VERTEX: common::Vector Coordinates of a (possibly not unique) affine vertex at which the maximum of the objective function is attained. Example: • The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for a maximal vertex:>$c = new Vector([0, 1, -1/2]);> $p = cube(2);>$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 $p->LP->MINIMAL_FACE; {0 2} • MINIMAL_VALUE: LinearProgram::Scalar Similar to MAXIMAL_VALUE. Examples: • The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for the minimal value:>$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 $p->LP->MINIMAL_VALUE; 1 • MINIMAL_VERTEX: common::Vector Similar to MAXIMAL_VERTEX. Example: • The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for a minimal vertex:>$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<Int>

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

##### Returns
 Array
•
VERTEX_OUT_DEGREES () → Array<Int>

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

##### Returns
 Array
•

### MixedIntegerLinearProgram

Category: Optimization

A mixed integer linear program specified by a linear or abstract objective function

#### Properties of MixedIntegerLinearProgram

•
INTEGER_VARIABLES: common::Set<Int>

Set of integers that indicate which entries of the solution should be integral. If no value is specified, all entries are required to be integral. If all entries should be rational, please use an LinearProgram instead.

Examples:
• The following defines a MixedIntegerLinearProgram together with a linear objective on a rational line segment embedded in two-dimensional space. > $l = new Polytope(INEQUALITIES=>[[0,1,0],[3/2,-1,0],[1,0,0]],EQUATIONS=>[[0,0,1]]);>$obj = new Vector([0,1,0]);> $intvar = new Set<Int>([0,1,2]);>$milp = $l->MILP(LINEAR_OBJECTIVE=>$obj, INTEGER_VARIABLES=>$intvar);> print$milp->INTEGER_VARIABLES; {0 1 2}
• Same as the previous example, but we do not require the first coordinate to be integral anymore.> $l = new Polytope(INEQUALITIES=>[[0,1,0],[3/2,-1,0],[1,0,0]],EQUATIONS=>[[0,0,1]]);>$obj = new Vector([0,1,0]);> $intvar = new Set<Int>([0,2]);>$milp = $l->MILP(LINEAR_OBJECTIVE=>$obj, INTEGER_VARIABLES=>$intvar);> print$milp->INTEGER_VARIABLES; {0 2}
•
LINEAR_OBJECTIVE: common::Vector

Linear objective funtion. 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 defines a MixedIntegerLinearProgram together with a linear objective on a rational line segment embedded in two-dimensional space. > $l = new Polytope(INEQUALITIES=>[[0,1,0],[3/2,-1,0],[1,0,0]],EQUATIONS=>[[0,0,1]]);>$obj = new Vector([0,-1,0]);> $intvar = new Set<Int>([0,1,2]);>$milp = $l->MILP(LINEAR_OBJECTIVE=>$obj, INTEGER_VARIABLES=>$intvar);> print$milp->LINEAR_OBJECTIVE; 0 -1 0
•
MAXIMAL_SOLUTION: common::Vector

Coordinates of a (possibly not unique) affine vertex at which the maximum of the objective function is attained.

Examples:
• The following defines a MixedIntegerLinearProgram together with a linear objective for the centered square with side length 2 and asks for a maximal solution:> $c = new Vector([0, 1, -1/2]);>$p = cube(2);> $p->MILP(LINEAR_OBJECTIVE=>$c);> print $p->MILP->MAXIMAL_SOLUTION; 1 1 -1 • The following defines a MixedIntegerLinearProgram together with a linear objective on a rational line segment embedded in two-dimensional space. Note that the maximal solution is not a vertex/endpoint of the line segment.>$l = new Polytope(INEQUALITIES=>[[0,1,0],[3/2,-1,0],[1,0,0]],EQUATIONS=>[[0,0,1]]);> $obj = new Vector([0,1,0]);>$intvar = new Set<Int>([0,1,2]);> $milp =$l->MILP(LINEAR_OBJECTIVE=>$obj, INTEGER_VARIABLES=>$intvar);> print $milp->MAXIMAL_SOLUTION; 1 1 0 • Same as the previous example, but we do not require the first coordinate to be integral anymore. Now the maximal solution is an endpoint of the line segment.>$l = new Polytope(INEQUALITIES=>[[0,1,0],[3/2,-1,0],[1,0,0]],EQUATIONS=>[[0,0,1]]);> $obj = new Vector([0,1,0]);>$intvar = new Set<Int>([0,2]);> $milp =$l->MILP(LINEAR_OBJECTIVE=>$obj, INTEGER_VARIABLES=>$intvar);> print $milp->MAXIMAL_SOLUTION; 1 3/2 0 • MAXIMAL_VALUE: MixedIntegerLinearProgram::Scalar Maximum value the objective funtion takes under the restriction given by INTEGER_VARIABLES. Examples: • The following defines a MixedIntegerLinearProgram together with a linear objective on a rational line segment embedded in two-dimensional space. Note that the maximal value is integral and not the same as the value of the objective function on any of the vertices.>$l = new Polytope(INEQUALITIES=>[[0,1,0],[3/2,-1,0],[1,0,0]],EQUATIONS=>[[0,0,1]]);> $obj = new Vector([0,1,0]);>$intvar = new Set<Int>([0,1,2]);> $milp =$l->MILP(LINEAR_OBJECTIVE=>$obj, INTEGER_VARIABLES=>$intvar);> print $milp->MAXIMAL_VALUE; 1 • Same as the previous example, but we do not require the first coordinate to be integral anymore.>$l = new Polytope(INEQUALITIES=>[[0,1,0],[3/2,-1,0],[1,0,0]],EQUATIONS=>[[0,0,1]]);> $obj = new Vector([0,1,0]);>$intvar = new Set<Int>([0,2]);> $milp =$l->MILP(LINEAR_OBJECTIVE=>$obj, INTEGER_VARIABLES=>$intvar);> print $milp->MAXIMAL_VALUE; 3/2 • MINIMAL_SOLUTION: common::Vector Similar to MAXIMAL_SOLUTION Example: • The following defines a MixedIntegerLinearProgram together with a linear objective for the centered square with side length 2 and asks for a maximal solution:>$c = new Vector([0, 1, -1/2]);> $p = cube(2);>$p->MILP(LINEAR_OBJECTIVE=>$c);> print$p->MILP->MINIMAL_SOLUTION; 1 -1 1
•
MINIMAL_VALUE: MixedIntegerLinearProgram::Scalar

Similar to MAXIMAL_VALUE.

Examples:
• The following defines a MixedIntegerLinearProgram together with a linear objective on a rational line segment embedded in two-dimensional space. Note that the maximal value is integral and not the same as the value of the objective function on any of the vertices.> $l = new Polytope(INEQUALITIES=>[[0,1,0],[3/2,-1,0],[1,0,0]],EQUATIONS=>[[0,0,1]]);>$obj = new Vector([0,-1,0]);> $intvar = new Set<Int>([0,1,2]);>$milp = $l->MILP(LINEAR_OBJECTIVE=>$obj, INTEGER_VARIABLES=>$intvar);> print$milp->MINIMAL_VALUE; -1
• Same as the previous example, but we do not require the first coordinate to be integral anymore.> $l = new Polytope(INEQUALITIES=>[[0,1,0],[3/2,-1,0],[1,0,0]],EQUATIONS=>[[0,0,1]]);>$obj = new Vector([0,-1,0]);> $intvar = new Set<Int>([0,2]);>$milp = $l->MILP(LINEAR_OBJECTIVE=>$obj, INTEGER_VARIABLES=>$intvar);> print$milp->MINIMAL_VALUE; -3/2
•

### PointConfiguration

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

•

### Combinatorics

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

•
faces_of_dim (p) → Array<Set<Int>>

Output the faces of a given dimension

##### Parameters
 PointConfiguration p the input point configuration
##### Returns
 Array>
•

### Geometry

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

•
AMBIENT_DIM () → Int

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

##### Returns
 Int
•
DIM () → Int

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

##### Returns
 Int
•

### Visualization

These methods are for visualization.

•
VISUAL () → Visual::PointConfiguration

Visualize a point configuration.

##### Options
 option list: Visual::Polygons::decorations option list: geometric_options
##### Returns
 Visual::PointConfiguration
•
VISUAL_POINTS () → Visual::Object

Visualize the POINTS of a point configuration.

##### Options
 option list: Visual::Polygons::decorations option list: geometric_options
##### Returns
 Visual::Object
•

### Polytope

Not necessarily bounded convex polyhedron, i.e., the feasible region of a linear program. Nonetheless, the name "Polytope" is used for two reasons: Firstly, as far as the combinatorics is concerned we always deal with polytopes; see the description of VERTICES_IN_FACETS for details. Note that a pointed polyhedron is projectively equivalent to a polytope. The second reason is historical. We use homogeneous coordinates, which is why Polytope is derived from Cone. Scalar is the numeric data type used for the coordinates.

Examples:
• To construct a polytope as the convex hull of three points in the plane use> $p=new Polytope(POINTS=>[[1,0,0],[1,1,0],[1,0,1]]);> print$p->N_FACETS 3 Note that homogeneous coordinates are used throughout.
• Many standard constructions are available directly. For instance, to get a regular 120-cell (which is 4-dimensional) use:> $c=regular_120_cell();> print$c->VOLUME; 1575+705r5 This is the exact volume 1575+705*\sqrt{5}. polymake has limited support for polytopes with non-rational coordinates.
derived from: Cone

#### Specializations of Polytope

•
Polytope<Rational>

A rational polyhedron realized in Q^d

•
Polytope<Float>

A pointed polyhedron with float coordinates realized in Rd.

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.

•
Polytope::Lattice

A polytope all of whose vertex coordinates are integral.

•
Symmetry

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

•

### Combinatorics

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

•
ALTSHULER_DET: common::Integer

Let M be the vertex-facet incidence matrix, then the Altshuler determinant is defined as max{det(M ∗ MT), det(MT ∗ 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::Int

Maximal 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::Bool

Dual 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; true
•
•
COCUBICAL: common::Bool

Dual 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; true
•
COCUBICALITY: common::Int

Dual 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
•
COMPLEXITY: common::Float

Parameter describing the shape of the face-lattice of a 4-polytope.

•
CUBICAL: common::Bool

True 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; true
• 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; true
•
CUBICALITY: common::Int

Maximal 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
•
CUBICAL_H_VECTOR: common::Vector<Integer>

Cubical h-vector. Defined for cubical polytopes.

•
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_GRAPH: Graph<Undirected> as Polytope::DUAL_GRAPH

Augmented subobject [[Cone::DUAL_GRAPH

•

### Geometry

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

•
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::Bool

True 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; true
• 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; true
•
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
•
EXCESS_VERTEX_DEGREE: common::Int

Measures the deviation of the cone from being simple in terms of the GRAPH. Alias for property Cone::EXCESS_RAY_DEGREE.

Example:
• The excess vertex degree of an egyptian pyramid is one.> print pyramid(cube(2))->EXCESS_VERTEX_DEGREE; 1
•
F2_VECTOR: common::Matrix<Integer, NonSymmetric>

fik 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::Int

Maximal 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

•
FATNESS: common::Float

Parameter describing the shape of the face-lattice of a 4-polytope.

•
FOLDABLE_MAX_SIGNATURE_UPPER_BOUND: common::Int

An 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.

•
F_VECTOR: common::Vector<Integer>

fk 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<Undirected> as Polytope::GRAPH

Augmented subobject [[Cone::GRAPH

#### Properties of GRAPH

•
G_VECTOR: common::Vector<Integer>

(Toric) g-vector, defined via the (generalized) h-vector as gi = hi - hi-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 (vi wi), for i=1, ..., k.

The Moebius strip in question is given by the quadrangles (vi, wi, wi+1,vi+1), for i=1, ..., k-1, and the quadrangle (v1, w1, vk, wk).

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

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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.

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NEIGHBORLINESS: common::Int

Maximal 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
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NEIGHBORLY: common::Bool

True 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)->NEIGHBORLY; true
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N_VERTEX_FACET_INC: common::Int

Number of pairs of incident vertices and facets. Alias for property Cone::N_RAY_FACET_INC.

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N_VERTICES: common::Int

Number 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
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SELF_DUAL: common::Bool

True 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; true
• 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; true
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SIMPLE: common::Bool

True if the polytope is simple. Dual to SIMPLICIAL.

Example:
• This determines if a 3-dimensional cube is simple or not:> print cube(3)->SIMPLE; true
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SIMPLEXITY_LOWER_BOUND: common::Int

A lower bound for the minimal number of simplices in a triangulation

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SIMPLICIAL: common::Bool

True 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; true
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SIMPLICIALITY: common::Int

Maximal 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
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SIMPLICITY: common::Int

Maximal 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
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SUBRIDGE_SIZES: common::Map<Int, Int>

Lists for each occurring size (= number of incident facets or ridges) of a subridge how many there are.

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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)}
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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
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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
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### Geometry

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

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AFFINE_HULL: common::Matrix

Dual 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.

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BOUNDED: common::Bool

True 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; true> $q = facet_to_infinity($p,4);> print $q->BOUNDED; false • CENTERED: common::Bool True if (1, 0, 0, ...) is in the relative interior. If full-dimensional then polar to BOUNDED. Example: • The cube [0,1]^3 is not centered, since the origin is on the boundary. By a small translation we can make it centered:>$p = cube(3,0,0);> print $p->CENTERED; false>$t = new Vector([-1/2,-1/2,-1/2]);> print translate($p,$t)->CENTERED; true
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CENTERED_ZONOTOPE: common::Bool

is 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.

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CENTRALLY_SYMMETRIC: common::Bool

True 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.> $p = cube(3);> print$p->CENTRALLY_SYMMETRIC; true> print stack($p,5)->CENTRALLY_SYMMETRIC; false> print stack($p,new Set<Int>(4,5))->CENTRALLY_SYMMETRIC; true
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CENTROID: common::Vector

Centroid (center of mass) of the polytope.

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CONE_AMBIENT_DIM: common::Int

One more than the dimension of the space in which the polyhedron lives. = dimension of the space in which the homogenization of the polyhedron lives

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CONE_DIM: common::Int

One 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
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CS_PERMUTATION: common::Array<Int>

The permutation induced by the central symmetry, if present.

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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.

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FAR_HYPERPLANE: common::Vector

Valid strict inequality for all affine points of the polyhedron.

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FEASIBLE: common::Bool

True if the polyhedron is not empty.

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GALE_TRANSFORM: common::Matrix

Coordinates of the Gale transform.

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LATTICE: common::Bool
Only defined for Polytope<Rational>

A rational polytope is lattice if each bounded vertex has integer coordinates.

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MINIMAL_VERTEX_ANGLE: common::Float

The minimal angle between any two vertices (seen from the VERTEX_BARYCENTER).

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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

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N_01POINTS: common::Int
Only defined for Polytope<Rational>

Number of points with 0/1-coordinates in a polytope.

Depends on: azove
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N_POINTS: common::Int

Number of POINTS. Alias for property Cone::N_INPUT_RAYS.

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ONE_VERTEX: common::Vector

A 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
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POINTED: common::Bool

True 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 does not change this, although it is no longer bounded. After removing two opposing facets, it contains infinitely many affine lines orthogonal to the removed facets.> $p = cube(2);> print$p->POINTED; true> print facet_to_infinity($p,0)->POINTED; true> print new Polytope(INEQUALITIES=>$p->FACETS->minor([0,1],All))->POINTED; false
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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.

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QUOTIENT_SPACE: QuotientSpace

A topological quotient space obtained from a polytope by identifying faces.

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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.

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SPLITS: common::Matrix

The splits of the polytope, i.e., hyperplanes cutting the polytope in two parts such that we have a regular subdivision.

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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.

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STEINER_POINT: common::Vector

Steiner point of the whole polytope.

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STEINER_POINTS: common::Matrix

A weighted inner point depending on the outer angle called Steiner point for all faces of dimensions 2 to d.

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TILING_LATTICE: AffineLattice

An affine lattice L such that P + L tiles the affine span of P

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VALID_POINT: common::Vector

Some 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
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VERTEX_BARYCENTER: common::Vector

The 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
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VERTEX_NORMALS: common::Matrix

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

Vertices 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);
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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.

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WEAKLY_CENTERED: common::Bool

True 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; true> print $p->CENTERED; false • ZONOTOPE_INPUT_POINTS: common::Matrix The 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. • ### Input property These properties are for input only. They allow redundant information. • EQUATIONS: common::Matrix Equations that hold for all points of the polyhedron. A vector (A0, A1, ..., Ad) describes the hyperplane of all points (1, x1, ..., xd) such that A0 + A1 x1 + ... + Ad xd = 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::Matrix Inequalities that describe half-spaces such that the polyhedron is their intersection. Redundancies are allowed. Dual to POINTS. A vector (A0, A1, ..., Ad) defines the (closed affine) half-space of points (1, x1, ..., xd) such that A0 + A1 x1 + ... + Ad xd >= 0. Input section only. Ask for FACETS and AFFINE_HULL if you want to compute an H-representation from a V-representation. • POINTS: common::Matrix Points such that the polyhedron is their convex hull. Redundancies are allowed. The vector (x0, x1, ... xd) represents a point in d-space given in homogeneous coordinates. Affine points are identified by x0 > 0. Points with x0 = 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 x0 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 • ### Lattice points in cones These properties capture information that depends on the lattice structure of the cone. polymake always works with the integer lattice. • ### Lattice points in polytopes These properties capture information that depends on the lattice structure of the polytope. polymake always works with the integer lattice. • ### Matroid properties Properties which belong to the corresponding (oriented) matroid • ### Optimization These properties provide tools from linear, integer and dicrete optimization. In particular, linear programs are defined here. • ### Symmetry These properties capture information of the object that is concerned with the action of permutation groups. • ### Triangulation and volume Everything in this group is defined for BOUNDED polytopes only. • MAHLER_VOLUME: Polytope::Scalar Mahler volume (or volume product) of the polytope. Defined as the volume of the polytope and the volume of its polar (for BOUNDED, CENTERED and FULL_DIM polytopes only). Often studied for centrally symmetric convex bodies, where the regular cubes are conjectured to be the global minimiers. Example: • The following prints the Mahler volume of the centered 2-cube:> print cube(2)->MAHLER_VOLUME; 8 • POLYTOPAL_SUBDIVISION: SubdivisionOfPoints as Polytope::POLYTOPAL_SUBDIVISION Polytopal Subdivision of the polytope using only its vertices. #### Properties of POLYTOPAL_SUBDIVISION • ### Combinatorics These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice. • REFINED_SPLITS: common::Set<Int> The splits that are coarsenings of the subdivision. If the subdivision is regular these form the unique split decomposition of the corresponding weight function. • 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::Array Array of the squared relative k-dimensional volumes of the simplices in a triangulation of a d-dimensional polytope. • TRIANGULATION: GeometricSimplicialComplex as Polytope::TRIANGULATION Augmented subobject [[Cone::TRIANGULATION #### Properties of TRIANGULATION • ### Triangulation and volume These properties collect information about triangulations of the object and properties usually computed from such, as the volume. • GKZ_VECTOR: common::Vector GKZ-vector See Chapter 7 in Gelfand, Kapranov, and Zelevinsky: Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994 • VOLUME: Polytope::Scalar Volume of the polytope. Example: • The following prints the volume of the centered 3-dimensional cube with side length 2:> print cube(3)->VOLUME; 8 • ### Unbounded polyhedra 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. • ### Visualization These properties are for visualization. #### User Methods of Polytope • ### Backward compatibility These methods are provided for backward compatibility with older versions of polymake only. They should not be used in new code. • N_EDGES () → Int The number of edges of the GRAPH ##### Returns  Int • ### Combinatorics 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 () → String Prettily print the cd-index given in CD_INDEX_COEFFICIENTS ##### Returns  String • N_RIDGES () → Int The number of ridges (faces of codimension 2) of the polytope equals the number of edges of the DUAL_GRAPH ##### Returns  Int • ### Geometry These methods capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets. • AMBIENT_DIM () → Int returns the dimension of the ambient space of the polytope ##### Returns  Int • DIM () → Int returns the dimension of the polytope ##### Returns  Int • 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> • labeled_vertices (label ...) → Set<Int> Find the vertices by given labels. ##### Parameters  String label ... vertex labels ##### Returns  Set vertex indices • 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 coeff coefficient vector to the rays of the Minkowski summand cone ##### Returns  Polytope • 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. ##### Parameters  Vector point point in the Minkowski summand cone ##### Returns  Polytope • 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> • ### Lattice points in cones These methods capture information that depends on the lattice structure of the cone. polymake always works with the integer lattice. • EHRHART_POLYNOMIAL_COEFF () → Vector<Rational> Only defined for Polytope::Lattice Vector containing the coefficients of the EHRHART_POLYNOMIAL, ordered by increasing degree of the corresponding term. ##### Returns  Vector • FACET_POINT_LATTICE_DISTANCES (v) → Vector<Integer> Only defined for Polytope::Lattice Vector containing the distances of a given point v from all facets ##### Parameters  Vector v point in the ambient space of the polytope ##### Returns  Vector • N_LATTICE_POINTS_IN_DILATION (n) → Int Only defined for Polytope::Lattice The number of LATTICE_POINTS in the n-th dilation of the polytope ##### Parameters  Int n dilation factor ##### Returns  Int • 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 ##### Parameters  Polytope P polytope ##### Returns  Polytope Pnew polytope • ### Lattice points in polytopes 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 • ### Symmetry These methods capture information of the object that is concerned with the action of permutation groups. • VISUAL_NOP (colors_ref, trans_ref) Only defined for Polytope::PointOrbit Visualizes all (nested) orbit polytopes contained in orb in one picture. ##### Parameters  ARRAY colors_ref the reference to an array of colors ARRAY trans_ref the reference to an array of transparency values • VISUAL_NOP_GRAPH (filename) Only defined for Polytope::PointOrbit 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  String filename the filename for the 'dot' file • ### Triangulation and volume 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 - +1/-1 array specifying the sign of the determinant of each simplex • ### Unbounded polyhedra 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> Indices of FACETS that are bounded. ##### Returns  Set • 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> Indices of VERTICES that are no rays. ##### Returns  Set • ### Visualization These methods are for visualization. • GALE () → Visual::Gale Generate the Gale diagram of a d-polyhedron with at most d+4 vertices. ##### Returns  Visual::Gale • SCHLEGEL () → Visual::SchlegelDiagram Create a Schlegel diagram and draw it. ##### Options  Visual::Graph::decorations proj_facet decorations for the edges of the projection face option list: schlegel_init option list: Visual::Wire::decorations ##### Returns  Visual::SchlegelDiagram • VISUAL () → Visual::Polytope Visualize 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::PolytopeGraph Visualize the BOUNDED_COMPLEX.GRAPH of a polyhedron. ##### Options  Int seed random seed value for the string embedder option list: Visual::Graph::decorations ##### Returns  Visual::PolytopeGraph • VISUAL_DUAL () → Visual::Object Visualize the dual polytope as a solid 3-d object. The polytope must be BOUNDED and CENTERED. ##### Options  option list: Visual::Polygons::decorations option list: geometric_options ##### Returns  Visual::Object • VISUAL_DUAL_FACE_LATTICE () → Visual::PolytopeLattice Visualize the dual face lattice of a polyhedron as a multi-layer graph. ##### Options  Int seed random seed value for the node placement option list: Visual::Lattice::decorations ##### Returns  Visual::PolytopeLattice • VISUAL_DUAL_GRAPH () → Visual::Graph Visualize the DUAL_GRAPH of a polyhedron. ##### Options  Int seed random seed value for the string embedder option list: Visual::Graph::decorations ##### Returns  Visual::Graph • VISUAL_FACE_LATTICE () → Visual::PolytopeLattice Visualize the HASSE_DIAGRAM of a polyhedron as a multi-layer graph. ##### Options  Int seed random seed value for the node placement option list: Visual::Lattice::decorations ##### Returns  Visual::PolytopeLattice • VISUAL_GRAPH () → Visual::PolytopeGraph Visualize the GRAPH of a polyhedron. ##### Options  Int seed random seed value for the string embedder option list: Visual::Graph::decorations ##### Returns  Visual::PolytopeGraph • VISUAL_ORBIT_COLORED_GRAPH () → Visual::PolytopeGraph Visualizes the graph of a symmetric cone: All nodes belonging to one orbit get the same color. ##### Options  option list: Visual::Graph::decorations ##### Returns  Visual::PolytopeGraph • write_stl (filename) Take a 3-polytope and write ASCII STL output. ##### Parameters  String filename Example: • > dodecahedron()->write_stl("/tmp/dodecahedron.stl");  • ### PropagatedPolytope 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 • ### QuotientSpace Category: Symmetry A topological quotient space obtained from a Polytope by identifying faces. This object will sit inside the polytope. #### Properties of QuotientSpace • ### SchlegelDiagram A Schlegel diagram of a polytope. ##### Type Parameters  Scalar default Rational #### Properties of SchlegelDiagram #### User Methods of SchlegelDiagram • VISUAL () → Visual::SchlegelDiagram Draw the Schlegel diagram. ##### Options  Visual::Graph::decorations proj_facet decorations for the edges of the projection face option list: Visual::Graph::decorations ##### Returns  Visual::SchlegelDiagram • ### SymmetrizedCocircuitEquations Category: Symmetry UNDOCUMENTED #### Properties of SymmetrizedCocircuitEquations • ### VectorConfiguration Category: Geometry 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 #### User Methods of VectorConfiguration • ### Visualization These methods are for visualization. • VISUAL () → Visual::PointConfiguration Visualize a vector configuration. ##### Options  option list: Visual::Polygons::decorations option list: geometric_options_linear ##### Returns  Visual::PointConfiguration • ### Visual::Cone Category: Visualization Visualization of a Cone as a graph (if 1d), or as a solid object (if 2d or 3d) derived from: Visual::Container • ### Visual::Gale Category: Visualization A gale diagram prepared for drawing. derived from: Visual::Object • ### Visual::PointConfiguration Category: Visualization Visualization of the point configuration. #### User Methods of Visual::PointConfiguration • POLYTOPAL_SUBDIVISION (index) → Visual::PointConfiguration Visualize a POLYTOPAL_SUBDIVISION of a point configuration. ##### Parameters  Int index Index of the subdivision to visualize ##### Options  option list: Visual::Polygons::decorations option list: geometric_options ##### Returns  Visual::PointConfiguration • TRIANGULATION (t) → Visual::PointConfiguration Visualize the TRIANGULATION of a point configuration ##### Parameters  Array> t facets of the triangulation ##### Options  option list: Visual::Polygons::decorations ##### Returns  Visual::PointConfiguration • ### Visual::Polytope Category: Visualization Visualization 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::Polytope Illustrate 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 Linear Program object attached to the polytope ##### Returns  Visual::Polytope 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::Polytope

Visualize the LATTICE_POINTS of a polytope

##### Options
 option list: Visual::PointSet::decorations
##### Returns
 Visual::Polytope

Example:
• Visualizes the lattice points of the threedimensional cube.> cube(3)->VISUAL->LATTICE;
•
LATTICE_COLORED () → Visual::Polytope

Visualize the LATTICE_POINTS of a polytope in different colors (interior / boundary / vertices)

##### Options
 option list: Visual::PointSet::decorations
##### Returns
 Visual::Polytope

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::Polytope

Illustrate 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:
•

### Visual::PolytopeGraph

Category: Visualization

Visualization of the graph of a polyhedron.

#### User Methods of Visual::PolytopeGraph

•
DIRECTED_GRAPH (lp) → Visual::PolytopeGraph

Show 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::PolytopeGraph

Produce an edge coloring of a bounded graph from local data in the Hasse diagram.

##### Returns
 Visual::PolytopeGraph
•
MIN_MAX_FACE (lp) → Visual::PolytopeGraph

Illustrate 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::PolytopeGraph

Illustrate 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
•

### Visual::PolytopeLattice

Category: Visualization

Visualization of the HASSE_DIAGRAM of a polyhedron as a multi-layer graph..

#### User Methods of Visual::PolytopeLattice

•
MIN_MAX_FACE (lp) → Visual::PolytopeLattice

Illustrate 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
•

### Visual::SchlegelDiagram

Category: Visualization

Visualization of the Schlegel diagram of a polytope.

#### User Methods of Visual::SchlegelDiagram

•
CONSTRUCTION () → Visual::SchlegelDiagram

Visualize the construction of a 3D Schlegel diagram, that is, the Viewpoint, the 3-polytope and the projection onto one facet.

##### Options
 option list: Visual::Polygons::decorations
##### Returns
 Visual::SchlegelDiagram
•
DIRECTED_GRAPH (lp) → Visual::SchlegelDiagram

Illustrate 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::SchlegelDiagram

Illustrate 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::SchlegelDiagram

Draw the facets of the Schlegel diagram as polytopes.

##### Options
 option list: Visual::Polygons::decorations
##### Returns
 Visual::SchlegelDiagram
•
STEINER ()

UNDOCUMENTED

##### Options
 option list: Visual::PointSet::decorations
•
TRIANGULATION_BOUNDARY () → Visual::SchlegelDiagram

Draw the edges of the TRIANGULATION_BOUNDARY

##### Options
 option list: Visual::Graph::decorations
##### Returns
 Visual::SchlegelDiagram
•
TRIANGULATION_BOUNDARY_SOLID () → Visual::SchlegelDiagram

Draw the boundary simplices of the triangulation as solid tetrahedra.

##### Options
 option list: Visual::Polygons::decorations
##### Returns
 Visual::SchlegelDiagram
•
VERTEX_COLORS (lp) → Visual::SchlegelDiagram

Illustrate 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
•

### VoronoiPolyhedron

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 VoronoiPolyhedron becomes a derived class from Polytope<Scalar>.

derived from: Polytope

•

### Visualization

These methods are for visualization.

•
VISUAL_CRUST () → Visual::Container

Draw a Voronoi diagram, its |dual graph and the crust. Use the interactive features of the viewer to select.

##### Options
 option list: Visual::Graph::decorations
##### Returns
 Visual::Container
•
VISUAL_NN_CRUST () → Visual::Container

Draw a Voronoi diagram, its dual graph and the nearest neighbor crust. Use the interactive features of the viewer to select.

##### Options
 option list: Visual::Graph::decorations
##### Returns
 Visual::Container
•
VISUAL_VORONOI () → Visual::Container

Visualize the Voronoi diagram together with its sites.

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

## User Functions

•

### Combinatorics

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,Set>> co /circuits a set of circuits or cocircuits
##### Returns
 SparseMatrix
•
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 ridge simplices symmetries of the cone are NOT taken into account

##### Parameters
 Cone C Array interior_ridge_simplices interior codimension 1 simplices Array interior_simplices interior simplices of maximal dimension
##### 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
•
cocircuit_equation_of_ridge (C, rho) → HashMap<Set,Rational>

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

##### Parameters
 Cone C Set rho the interior ridge
##### Returns
 HashMap
•
codegree <Scalar> (P)

Calculate the codegree of a cone or polytope P. This is the maximal positive integer c such that every subset of size < c lies in a common facet of conv P. Moreover, the relation degree(P) + codegree(P) = dim(P) + 1 holds.

##### Type Parameters
 Scalar the underlying number type,
##### Parameters
 Cone P

Example:
• To find the codegree of the 3-cube, type> print codegree(cube(3)); 1
•
codegree <Scalar> (P)

Calculate the codegree of a point configuration P. This is the maximal positive integer c such that every subset of size < c lies in a common facet of conv P. Moreover, the relation degree(P) + codegree(P) = dim(P) + 1 holds.

##### Type Parameters
 Scalar the underlying number type,
##### Parameters
 PointConfiguration P
•
contraction (C, v)

Contract a vector configuration C along a specified vector v.

##### Parameters
 VectorConfiguration C Int v index of the vector to contract
•
degree <Scalar> (P)

Calculate the degree of a cone, polytope or point configuration P. This is the maximal dimension of an interior face of P, where an interior face is a subset of the points of P whose convex hull does not lie on the boundary of P. Moreover, the relation degree(P) + codegree(P) = dim(P) + 1 holds.

##### Type Parameters
 Scalar the underlying number type,
##### Parameters
 PointConfiguration P (or Cone or Polytope)

Example:
• To find the degree of the 3-cube, type> print degree(cube(3)); 3
•
deletion (C, v)

Delete a specified vector v from a vector configuration C.

##### Parameters
 VectorConfiguration C Int v index of the vector to delete
•

### Comparing

Functions based on graph isomorphisms.

•
congruent (P1, P2) → Scalar

Check 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 polytope Polytope 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,$tn); 0 • equal_polyhedra (P1, P2) → Bool UNDOCUMENTED ##### Parameters  Polytope P1 the first polytope Polytope P2 the second polytope ##### Options  Bool verbose Prints information on the difference between P1 and P2 if they are not equal. ##### Returns  Bool true if the two polyhedra are equal, false otherwise Example: • >$p = new Polytope(VERTICES => [[1,-1,-1],[1,1,-1],[1,-1,1],[1,1,1]]);> print equal_polyhedra($p,cube(2)); true To see why two polytopes are unequal, try this:> print equal_polyhedra($p,cube(3),verbose => 1); Cones/Polytopes do no live in the same ambient space. false> print equal_polyhedra($p,simplex(2),verbose => 1); Inequality 1 -1 -1 not satisfied by point 1 1 1. false • find_facet_vertex_permutations (P1, P2) → Pair<Array<Int>, Array<Int>> Find the permutations of facets and vertices which maps the cone or polyhedron P1 to P2. The facet permutation is the first component, the vertex permutation is the second component of the return value. Only the combinatorial isomorphism is considered. If the polytopes are not isomorphic, an exception is thrown. ##### Parameters  Cone P1 the first cone/polytope Cone P2 the second cone/polytope ##### Returns  Pair, Array> the facet and the vertex permutations Example: • To print the vertex permutation that maps the 3-simplex to its mirror image, type this:>$p = find_facet_vertex_permutations(simplex(3),scale(simplex(3),-1));> print $p->first; 1 2 3 0 • included_polyhedra (P1, P2) → Bool UNDOCUMENTED ##### Parameters  Polytope P1 the first polytope Polytope P2 the second polytope ##### Options  Bool verbose Prints information on the difference between P1 and P2 if none is included in the other. ##### Returns  Bool 'true' if P1 is included in P2, 'false' otherwise Example: • > print included_polyhedra(simplex(3),cube(3)); true To see in what way the two polytopes differ, try this:> print included_polyhedra(cube(2),cube(3),verbose=>1); Cones/Polytopes do no live in the same ambient space. false • isomorphic (P1, P2) → Bool Check whether the face lattices of two cones or polytopes are isomorphic. The problem is reduced to graph isomorphism of the vertex-facet incidence graphs. ##### Parameters  Cone P1 the first cone/polytope Cone P2 the second cone/polytope ##### Returns  Bool 'true' if the face lattices are isomorphic, 'false' otherwise Example: • The following compares the standard 2-cube with a polygon generated as the convex hull of five points. The return value is true since both polygons are quadrangles.>$p = new Polytope(POINTS=>[[1,-1,-1],[1,1,-1],[1,-1,1],[1,1,1],[1,0,0]]);> print isomorphic(cube(2),$p); true • lattice_isomorphic_smooth_polytopes (P1, P2) → Bool Tests whether two smooth lattice polytopes are lattice equivalent by comparing lattice distances between vertices and facets. ##### Parameters  Polytope P1 the first lattice polytope Polytope P2 the second lattice polytope ##### Returns  Bool 'true' if the polytopes are lattice equivalent, 'false' otherwise Example: • >$t = new Vector(2,2);> print lattice_isomorphic_smooth_polytopes(cube(2),translate(cube(2),$t)); true • ### Consistency check These functions are for checking the consistency of some properties. • check_inc (points, hyperplanes, sign, verbose) → Bool Check coordinate data. For each pair of vectors from two given matrices their inner product must satisfy the given relation. ##### Parameters  Matrix points Matrix hyperplanes String sign composed of one or two characters from [-+0], representing the allowed domain of the vector inner products. Bool verbose print all products violating the required relation ##### Returns  Bool 'true' if all relations are satisfied, 'false' otherwise Example: • Let's check which vertices of the square lie in its zeroth facet:>$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 #points==4, #hyperplanes==1, -:0, 0:2, +:2, total:4 false 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) → Polytope Try 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 matrix Bool verbose prints information about the check. ##### Returns  Polytope the resulting polytope under the assumption that VIF actually defines a polytope • validate_moebius_strip (P) → Bool Validates 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. ##### Parameters  Polytope P the given polytope ##### Returns  Bool 'true' if the Moebius strip edges form such a Moebius strip, 'false' otherwise • 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 the Moebius strip edges • ### Coordinate conversions 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 ##### Parameters  Polytope P source object ##### Returns  Matrix the dehomogenized vertices converted 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>($cone) ##### Type Parameters  Coord target coordinate type ##### Parameters  Cone c the input cone ##### Returns  Cone a new cone object or C itself it has the requested type • convert_to <Coord> (P) → Polytope<Coord> provide a Polytope object with desired coordinate type ##### Type Parameters  Coord target coordinate type ##### Parameters  Polytope P source object ##### Returns  Polytope P if it already has the requested type, a new object otherwise Example: • > print cube(2)->type->full_name; Polytope<Rational>>$pf = convert_to<Float>(cube(2));> print $pf->type->full_name; Polytope<Float> • ### Finite metric spaces Tight spans and their connections to polyhedral geometry • tight_span_envelope (sd) → Polytope Computes the envelope for a given subdivision of points. ##### Parameters  fan::SubdivisionOfPoints sd ##### Options  Bool extended If True, the envelope of the extended tight span is computed. ##### Returns  Polytope • ### Geometry These functions capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets. • all_steiner_points (P) → Matrix Compute the Steiner points of all faces of a polyhedron P using a randomized approximation of the angles. P must be BOUNDED. ##### Parameters  Polytope P ##### Options  Float eps controls the accuracy of the angles computed Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome. ##### Returns  Matrix • center_distance (p) → Float Compute the mean or median distance of the VERTICES to the VERTEX_BARYCENTER. ##### Parameters  Polytope p ##### Options  Bool median use median instead of arithmetic mean ##### Returns  Float • circuit_completions (L, R) → Array<Set> Given two matrices L (n x d) and R (m x d) such that (L/R) has rank r, select all (r+1-n)-subsets C of rows of R such that (L,S) or (S,L) is a circuit. Optionally, if d > r, a basis H for the orthogonal span of the affine hull of (L/R) may be given. ##### Parameters  Matrix L Matrix R ##### Options  Matrix H ##### Returns  Array Example: • Divide the vertex set of the 3-cube into a body diagonal L and six remaining vertices R. To find the subsets of R that complete L to a circuit, type>$c = cube(3);> $L =$c->VERTICES->minor([0,7],All);> $R =$c->VERTICES->minor([1,2,3,4,5,6],All);> print circuit_completions($L,$R); {0 1 3} {2 4 5}
•
containing_normal_cone (P, q) → Set

Return the vertices of the face of P whose normal cone contains a point q.

##### Parameters
 Cone P Vector q
##### Returns
 Set

Example:
• To find the face whose normal cone contains a given point, type> $p = new Polytope(VERTICES=>[[1,-1,0],[1,0,-1],[1,0,1],[1,100,0]]);> print containing_normal_cone($p, new Vector([1,1,2])); {2 3}
•
containing_outer_cone (P, q) → Set

Return the vertices of the face of P whose outer cone contains a point q.

##### Parameters
 Polytope P Vector q
##### Returns
 Set

Example:
• To find the face whose outer cone contains a given point, type> print containing_outer_cone(cube(3), new Vector([1,2,2,2])); {7}
•
dihedral_angle (H1, H2) → Float

Compute the dihedral angle between two (oriented) affine or linear hyperplanes.

##### Parameters
 Vector H1 : first hyperplane Vector H2 : second hyperplane
##### Options
 Bool deg output in degrees rather than radians, default is false Bool 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

##### Parameters
 Polytope p the input polytope
##### Returns
 Matrix - the lattice basis.

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.

##### Parameters
 Polytope P
##### Returns
 Matrix

Example:
• > $p = new Polytope(VERTICES=>[[1,13/10,1/2],[1,1/5,6/5],[1,1/10,-3/2],[1,-7/5,1/5]]);> print integer_points_bbox($p); 1 0 -1 1 -1 0 1 0 0 1 1 0 1 0 1
•
minimal_vertex_angle (P) → Float

Computes the minimal angle between two vertices of the input polytope P.

##### Parameters
 Polytope P
##### Returns
 Float

Example:
• > print minimal_vertex_angle(simplex(3)); 3.14159265358979
•
normaliz_compute (C) → List

Compute 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 libnormaliz.
##### Parameters
 Cone C
##### Options
 Bool from_facets supply facets instead of rays to normaliz Bool degree_one_generators compute the generators of degree one, i.e. lattice points of the polytope Bool hilbert_basis compute Hilbert basis of the cone C Bool h_star_vector compute Hilbert h-vector of the cone C Bool hilbert_series compute Hilbert series of the monoid Bool ehrhart_quasi_polynomial compute Ehrhart quasi polynomial of a polytope Bool facets compute support hyperplanes (=FACETS,LINEAR_SPAN) Bool rays compute extreme rays (=RAYS) Bool dual_algorithm use the dual algorithm by Pottier Bool skip_long do not try to use long coordinates first Bool verbose libnormaliz debug output
##### Returns
 List (Matrix degree one generators, Matrix Hilbert basis, Vector Hilbert h-vector, RationalFunction Hilbert series, Matrix facets, Matrix linear_span, Matrix rays) (only requested items)
•
occluding_cone (P, F) → Cone

For a face F of a cone or polytope P, return the polyhedral cone C such that taking the convex hull of P and any point in C destroys the face F

##### Parameters
 Cone P Set F
##### Returns
 Cone

Example:
• To find the occluding cone of an edge of the 3-cube, type> $c=occluding_cone(cube(3), [0,1]);> print$c->FACETS; -1 0 -1 0 -1 0 0 -1
•
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}}
•
separable (q, P) → Bool

Checks whether there exists a hyperplane separating a given point q from a polytope/cone P by solving a suitable LP. If true, q is a vertex of the polytope defined by q and the vertices of P. To get the separating hyperplane, use separating_hyperplane. Works without knowing the facets of P!

##### Parameters
 Vector q the vertex (candidate) which is to be separated from P Cone P the polytope/cone from which q is to be separated
##### Options
 Bool strong Test for strong separability. default: true
##### Returns
 Bool 'true' if q is separable from p

Example:
• > $q = cube(2)->VERTICES->row(0);> print separable(cube(2),$q, strong=>0); true
•
steiner_point (P) → Vector

Compute the Steiner point of a polyhedron P using a randomized approximation of the angles.

##### Parameters
 Polytope P
##### Options
 Float eps controls the accuracy of the angles computed Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome.
##### Returns
 Vector
•
violations (P, q) → Set

Check which relations, if any, are violated by a point.

##### Parameters
 Cone P Vector q
##### Options
 String section Which section of P to test against q Int 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}
•
visible_facet_indices (P, q) → Set

Return the indices of all facets that are visible from a point q.

##### Parameters
 Cone P Vector q
##### Returns
 Set

Example:
• This prints the facets of a square with the origin as its center and side length 2 that are visible from a certain point:> $p = cube(2);>$v = new Vector([1,2,2]);> map { print $p->VERTICES_IN_FACETS->[$_], "\n" } @{visible_facet_indices($p,$v)}; {1 3} {2 3}
•
visible_face_indices (P, q) → Set

Return the indices (in the HASSE_DIAGRAM) of all faces that are visible from a point q.

##### Parameters
 Cone P Vector q
##### Returns
 Set

Example:
• This prints the faces of a square with the origin as its center and side length 2 that are visible from a certain point:> $p = cube(2);>$v = new Vector([1,2,2]);> map { print $p->HASSE_DIAGRAM->FACES->[$_], "\n" } @{visible_face_indices($p,$v)}; {} {1} {2} {3} {1 3} {2 3}
•
zonotope_tiling_lattice (P) → AffineLattice

Calculates 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
•

### Optimization

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) → Polytope

Computes the inequalities and the linear objective for an LP to lift a simplicial d-ball embedded starshaped in Rd.

Contained in extension local.
##### Parameters
 topaz::GeometricSimplicialComplex c Int facet_index index of the facet to be lifted Rational conv_eps some epsilon > 0
##### Returns
 Polytope
•
core_point_algo (p, optLPvalue) → List

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).

##### Parameters
 Polytope p Rational optLPvalue optimal value of LP approximation
##### Options
 Bool verbose
##### Returns
 List (Vector optimal solution, Rational optimal value) may be empty
•
core_point_algo_Rote (p, optLPvalue) → List

Version 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).

##### Parameters
 Polytope p Rational optLPvalue optimal value of LP approximation
##### Options
 Bool verbose
##### Returns
 List (Vector optimal solution, Rational optimal value) may be empty
•
find_transitive_lp_sol (Inequalities) → List

Algorithm to solve symmetric linear programs (LP) of the form max ctx , 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 optimal solution, Rational optimal value, Bool feasible, Bool max_bounded)

Example:
• Consider the LP described by the facets of the 3-cube:> @sol=find_transitive_lp_sol(cube(3)->FACETS);> print $_, "\n" for @sol; 1 1 1 1 3 true true The optimal solution is [1,1,1,1], its value under c is 3, and the LP is feasible and bounded in direction of c. • inner_point (points) → Vector Compute a true inner point of a convex hull of the given set of points. ##### Parameters  Matrix points ##### Returns  Vector Example: • To print an inner point of the square, do this:> print inner_point(cube(2)->VERTICES); 1 -1/3 -1/3 • lp2poly <Scalar> (file, testvec, prefix) → Polytope<Scalar> Read a linear programming problem given in LP-Format (as used by cplex & Co.) and convert it to a Polytope<Scalar> object. WARNING The property FEASIBLE is NOT computed upon creation. This is done to avoid possibly long computation times before the object becomes available to the caller. This is NOT in keeping with standard practice in polymake, but after, all, these are linear programs and not polytopes. ##### Type Parameters  Scalar coordinate type of the resulting polytope; default is Rational. ##### Parameters  String file filename of a linear programming problem in LP-Format Vector testvec If present, after reading in each line of the LP it is checked whether testvec fulfills it String prefix If testvec is present, all variables in the LP file are assumed to have the form$prefix$i ##### Options  Int nocheck Do not automatically compute FEASIBLE for the polytope (recommended for very large LPs) ##### Returns  Polytope • 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->LP Bool maximize produces a maximization problem; default value: 0 (minimize) String file default value: standard output • poly2porta (p, file) take a rational polytope and write a porta input file (.ieq or .poi) ##### Parameters  Polytope p String file filename for the porta file (.ieq or .poi) or an open IO handle ##### Options  Bool primal true if points should be written, false if inequalities should be written (default is true) • 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 ##### Parameters  String file filename of a porta file (.ieq or .poi) ##### Returns  Polytope • 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 C the given polytope or cone ##### Options  Array ineq_labels changes the labels of the inequality rows Array 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. ##### Parameters  Graph G a directed graph ##### Returns  Vector • 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 polytope Int 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 • separating_hyperplane (q, points) → Vector Computes (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 throws an infeasible exception. 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  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);> $points = cube(2)->VERTICES->minor(sequence(1,3),All);> print separating_hyperplane($q,$points); 0 1/2 1/2 • separating_hyperplane (p1, p2) → Vector Computes (the normal vector of) a hyperplane which separates two given polytopes p1 and p2 if possible. Works by solving a linear program, not by facet enumeration. ##### Parameters  Polytope p1 the first polytope, will be on the positive side of the separating hyperplane Polytope p2 the second polytope ##### Options  Bool strong If this is set to true, the resulting hyperplane will be strongly separating, i.e. it wont touch either of the polytopes. If such a plane does not exist, an exception will be thrown. default: true ##### Returns  Vector a hyperplane separating p1 from p2 • totally_dual_integral (inequalities) → Bool Checks weather a given system of inequalities is totally dual integral or not. The inequalities should describe a full dimensional polyhedron ##### Parameters  Matrix inequalities ##### Returns  Bool Example: • > print totally_dual_integral(cube(2)->FACETS); true • vertex_colors (P, LP) → Array<RGB> Calculate RGB-color-values for each vertex depending on a linear or abstract objective function. Maximal and minimal affine vertices are colored as specified. Far vertices (= rays) orthogonal to the linear function normal vector are white. The colors for other affine vertices are linearly interpolated in the HSV color model. If the objective function is linear and the corresponding LP problem is unbounded, then the affine vertices that would become optimal after the removal of the rays are painted pale. ##### Parameters  Polytope P LinearProgram LP ##### Options  RGB min the minimal RGB value RGB max the maximal RGB value ##### Returns  Array Example: • This calculates a vertex coloring with respect to a linear program. For a better visualization, we also set the vertex thickness to 2.>$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)

construct a linear program whose optimal value is an upper bound for the algebraic signature of a triangulation of P. This is the absolute value of the difference of normalized volumes of black minus white simplices (counting only those with odd normalized volume) in a triangulation of P whose dual graph is bipartite. If P has a GROUP, it will be used to construct the linear program.

##### Parameters
 Polytope P String outfile_name

Examples:
• For the 0/1 2-cube without a GROUP, the foldable max signature lp is computed as follows:> write_foldable_max_signature_ilp(cube(2,0)); MINIMIZE obj: +1 x1 -1 x2 +1 x3 -1 x4 +1 x5 -1 x6 +1 x7 -1 x8 Subject To ie0: +1 x1 >= 0 ie1: +1 x2 >= 0 ie2: +1 x3 >= 0 ie3: +1 x4 >= 0 ie4: +1 x5 >= 0 ie5: +1 x6 >= 0 ie6: +1 x7 >= 0 ie7: +1 x8 >= 0 ie8: -1 x1 -1 x2 >= -1 ie9: -1 x3 -1 x4 >= -1 ie10: -1 x5 -1 x6 >= -1 ie11: -1 x7 -1 x8 >= -1 eq0: -1 x4 +1 x5 = 0 eq1: +1 x3 -1 x6 = 0 eq2: -1 x2 +1 x7 = 0 eq3: +1 x1 -1 x8 = 0 eq4: +1 x1 +1 x2 +1 x3 +1 x4 +1 x5 +1 x6 +1 x7 +1 x8 = 2 BOUNDS x1 free x2 free x3 free x4 free x5 free x6 free x7 free x8 free GENERAL x1 x2 x3 x4 x5 x6 x7 x8 END There are eight variables, one for each possible black or white maximal interior simplex. The optimal value of this LP is zero, because any triangulation has exactly one black and one white simplex of odd normalized volume. Notice that the objective function becomes empty for cube(2), because in the +1/-1 cube, each simplex has even volume.
• For the 0/1 3-cube, we use a GROUP property:> write_foldable_max_signature_ilp(cube(3,0,group=>1)); MINIMIZE obj: +1 x1 -1 x2 +1 x3 -1 x4 +1 x5 -1 x6 Subject To ie0: +1 x1 >= 0 ie1: +1 x2 >= 0 ie2: +1 x3 >= 0 ie3: +1 x4 >= 0 ie4: +1 x5 >= 0 ie5: +1 x6 >= 0 ie6: +1 x7 >= 0 ie7: +1 x8 >= 0 ie8: -1 x1 -1 x2 >= -8 ie9: -1 x3 -1 x4 >= -24 ie10: -1 x5 -1 x6 >= -24 ie11: -1 x7 -1 x8 >= -2 eq0: +2 x3 -2 x4 +2 x5 -2 x6 = 0 eq1: -2 x3 +2 x4 -2 x5 +2 x6 = 0 eq2: -6 x2 +6 x5 +24 x7 = 0 eq3: -6 x1 +6 x6 +24 x8 = 0 eq4: +1 x1 +1 x2 +1 x3 +1 x4 +1 x5 +1 x6 +2 x7 +2 x8 = 6 BOUNDS x1 free x2 free x3 free x4 free x5 free x6 free x7 free x8 free GENERAL x1 x2 x3 x4 x5 x6 x7 x8 END There are again 8 variables, but now they correspond to the black and white representatives of the four symmetry classes of maximal interior simplices. The optimal value of this linear program is 4, because the most imbalanced triangulation is the one with 5 simplices, in which the volume of the big interior simplex is even and doesn't get counted in the objective function.
•
write_simplexity_ilp (P)

construct a linear program whose optimal value is a lower bound for the minimal number of simplices in a triangulation of P.

##### Parameters
 Polytope P
##### Options
 String outfile_name . If the string is '-' (as is the default), the linear program is printed to STDOUT.

Example:
• To print the linear program for the 2-dimensional cube, write> write_simplexity_ilp(cube(2)); MINIMIZE obj: +1 x1 +1 x2 +1 x3 +1 x4 Subject To ie0: +1 x1 >= 0 ie1: +1 x2 >= 0 ie2: +1 x3 >= 0 ie3: +1 x4 >= 0 eq0: +4 x1 +4 x2 +4 x3 +4 x4 = 8 eq1: -1 x2 +1 x3 = 0 eq2: -1 x1 +1 x4 = 0 BOUNDS x1 free x2 free x3 free x4 free GENERAL x1 x2 x3 x4 END
•
write_simplexity_ilp_with_angles (P, outfile_name)

construct a linear program whose optimal value is a lower bound for the minimal number of simplices in a triangulation of P, and that takes into account the angle constraint around codimension 2 faces. The first set of variables correspond to possible maximal internal simplices, the second set to the simplices of codimension 2. See the source file polytope/src/symmetrized_codim_2_angle_sums.cc for details.

##### Parameters
 Polytope P String outfile_name

Example:
• To print the linear program for the 2-dimensional cube, write> write_simplexity_ilp_with_angles(cube(2)); MINIMIZE obj: +1 x1 +1 x2 +1 x3 +1 x4 Subject To ie0: +1 x1 >= 0 ie1: +1 x2 >= 0 ie2: +1 x3 >= 0 ie3: +1 x4 >= 0 ie4: +1 x5 >= 0 ie5: +1 x6 >= 0 ie6: +1 x7 >= 0 ie7: +1 x8 >= 0 eq0: -1 x2 +1 x3 = 0 eq1: -1 x1 +1 x4 = 0 eq2: +0.5 x1 +0.25 x2 +0.2500000000000001 x3 -0.5 x5 = 0 eq3: +0.25 x1 +0.5 x3 +0.2500000000000001 x4 -0.5 x6 = 0 eq4: +0.25 x1 +0.5 x2 +0.2500000000000001 x4 -0.5 x7 = 0 eq5: +0.25 x2 +0.2500000000000001 x3 +0.5 x4 -0.5 x8 = 0 eq6: +1 x5 = 1 eq7: +1 x6 = 1 eq8: +1 x7 = 1 eq9: +1 x8 = 1 eq10: +4 x1 +4 x2 +4 x3 +4 x4 = 8 BOUNDS x1 free x2 free x3 free x4 free x5 free x6 free x7 free x8 free GENERAL x1 x2 x3 x4 x5 x6 x7 x8 END
•
write_symmetrized_simplexity_ilp (P, isotypic_components, outfile_name)

construct a linear program whose optimal value is a lower bound for the minimal number of simplices in a triangulation of P. The symmetry group of P is taken into account, in that the variables in the linear program are projections of the indicator variables of the maximal interior simplices to a given direct sum of isotypic components of the symmetry group of P acting on these simplices.

##### Parameters
 Polytope P Set isotypic_components the set of indices of isotypic components to project to; default [0] String outfile_name . Setting this to '-' (as is the default) prints the LP to stdout.

Example:
• For the 3-cube, the symmetrized LP for isotypic component 0 reads as follows:> write_symmetrized_simplexity_ilp(cube(3,group=>1)); MINIMIZE obj: +1 x1 +1 x2 +1 x3 +1 x4 Subject To ie0: +1 x1 >= 0 ie1: +1 x2 >= 0 ie2: +1 x3 >= 0 ie3: +1 x4 >= 0 eq0: +8 x1 +8 x2 +8 x3 +16 x4 = 48 eq1: -6 x1 +6 x3 +24 x4 = 0 BOUNDS x1 free x2 free x3 free x4 free GENERAL x1 x2 x3 x4 END The interpretation is as follows: The variables x1,...,x4 correspond to the representatives of interior simplices:> print cube(3,group=>1)->GROUP->REPRESENTATIVE_MAX_INTERIOR_SIMPLICES; {0 1 2 4} {0 1 2 5} {0 1 2 7} {0 3 5 6} The solution (x1,x2,x3,x4) = (4,0,0,1) of the LP says that in a minimal triangulation of the 3-cube, there are 4 simplices in the same symmetry class as {0,1,2,4}, and one in the class of {0,3,5,6}.
•

### Other

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
•
face_pair (P, S) → Pair<Set,Set>

For a given set S of rays compute the smallest face F of a cone P containing them all; see also face.

##### Parameters
 Cone P Set S
##### Returns
 Pair where the first is the set of vertices of F, while the second is the set of facets containing F.

Example:
• computes the dimension of the face of the 3-cube which is spanned by the vertices 0 and 1> $c=cube(3);> print rank($c->VERTICES->minor(face_pair($c,[0,1])->first(),All))-1; 1 • lawrence_matrix (M) → Matrix UNDOCUMENTED ##### Parameters  Matrix M Create the Lawrence matrix Lambda(M) corresponding to M. If M has n rows and d columns, then Lambda(M) equals ( M I_n ) ( 0_{n,d} I_n ). ##### Returns  Matrix • matroid_indices_of_hypersimplex_vertices () → Set<Int> For a given matroid return the bases as a subset of the vertices of the hypersimplex ##### Options  matroid::Matroid m the matroid ##### Returns  Set • m_sequence (h) → Bool Test if the given vector is an M-sequence. ##### Parameters  Vector h ##### Returns  Bool Example: • The h-vector of a simplicial or simple polytope is an M-sequence.> print m_sequence(cyclic(7,23)->H_VECTOR); true • 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 L lattice points Vector 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 M points (in homogeneous coordinates); affinely span the space Vector lambda height function on lattice points Array coeff coefficients Rational s additional Parameter in the polynomial ##### Options  topaz::SimplicialComplex triangulation The triangulation of the pointset corresponding to the lifting function • wronski_system (M, lambda, coeff_array, s) Returns a Wronski system of a topaz::FOLDABLE triangulation of a lattice polytope ##### Parameters  Matrix M points (in homogeneous coordinates); affinely span the space Vector lambda height function on lattice points Array> coeff_array coefficients Rational s additional Parameter in the polynomial ##### Options  topaz::SimplicialComplex triangulation The triangulation of the pointset corresponding to the lifting function • ### Producing a cone Various constructions of cones. • inner_cone (p, F) → Cone Computes the inner cone of p at a face F (or a vertex v). ##### Parameters  Cone p Set F (or Int v) vertex indices which are not contained in the far face ##### Options  Bool outer Make it point outside the polytope? Default value is 0 (= point inside) Bool attach Attach the cone to F? Default 0 (ie, return the cone inside the hyperplane at infinity) ##### Returns  Cone Examples: • To compute the inner cone at a vertex of the 3-cube, do this:>$c = inner_cone(cube(3), 1);> print $c->RAYS; -1 0 0 0 1 0 0 0 1 • To compute the inner cone along an edge of the 3-cube, and make it point outside the polytope, do this:> print inner_cone(cube(3), [0,1], outer=>1)->RAYS; 0 0 -1 0 -1 0 • If you want to attach the cone to the polytope, specify the corresponding option:> print normal_cone(cube(3), [0,1], attach=>1)->RAYS; 1 -1 -1 -1 1 1 -1 -1 0 0 1 0 0 0 0 1 • normal_cone (p, F) → Cone Computes the normal cone of p at a face F (or vertex v). By default this is the inner normal cone. ##### Parameters  PointConfiguration p Set F (or Int v) point indices which are not contained in the far face ##### Options  Bool outer Calculate outer normal cone? Default value is 0 (= inner) Bool attach Attach the cone to F? Default 0 (ie, return the cone inside the hyperplane at infinity) ##### Returns  Cone Example: • To compute the outer normal cone of a doubled 2-cube, do this:>$v = cube(2)->VERTICES;> $p = new PointConfiguration(POINTS=>($v/$v));> print normal_cone($p, 4, outer=>1)->RAYS; 0 -1 -1 0
•
normal_cone (p, F) → Cone

Computes the normal cone of p at a face F (or a vertex v). By default this is the inner normal cone.

##### Parameters
 Cone p Set F (or Int v) vertex indices which are not contained in the far face
##### Options
 Bool outer Calculate outer normal cone? Default value is 0 (= inner) Bool attach Attach the cone to F? Default 0 (ie, return the cone inside the hyperplane at infinity)
##### Returns
 Cone

Examples:
• To compute the outer normal cone at a vertex of the 3-cube, do this:> $c = normal_cone(cube(3), 0, outer=>1);> print$c->RAYS; -1 0 0 0 -1 0 0 0 -1
• To compute the outer normal cone along an edge of the 3-cube, do this:> print normal_cone(cube(3), [0,1], outer=>1)->RAYS; 0 -1 0 0 0 -1
• If you want to attach the cone to the polytope, specify the corresponding option:> print normal_cone(cube(3), [0,1], outer=>1, attach=>1)->RAYS; 1 -1 -1 -1 1 1 -1 -1 0 0 -1 0 0 0 0 -1
•
recession_cone (P) → Cone<Scalar>

retuns the recession cone (tail cone, characteristic cone) of a polytope

##### Parameters
 Polytope P polytope
##### Returns
 Cone
•
subcone (C) → Cone

Make a subcone from a cone.

##### Parameters
 Cone C the input cone
##### Options
 Bool no_labels Do not create RAY_LABELS. default: 0
##### Returns
 Cone
•

### Producing a point configuration

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

•
minkowski_sum (P1, P2) → PointConfiguration

Produces the Minkowski sum of P1 and P2.

##### Parameters
 PointConfiguration P1 PointConfiguration P2
##### Returns
 PointConfiguration

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) → PointConfiguration

Produces the polytope lambda*P1+mu*P2, where * and + are scalar multiplication and Minkowski addition, respectively.

##### Parameters
 Scalar lambda PointConfiguration P1 Scalar mu PointConfiguration P2
##### Returns
 PointConfiguration

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]]);> $m = minkowski_sum(1,$P1,3,$P2);> print$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
•

### Producing a polytope from graphs

Polytope constructions which take graphs as input.

•
flow_polytope <Scalar> (G, Arc_Bounds, source, sink) → Polytope

Produces 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 = 0

sum_{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 G EdgeMap Arc_Bounds Int source Int sink
##### Returns
 Polytope
•
flow_polytope <Scalar> (G, Arc_Bounds, source, sink) → Polytope

Produces 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 = 0

sum_{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 G Array Arc_Bounds Int source Int sink
##### Returns
 Polytope
•
fractional_cut_polytope (G) → Polytope

Cut polytope of an undirected graph.

##### Parameters
 Graph G
##### Returns
 Polytope
•
fractional_matching_polytope (G) → Polytope

Matching polytope of an undirected graph.

##### Parameters
 Graph G
##### Returns
 Polytope
•
tutte_lifting (G) → Polytope

Let G be a 3-connected planar graph. If the corresponding polytope contains a triangular facet (ie. the graph contains a non- separating cycle of length 3), the client produces a realization in R3.

##### Parameters
 Graph G
##### Returns
 Polytope
•
weighted_digraph_polyhedron (encoding) → polytope::Polytope

Weighted digraph polyhedron of a directed graph with a weight function. The graph and the weight function are combined into a matrix.

##### Parameters
 Matrix encoding weighted digraph
##### Returns
 polytope::Polytope
•

### Producing a polytope from other objects

Polytope constructions which take other big objects as input.

•
billera_lee (H) → Polytope

Produces a simplicial polytope whose H-vector is the given input vector. The G-vector coming from the given vector must be an M-sequence. This is an implementation of the algorithm described in the paper "A Proof of the Sufficiency of McMullen’s Conditions of Simplicial Convex Polytopes" by Louis Billera and Carl Lee, DOI: 10.1016/0097-3165(81)90058-3

##### Parameters
 Vector H
##### Returns
 Polytope

Example:
• > $p = billera_lee([1,5,15,15,5,1]);> print$p->H_VECTOR; 1 5 15 15 5 1
•

### Producing a polytope from polytopes

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) → Polytope

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 no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0 label the new vertices with "Apex" and "Apex'".
##### Returns
 Polytope

Example:
• Here's a way to construct the 3-dimensional cross polytope:> $p = bipyramid(bipyramid(cube(1)));> print equal_polyhedra($p,cross(3)); true
•
blending (P1, v1, P2, v2) → Polytope

Compute 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.

##### Parameters
 Polytope P1 Int v1 the index of the first vertex Polytope P2 Int v2 the index of the second vertex
##### Options
 Array permutation Bool no_labels Do not copy VERTEX_LABELS from the original polytopes. default: 0
##### Returns
 Polytope

Example:
• The following gives the smallest EVEN 3-polytope which is not a zonotope.> $c = cube(3);$bc = blending($c,0,$c,0);> print $bc->EVEN true> print$bc->F_VECTOR 14 21 9
•
cayley_embedding (P_0, P_1, t_0, t_1) → Polytope

Create 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.

##### Parameters
 Polytope P_0 the first polytope Polytope P_1 the second polytope Scalar t_0 the extra coordinate for the vertices of P_0 Scalar t_1 the extra coordinate for the vertices of P_1
##### Options
 Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0
##### Returns
 Polytope
•
cayley_embedding (A) → Polytope

Create 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.

##### Parameters
 Array A the input polytopes
##### Options
 Array factors array of scaling factors for the Cayley embedding; defaults to the all-1 vector Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0
##### Returns
 Polytope
•
cayley_polytope (P_Array) → Polytope

Construct the cayley polytope of a set of pointed lattice polytopes contained in P_Array which is the convex hull of P1×e1, ..., Pk×ek where e1, ...,ek are the standard unit vectors in Rk. 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 P_Array an array containing the lattice polytopes P1,...,Pk
##### 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 P Set cells
##### Options
 Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0
##### Returns
 Polytope

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]);> 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) → Polytope

Extract the given cell of the subdivision of a polyhedron and write it as a new polyhedron.

##### Parameters
 Polytope P Int cell
##### Options
 Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0
##### 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 [1,2,3]-cell, copying the vertex labels.> $c = cell_from_subdivision($p,1);> print $c->VERTICES; 1 1 -1 1 -1 1 1 1 1> print$c->VERTEX_LABELS; 1 2 3
•
conv (P_Array) → PropagatedPolytope

Construct a new polyhedron as the convex hull of the polyhedra given in P_Array.

##### Parameters
 Array P_Array
##### Returns
 PropagatedPolytope

Example:
• > $p = conv([cube(2,1,0),cube(2,6,5)]);> print$p->VERTICES; 1 0 0 1 1 0 1 0 1 1 6 5 1 5 6 1 6 6
•
dual_linear_program (P, maximize) → Polytope

Produces 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.

##### Parameters
 Polytope P = {x | Ax >= b, Bx = d} Bool maximize tells if the primal lp is a maximization problem. Default value is 0 (= minimize)
##### Returns
 Polytope
•
edge_middle (P) → Polytope

Produce the convex hull of all edge middle points of some polytope P. The polytope must be BOUNDED.

##### Parameters
 Polytope P
##### Returns
 Polytope
•
face (P, S) → Cone

For a given set S of rays compute the smallest face F of a cone P containing them all; see also face_pair.

##### Parameters
 Cone P Set S
##### Options
 Bool no_coordinates don't copy the coordinates, produce purely combinatorial description. Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0
##### Returns
 Cone

Example:
• To create a cone from the vertices of the zeroth facet of the 3-cube, type this:> $p = face(cube(3),[0,1]); • facet (P, facet) → Cone Extract the given facet of a polyhedron and write it as a new polyhedron. ##### Parameters  Cone P Int facet ##### Options  Bool no_coordinates don't copy the coordinates, produce purely combinatorial description. Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0 ##### Returns  Cone Example: • To create a cone from the vertices of the zeroth facet of the 3-cube, type this:>$p = facet(cube(3),0);
•
facet_to_infinity (P, i) → Polytope

Make an affine transformation such that the i-th facet is transformed to infinity

##### Parameters
 Polytope P Int i the facet index
##### Returns
 Polytope

Example:
• This generates the polytope that is the positive quadrant in 2-space:> $q = new Polytope(VERTICES=>[[1,-1,-1],[1,0,1],[1,1,0]]);>$pf = facet_to_infinity($q,2);> print$pf->VERTICES; 0 -1 -1 0 0 1 1 0 1
•
free_sum (P1, P2) → Polytope

Construct 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. Defaults 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

##### Parameters
 Polytope P
##### Returns
 Array
•
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

##### Parameters
 Polytope P
##### Returns
 Array

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($p); {0 1} {2 3} • gc_closure (P) → Polytope Produces the gomory-chvatal closure of a full dimensional polyhedron ##### Parameters  Polytope P ##### Returns  Polytope • integer_hull (P) → Polytope Produces the integer hull of a polyhedron ##### Parameters  Polytope P ##### Returns  Polytope Example: • >$p = new Polytope(VERTICES=>[[1,13/10,1/2],[1,1/5,6/5],[1,1/10,-3/2],[1,-7/5,1/5]]);> $ih = integer_hull($p);> print $ih->VERTICES; 1 -1 0 1 0 -1 1 0 1 1 1 0 • intersection (C ...) → Cone Construct a new polyhedron or cone as the intersection of given polyhedra and/or cones. Works only if all CONE_AMBIENT_DIM values are equal. If the input contains both cones and polytopes, the output will be a polytope. ##### Parameters  Cone C ... polyhedra and cones to be intersected ##### Returns  Cone Example: • >$p = intersection(cube(2), cross(2,3/2));> print $p->VERTICES; 1 -1/2 1 1 -1 1/2 1 1/2 1 1 1 1/2 1 1/2 -1 1 1 -1/2 1 -1/2 -1 1 -1 -1/2 • join_polytopes (P1, P2) → Polytope Construct a new polyhedron as the join of two given bounded ones. ##### Parameters  Polytope P1 Polytope P2 ##### Options  Bool no_coordinates produces a pure combinatorial description. Bool group Compute the canonical group induced by the groups on P1 and P2 Throws an exception if the GROUPs of the input polytopes are not provided. default 0 ##### Returns  Polytope Example: • To join two squares, use this:>$p = join_polytopes(cube(2),cube(2));> print $p->VERTICES; 1 -1 -1 -1 0 0 1 1 -1 -1 0 0 1 -1 1 -1 0 0 1 1 1 -1 0 0 1 0 0 1 -1 -1 1 0 0 1 1 -1 1 0 0 1 -1 1 1 0 0 1 1 1 • lattice_bipyramid (P, v, v_prime, z, z_prime) → Polytope Make a lattice bipyramid over a polyhedron. The bipyramid is the convex hull of the input polyhedron P and two points (v, z), (v_prime, z_prime) on both sides of the affine span of P. ##### Parameters  Polytope P Vector v basis point for the first apex Vector v_prime basis for the second apex If v_prime is omitted, v will be used for both apices. If both v and v_prime are omitted, it tries to find two vertices which don't lie in a common facet. If no such vertices can be found or P is a simplex, it uses an interior lattice point as both v and v_prime. Rational z height for the first apex, default value is 1 Rational z_prime height for the second apex, default value is -z ##### Options  Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0 label the new vertices with "Apex" and "Apex'". ##### Returns  Polytope Example: • To create the bipyramid over a square and keep the vertex labels, do this:>$p = lattice_bipyramid(cube(2),new Vector(1,0,0));> 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) → Polytope

Make 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 no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0 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:> $p = lattice_pyramid(cube(2),5,new Vector(1,0,0));> print$p->VERTICES; 1 -1 -1 0 1 1 -1 0 1 -1 1 0 1 1 1 0 1 0 0 5> print $p->VERTEX_LABELS; 0 1 2 3 Apex • lawrence (P) → Cone Create the Lawrence polytope Lambda(P) corresponding to P. Lambda(P) has the property that Gale(Lambda(P)) = Gale(P) union -Gale(P). ##### Parameters  Cone P an input cone or polytope ##### Returns  Cone the Lawrence cone or polytope to P • make_totally_dual_integral (P) → Polytope Produces a polyhedron with an totally dual integral inequality formulation of a full dimensional polyhedron ##### Parameters  Polytope P ##### Returns  Polytope • mapping_polytope (P1, P2) → Polytope Construct a new polytope as the mapping polytope of two polytopes P1 and P2. The mapping polytope is the set of all affine maps from Rp to Rq, that map P1 into P2. The label of a new facet corresponding to v1 and h1 will have the form "v1*h1". ##### Parameters  Polytope P1 Polytope P2 ##### Options  Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0 ##### Returns  Polytope • minkowski_sum (P1, P2) → Polytope Produces the Minkowski sum of P1 and P2. ##### Parameters  Polytope P1 Polytope P2 ##### Returns  Polytope Example: • The following stores the minkowski sum of a square and a triangle in the variable$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) → Polytope

Produces the polytope lambda*P1+mu*P2, where * and + are scalar multiplication and Minkowski addition, respectively.

##### Parameters
 Scalar lambda Polytope P1 Scalar mu Polytope P2
##### Returns
 Polytope

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 $p->VERTICES; 1 -2 -2 1 3 -2 1 -2 3 1 3 2 1 2 3 • minkowski_sum_fukuda (summands) → Polytope<Scalar> Computes the (VERTICES of the) Minkowski sum of a list of polytopes using the algorithm by Fukuda described in Komei Fukuda, From the zonotope construction to the Minkowski addition of convex polytopes, J. Symbolic Comput., 38(4):1261-1272, 2004. ##### Parameters  Array> summands ##### Returns  Polytope Example: • >$p = minkowski_sum_fukuda([cube(2),simplex(2),cross(2)]);> print $p->VERTICES; 1 3 -1 1 3 1 1 -1 -2 1 1 3 1 -1 3 1 2 -2 1 -2 2 1 -2 -1 • mixed_integer_hull (P, int_coords) → Polytope Produces the mixed integer hull of a polyhedron ##### Parameters  Polytope P Array int_coords the coordinates to be integral; ##### Returns  Polytope • pointed_part (P) → Polytope Produces the pointed part of a polyhedron ##### Parameters  Polytope P ##### Returns  Polytope Example: • >$p = new Polytope(POINTS=>[[1,0,0],[1,0,1],[1,1,0],[1,1,1],[0,1,0],[0,0,1]]);> $pp = pointed_part($p);> print $pp->VERTICES; 1 0 0 0 1 0 0 0 1 • prism (P, z1, z2) → Polytope Make a prism over a pointed polyhedron. The prism is the product of the input polytope P and the interval [z1, z2]. ##### Parameters  Polytope P the input polytope Scalar z1 the left endpoint of the interval; default value: -1 Scalar z2 the right endpoint of the interval; default value: -z1 ##### Options  Bool group Compute the canonical group induced by the group on P with an extra generator swapping the upper and lower copy. throws an exception if GROUP of P is not provided. default 0 Bool no_coordinates only combinatorial information is handled Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0 the bottom facet vertices get the labels from the original polytope; the labels of their clones in the top facet get a tick (') appended. ##### Returns  Polytope Example: • The following saves the prism over the square and the interval [-2,2] to the variable$p, and then prints a nice representation of its vertices.> $p = prism(cube(2),-2);> print rows_labeled($p->VERTICES,$p->VERTEX_LABELS); 0:1 -1 -1 -2 1:1 1 -1 -2 2:1 -1 1 -2 3:1 1 1 -2 0':1 -1 -1 2 1':1 1 -1 2 2':1 -1 1 2 3':1 1 1 2 • product (P1, P2) → Polytope Construct a new polytope as the product of two given polytopes P1 and P2. As little additional properties of the input polytopes as possible are computed. You can control this behaviour using the option flags. ##### Parameters  Polytope P1 Polytope P2 ##### Options  Bool no_coordinates only combinatorial information is handled Bool no_vertices do not compute vertices Bool no_facets do not compute facets Bool no_labels Do not copy VERTEX_LABELS from the original polytopes, if present. the label of a new vertex corresponding to v1 ⊕ v2 will have the form LABEL_1*LABEL_2. default: 0 Bool group Compute the canonical group of the product, as induced by the groups on FACETS of VERTICES of P1 and P2. If neither FACETS_ACTION nor VERTICES_ACTION of the GROUPs of the input polytopes are provided, an exception is thrown. default 0 ##### Returns  Polytope Example: • The following builds the product of a square and an interval, without computing vertices of neither the input nor the output polytopes.>$p = product(cube(2),cube(1), no_vertices=>1);
•
projection (P, indices) → Cone

Orthogonally 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 indices
##### Options
 Bool revert inverts the coordinate list Bool 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_preimage (P_Array) → Cone

Construct a new polyhedron that projects to a given array of polyhedra. If the n polyhedra are d_1, d_2, ..., d_n-dimensional and all have m vertices, the resulting polyhedron is (d_1+...+d_n)-dimensional, has m vertices, and the projection to the i-th d_i coordinates gives the i-th input polyhedron.

##### Parameters
 Array P_Array
##### Returns
 Cone

Example:
• > $p = projection_preimage(cube(2),cube(2));> print$p->VERTICES; 1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 1 1 1 1 1
•
project_full (P) → Cone

Orthogonally 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.

##### Parameters
 Cone P
##### Options
 Bool nofm suppresses Fourier-Motzkin elimination Bool no_labels Do not copy VERTEX_LABELS to the projection. default: 0
##### Returns
 Cone
•
pyramid (P, z) → Polytope

Make 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 group compute the group induced by the GROUP of P and leaving the apex fixed. throws an exception if GROUP of P is not provided. default 0 Bool no_coordinates don't compute new coordinates, produce purely combinatorial description. Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0 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); 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) → Polytope

Produce 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.

##### Parameters
 Polytope P the input polytope Int n the number of random points
##### Options
 Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome.
##### Returns
 Polytope
•
rand_vert (V, n) → Matrix

Selects 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.

##### Parameters
 Matrix V the vertices of a polytope Int n the number of random points
##### Options
 Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome.
##### Returns
 Matrix
•
spherize (P) → Polytope

Project all vertices of a polyhedron P on the unit sphere. P must be CENTERED and BOUNDED.

##### Parameters
 Polytope P
##### Returns
 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) → Polytope

Stack 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 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: 1/2 Bool no_coordinates produces a pure combinatorial description (in contrast to lift) Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0 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:
• 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:> $p = unirand(cube(2),5,boundary=>1); • vertex_figure (p, n) → Polytope Construct the vertex figure of the vertex n of a polyhedron. The vertex figure is dual to a facet of the dual polytope. ##### Parameters  Polytope p Int n number of the chosen vertex ##### Options  Scalar cutoff controls the exact location of the cutting hyperplane. It should lie between 0 and 1. Value 0 would let the hyperplane go through the chosen vertex, thus degenerating the vertex figure to a single point. Value 1 would let the hyperplane touch the nearest neighbor vertex of a polyhedron. Default value is 1/2. Bool no_coordinates skip the coordinates computation, producing a pure combinatorial description. Bool no_labels Do not copy VERTEX_LABELS from the original polytope. default: 0 by default, the labels are produced from the corresponding neighbor vertices of the original polytope. ##### Returns  Polytope Example: • This produces a vertex figure of one vertex of a 3-dimensional cube with the origin as its center and side length 2. The result is a 2-simplex.>$p = vertex_figure(cube(3),5);> print $p->VERTICES; 1 1 -1 0 1 0 -1 1 1 1 0 1 • wedge (P, facet, z, z_prime) → Polytope Make a wedge from a polytope over the given facet. The polytope must be bounded. The inclination of the bottom and top side facet is controlled by z and z_prime, which are heights of the projection of the old vertex barycenter on the bottom and top side facet respectively. ##### Parameters  Polytope P , must be bounded Int facet the cutting edge'. Rational z default value is 0. Rational z_prime default value is -z, or 1 if z==0. ##### Options  Bool no_coordinates don't compute coordinates, pure combinatorial description is produced. Bool no_labels Do not copy VERTEX_LABELS from the original polytopes. default: 0 By default, the vertices get labelled as follows: The bottom facet vertices obtain the labels from the original polytope; the labels of their clones in the top facet get a tick (') appended. ##### Returns  Polytope Example: • This produces the wedge from a square (over the facet 0), which yields a prism over a triangle:>$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 • wreath (P1, P2) → Polytope Construct a new polytope as the wreath product of two input polytopes P1, P2. P1 and P2 have to be BOUNDED. ##### Parameters  Polytope P1 Polytope P2 ##### Options  Bool dual invokes the computation of the dual wreath product Bool no_labels Do not copy VERTEX_LABELS from the original polytopes. default: 0 the label of a new vertex corresponding to v1 ⊕ v2 will have the form LABEL_1*LABEL_2. ##### Returns  Polytope • ### Producing a polytope from scratch 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 Produce a d-dimensional associahedron (or Stasheff polytope). We use the facet description given in Ziegler's book on polytopes, section 9.2. ##### Parameters  Int d the dimension ##### Options  Bool group Compute the combinatorial symmetry group of the polytope. It has two generators, as it is induced by the symmetry group of an d+3-gon, the dihedral group of degree d+3. See arXiv 1109.5544v2 for details. ##### Returns  Polytope • binary_markov_graph (observation) → PropagatedPolytope Defines 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 biology by L. Pachter and B. Sturmfels (eds.), Cambridge, 2005. ##### Parameters  Array observation ##### Returns  PropagatedPolytope • binary_markov_graph (observation) ##### Parameters  String observation encoded as a string of "0" and "1". • birkhoff (n, even) → Polytope Constructs the Birkhoff polytope of dimension n2. It is the polytope of nxn stochastic matrices (encoded as n2 row vectors), thus matrices with non-negative entries whose row and column entries sum up to one. Its vertices are the permutation matrices. ##### Parameters  Int n Bool even Defaults to '0'. Set this to '1' to get vertices only for even permutation matrices. ##### Options  Bool group add the symmetry group induced by the symmetric group to the resulting polytope ##### Returns  Polytope • cyclic (d, n) → Polytope<Rational> Produce 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 dimension Int 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 $p->VERTICES; 1 1/10 3/10 1 1/17 4/17 1 1/26 5/26 1 1/37 6/37 1 1/50 7/50 1 1/65 8/65 • cyclic_caratheodory (d, n) → Polytope Produce a d-dimensional cyclic polytope with n points. Prototypical example of a neighborly polytope. Combinatorics completely known due to Gale's evenness criterion. Coordinates are chosen on the trigonometric moment curve. For cyclic polytopes from other curves, see polytope::cyclic. ##### Parameters  Int d the dimension. Required to be even. Int n the number of points ##### Options  Bool group add a symmetry group description. If 0 (default), the return type is Polytope, else Polytope so that the matrices corresponding to the symmetry action may be approximated ##### Returns  Polytope • delpezzo (d, scale) → Polytope<Scalar> Produce a d-dimensional del-Pezzo polytope, which is the convex hull of the cross polytope together with the all-ones and minus all-ones vector. All coordinates are +/- scale or 0. ##### Parameters  Int d the dimension Scalar scale the absolute value of each non-zero vertex coordinate. Needs to be positive. The default value is 1. ##### Returns  Polytope • dwarfed_cube (d) → Polytope Produce a d-dimensional dwarfed cube. ##### Parameters  Int d the dimension ##### Returns  Polytope • dwarfed_product_polygons (d, s) → Polytope Produce a d-dimensional dwarfed product of polygons of size s. ##### Parameters  Int d the dimension Int s the size ##### Returns  Polytope • explicit_zonotope (zones) → Polytope Produce the POINTS of a zonotope as the iterated Minkowski sum of all intervals [-x,x], where x ranges over the rows of the input matrix zones. ##### Parameters  Matrix zones the input vectors ##### Options  Bool rows_are_points the rows of the input matrix represent affine points(true, default) or linear vectors(false) ##### Returns  Polytope Example: • >$M = new Matrix([1,1],[1,-1]);> $p = explicit_zonotope($M,rows_are_points=>0);> print $p->VERTICES; 1 2 0 1 0 -2 1 0 2 1 -2 0 • fano_simplex (d) → Polytope Produce a Fano d-simplex. ##### Parameters  Int d the dimension ##### Options  Bool group ##### Returns  Polytope Example: • To create the 2-dimensional fano simplex and compute its symmetry group, type this: and print ints generators, do this:>$p = fano_simplex(2,group=>1);> print $p->GROUP->RAYS_ACTION->GENERATORS; 1 0 2 2 0 1 • fractional_knapsack (b) → Polytope Produce a knapsack polytope defined by one linear inequality (and non-negativity constraints). ##### Parameters  Vector b linear inequality ##### Returns  Polytope • goldfarb (d, e, g) → Polytope Produces a d-dimensional Goldfarb cube if e<1/2 and g<=e/4. The Goldfarb cube is a combinatorial cube and yields a bad example for the Simplex Algorithm using the Shadow Vertex Pivoting Strategy. Here we use the description as a deformed product due to Amenta and Ziegler. For e<1/2 and g=0 we obtain the Klee-Minty cubes. ##### Parameters  Int d the dimension Scalar e Scalar g ##### Returns  Polytope • goldfarb_sit (d, eps, delta) → Polytope Produces a d-dimensional variation of the Klee-Minty cube if eps<1/2 and delta<=1/2. This Klee-Minty cube is scaled in direction x_(d-i) by (eps*delta)^i. This cube is a combinatorial cube and yields a bad example for the Simplex Algorithm using the 'steepest edge' Pivoting Strategy. Here we use a scaled description of the construction of Goldfarb and Sit. ##### Parameters  Int d the dimension Scalar eps Scalar delta ##### Returns  Polytope • hypersimplex (k, d) → Polytope Produce the hypersimplex Δ(k,d), that is the the convex hull of all 0/1-vector in Rd with exactly k 1s. Note that the output is never full-dimensional. ##### Parameters  Int k number of 1s Int d ambient dimension ##### Options  Bool group Bool no_vertices do not compute vertices Bool no_facets do not compute facets Bool no_vif do not compute vertices in facets ##### Returns  Polytope Example: • This creates the hypersimplex in dimension 4 with vertices with exactly two 1-entries and computes its symmetry group:>$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 dimension Scalar k cutoff parameter Scalar lambda scaling of extra vertex
##### Returns
 Polytope
•
klee_minty_cube (d, e) → Polytope

Produces a d-dimensional Klee-Minty-cube if e<1/2. Uses the goldfarb client with g=0.

##### Parameters
 Int d the dimension Scalar e
##### Returns
 Polytope
•
k_cyclic (n, s) → Polytope

Produce 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
##### Parameters
 Int n the number of points Vector s s=(s_1,...,s_k)
##### Returns
 Polytope

Example:
• This yields a rational 4-polytope with the same combinatorics.> $p = long_and_winding(2,eval_ratio=>2); • max_GC_rank (d) → Polytope Produce a d-dimensional polytope of maximal Gomory-Chvatal rank Omega(d/log(d)), integrally infeasible. With symmetric linear objective function (0,1,1..,1). Construction due to Pokutta and Schulz. ##### Parameters  Int d the dimension ##### Returns  Polytope • multiplex (d, n) → Polytope Produce a combinatorial description of a multiplex with parameters d and n. This yields a self-dual d-dimensional polytope with n+1 vertices. They are introduced by T. Bisztriczky, On a class of generalized simplices, Mathematika 43:27-285, 1996, see also M.M. Bayer, A.M. Bruening, and J.D. Stewart, A combinatorial study of multiplexes and ordinary polytopes, Discrete Comput. Geom. 27(1):49--63, 2002. ##### Parameters  Int d the dimension Int n ##### Returns  Polytope • neighborly_cubical (d, n) → Polytope Produce the combinatorial description of a neighborly cubical polytope. The facets are labelled in oriented matroid notation as in the cubical Gale evenness criterion. See Joswig and Ziegler, Discr. Comput. Geom. 24:315-344 (2000). ##### Parameters  Int d dimension of the polytope Int n dimension of the equivalent cube ##### Returns  Polytope • newton (p) → Polytope<Rational> Produce the Newton polytope of a polynomial p. ##### Parameters  Polynomial p ##### Returns  Polytope Example: • Create the newton polytope of 1+x^2+y like so:> local_var_names<Polynomial>(qw(x y));$p=new Polynomial('1+x^2+y');> $n = newton($p);> print $n->VERTICES; 1 0 0 1 0 1 1 2 0 • n_gon (n, r, alpha_0) → Polytope Produce a regular n-gon. All vertices lie on a circle of radius r. The radius defaults to 1. ##### Parameters  Int n the number of vertices Rational r the radius (defaults to 1) Rational alpha_0 the initial angle divided by pi (defaults to 0) ##### Options  Bool group ##### Returns  Polytope Example: • To store the regular pentagon in the variable$p and calculate its symmetry group, do this:> $p = n_gon(5,group=>1);> print$p->GROUP->RAYS_ACTION->GENERATORS; 0 4 3 2 1 1 2 3 4 0
•
perles_irrational_8_polytope () → Polytope

Create an 8-dimensional polytope without rational realizations due to Perles

##### Returns
 Polytope
•
permutahedron (d) → Polytope

Produce a d-dimensional permutahedron. The vertices correspond to the elements of the symmetric group of degree d+1.

##### Parameters
 Int d the dimension
##### Options
 Bool group
##### Returns
 Polytope

Example:
• To create the 3-permutahedron and also compute its symmetry group, do this:> $p = permutahedron(3,group=>1);> print$p->GROUP->COORDINATE_ACTION->GENERATORS; 1 0 2 3 3 0 1 2
•
perturbed_long_and_winding (r) → Polytope<PuiseuxFraction<Max, Rational, Rational> >

Produce polytope in dimension 2r with 3r+2 facets such that the total curvature of the central path is at least Omega(2^r). This is a perturbed version of long_and_winding, which yields simple polytopes.

##### Parameters
 Int r defining parameter
##### Options
 Rational eval_ratio parameter for evaluating the puiseux rational functions Int eval_exp to evaluate at eval_ratio^eval_exp, default: 1 Float eval_float parameter for evaluating the puiseux rational functions
##### Returns
 Polytope >

Example:
• This yields a simple 4-polytope over the field of Puiseux fractions.> $p = perturbed_long_and_winding(2); • pile (sizes) → Polytope Produce 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 sizes a vector (s1,...,sd, where si 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 dimension Scalar scale the absolute value of each non-zero vertex coordinate. Needs to be positive. The default value is 1. ##### Returns  Polytope • rand01 (d, n) → Polytope Produce a d-dimensional 0/1-polytope with n random vertices. Uniform distribution. ##### Parameters  Int d the dimension Int n the number of random vertices ##### Options  Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome. ##### Returns  Polytope • rand_box (d, n, b) → Polytope Computes the convex hull of n points sampled uniformly at random from the integer points in the cube [0,b]d. ##### Parameters  Int d the dimension of the box Int n the number of random points Int b the size of the box ##### Options  Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome. ##### Returns  Polytope • rand_cyclic (d, n) → Polytope Computes a random instance of a cyclic polytope of dimension d on n vertices by randomly generating a Gale diagram whose cocircuits have alternating signs. ##### Parameters  Int d the dimension Int n the number of vertices ##### Options  Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome. ##### Returns  Polytope • rand_metric <Scalar> (n) → Matrix Produce an n-point metric with random distances. The values are uniformily distributed in [1,2]. ##### Type Parameters  Scalar element type of the result matrix ##### Parameters  Int n ##### Options  Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome. ##### Returns  Matrix • rand_metric_int <Scalar> (n) → Matrix Produce an n-point metric with random distances. The values are uniformily distributed in [1,2]. ##### Type Parameters  Scalar element type of the result matrix ##### Parameters  Int n ##### Options  Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome. ##### Returns  Matrix • rand_sphere (d, n) → Polytope Produce a d-dimensional polytope with n random vertices uniformly distributed on the unit sphere. ##### Parameters  Int d the dimension Int n the number of random vertices ##### Options  Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome. Int precision Number of bits for MPFR sphere approximation ##### Returns  Polytope • rss_associahedron (l) → Polytope Produce a polytope of constrained expansions in dimension l according to Rote, Santos, and Streinu: Expansive motions and the polytope of pointed pseudo-triangulations. Discrete and computational geometry, 699--736, Algorithms Combin., 25, Springer, Berlin, 2003. ##### Parameters  Int l ambient dimension ##### Returns  Polytope • signed_permutahedron (d) → Polytope Produce a d-dimensional signed permutahedron. ##### Parameters  Int d the dimension ##### Options  Bool group ##### Returns  Polytope • simplex (d, scale) → Polytope Produce the standard d-simplex. Combinatorially equivalent to a regular polytope corresponding to the Coxeter group of type Ad-1. Optionally, the simplex can be scaled by the parameter scale. ##### Parameters  Int d the dimension Scalar scale default value: 1 ##### Options  Bool group ##### Returns  Polytope 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 Produces the stable set polytope from an undirected graph G=(V,E). The stable set Polytope has the following inequalities: x_i + x_j <= 1 forall {i,j} in E x_i >= 0 forall i in V x_i <= 1 forall i in V with deg(i)=0 ##### Parameters  Graph G ##### Returns  Polytope • transportation (r, c) → Polytope Produce the transportation polytope from two vectors r of length m and c of length n, i.e. all positive m×n Matrizes with row sums equal to r and column sums equal to c. ##### Parameters  Vector r Vector c ##### Returns  Polytope • upper_bound_theorem (d, n) → Polytope Produce combinatorial data common to all simplicial d-polytopes with n vertices with the maximal number of facets as given by McMullen's Upper-Bound-Theorem. Essentially this lets you read off all possible entries of the H_VECTOR and the F_VECTOR. ##### Parameters  Int d the dimension Int n the number of points ##### Returns  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 $p->H_VECTOR; 1 2 2 1 • zonotope (M) → Polytope<Scalar> Create a zonotope from a matrix whose rows are input points or vectors. This method merely defines a Polytope object with the property ZONOTOPE_INPUT_POINTS. ##### Parameters  Matrix M input points or vectors ##### Options  Bool rows_are_points true if M are points instead of vectors; default true Bool centered true if output should be centered; default true ##### Returns  Polytope the zonotope generated by the input points or vectors Examples: • The following produces a parallelogram with the origin as its vertex barycenter:>$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 $p->VERTICES; 1 1 0 1 0 0 1 1 1 1 2 1 • zonotope_vertices_fukuda (M) → Matrix<E> Create the vertices of a zonotope from a matrix whose rows are input points or vectors. ##### Parameters  Matrix M ##### Options  Bool centered_zonotope default 1 ##### Returns  Matrix Example: • The following stores the vertices of a parallelogram with the origin as its vertex barycenter and prints them:>$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 • ### Producing a vector configuration A way of constructing vector configurations is to modify an already existing vector configuration. • free_sum (P1, P2) → VectorConfiguration Construct the free sum of two vector configurations. ##### Parameters  VectorConfiguration P1 VectorConfiguration P2 ##### Options  Bool force_centered if the input polytopes must be centered. Defaults to true. Bool no_coordinates produces a pure combinatorial description. Defaults to false. ##### Returns  VectorConfiguration • projection <Scalar> (P, indices) → VectorConfiguration Orthogonally 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. ##### Type Parameters  Scalar coordinate type ##### Parameters  VectorConfiguration P Array indices ##### Options  Bool revert inverts the coordinate list ##### Returns  VectorConfiguration • projection_preimage <Scalar> (P_Array) → VectorConfiguration Construct a new vector configuration that projects to a given array of vector configurations If the n vector configurations are d_1, d_2, ..., d_n-dimensional and all have m vectors, the resulting vector configuration is (d_1+...+d_n)-dimensional, has m vectors, and the projection to the i-th d_i coordinates gives the i-th input vector configuration. ##### Type Parameters  Scalar coordinate type ##### Parameters  Array P_Array ##### Returns  VectorConfiguration • project_full <Scalar> (P) → VectorConfiguration Orthogonally 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. ##### Type Parameters  Scalar coordinate type ##### Parameters  VectorConfiguration P ##### Options  Bool no_labels Do not copy VERTEX_LABELS to the projection. default: 0 ##### Returns  VectorConfiguration • project_out <Scalar> (V, B) → VectorConfiguration Project a vector configuration V along the subspace with the given basis B. The result is still expressed in the original ambient basis. If V is a PointConfiguration and the first column of B is zero, the result is a PointConfiguration, else a VectorConfiguration. ##### Type Parameters  Scalar coordinate type ##### Parameters  VectorConfiguration V Matrix B a matrix whose rows contain the basis of the space to be projected out ##### Returns  VectorConfiguration • project_out <Scalar> (C, B) → Cone Project a Cone C along the subspace with the given basis B The result is still expressed in the original ambient basis. If V is a Polytope and the first column of B is zero, the result is a Polytope, else a Cone. ##### Type Parameters  Scalar coordinate type ##### Parameters  Cone C Matrix B a matrix whose rows contain the basis of the space to be projected out ##### Returns  Cone • project_to <Scalar> (V, B) → VectorConfiguration Project a vector configuration V to the subspace with a given basis B and express the result in that basis. A boolean flag make_affine (by default undef) determines whether the resulting coordinates acquire an extra leading '1'. The return type is a VectorConfiguration, unless (i) P is a PointConfiguration, (ii) the first column of B is zero, (iii) make_affine is not 0, in which case it is a PointConfiguration. The return type depends on the input: (1) If V is a VectorConfiguration, the result is also a VectorConfiguration. (2a) If V is a PointConfiguration and all rows in B start with a 0, the result is a PointConfiguration. If, furthermore, make_affine is undef, it is set to 1. (2b) If V is a PointConfiguration and some row of B has a non-zero first entry, the result is a VectorConfiguration. The reasoning here is that B having a zero first column or not influences the hyperplane at infinity. ##### Type Parameters  Scalar coordinate type ##### Parameters  VectorConfiguration V Matrix B a matrix whose rows contain the basis of the image space ##### Options  Bool make_affine . If undef (default), apply the above reasoning. If 1 (0), unconditionally (don't) add leading 1's. ##### Returns  VectorConfiguration or PointConfiguration • project_to <Scalar> (C, B) → Cone Project a Polytope or Cone to the subspace with a given basis, and express the result in that basis A boolean flag make_affine (by default undef) determines whether the resulting coordinates acquire an extra leading '1'. The return type is a Cone, unless (i) P is a Polytope, (ii) the first column of B is zero, (iii) make_affine is not 0, in which case it is a Polytope. If make_affine is undef and (ii) is true, make_affine is set to 1. The reasoning here is that B having a zero first column or not influences the hyperplane at infinity. ##### Type Parameters  Scalar coordinate type ##### Parameters  Cone C Matrix B a matrix whose rows contain the basis of the image space ##### Returns  Cone or Polytope • ### Producing other objects Functions producing big objects which are not contained in application polytope. • coxeter_group (type) → group::Group Produces 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::Group the Coxeter group of type type • crosscut_complex (p) → topaz::SimplicialComplex Produce 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 • ### Producing regular polytopes and their generalizations 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) → Polytope Create 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) → Polytope Create 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 Bd-1 = Cd-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 dimension Scalar 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 Bool character_table add the character table to the symmetry group description, if 0 ##### Returns  Polytope 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 $p->GROUP->RAYS_ACTION->GENERATORS; 1 0 2 3 4 5 2 3 0 1 4 5 0 1 4 5 2 3 • cube <Scalar> (d, x_up, x_low) → Polytope<Scalar> Produce a d-dimensional cube. Regular polytope corresponding to the Coxeter group of type Bd-1 = Cd-1. The bounding hyperplanes are xi <= x_up and xi >= x_low. ##### Type Parameters  Scalar Coordinate type of the resulting polytope. Unless specified explicitly, deduced from the type of bound values, defaults to Rational. ##### Parameters  Int d the dimension Scalar x_up upper bound in each dimension Scalar x_low lower bound in each dimension ##### Options  Bool group add a symmetry group description to the resulting polytope Bool character_table add the character table to the symmetry group description, if 0 ##### Returns  Polytope Examples: • This yields a +/-1 cube of dimension 3 and stores it in the variable$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 Create cuboctahedron. An Archimedean solid. ##### Returns  Polytope • dodecahedron () → Polytope Create exact regular dodecahedron in Q(sqrt{5}). A Platonic solid. ##### Returns  Polytope • icosahedron () → Polytope Create exact regular icosahedron in Q(sqrt{5}). A Platonic solid. ##### Returns  Polytope • icosidodecahedron () → Polytope Create exact icosidodecahedron in Q(sqrt{5}). An Archimedean solid. ##### Returns  Polytope • johnson_solid (n) → Polytope Create Johnson solid number n, where 1 <= n <= 92. A Johnson solid is a 3-polytope all of whose facets are regular polygons. Some are realized with floating point numbers and thus not exact; yet VERTICES_IN_FACETS is correct in all cases. ##### Parameters  Int n the index of the desired Johnson polytope ##### Returns  Polytope • johnson_solid (s) → Polytope Create Johnson solid with the given name. A Johnson solid is a 3-polytope all of whose facets are regular polygons. Some are realized with floating point numbers and thus not exact; yet VERTICES_IN_FACETS is 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. 'tridiminished_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 • octahedron () → Polytope Produce a regular octahedron, which is the same as the 3-dimensional cross polytope. ##### Returns  Polytope • platonic_solid (s) → Polytope Create 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 • reduced (t, x, s, h, r) → Polytope<Rational> Produce a 3-dimensional reduced polytope (for suitably chosen parameters). That is, a polytope of constant width which does not properly contain a subpolytope of the same width. Construction due to Bernardo González Merino, Thomas Jahn, Alexandr Polyanskii and Gerd Wachsmuth, arXiv:1701.08629 ##### Parameters  Rational t Rational x Rational s Rational h Rational r ##### Returns  Polytope Example: • These values yield a reduced 3-polytope (approximately). The width equals 1.>$r = reduced(0.55, 0.6176490959800, 0.1351384931026, 0.0984300252409, 0.3547183586709);
•
regular_120_cell () → Polytope

Create exact regular 120-cell in Q(sqrt{5}).

##### Returns
 Polytope
•
regular_24_cell () → Polytope

Create regular 24-cell.

##### Returns
 Polytope
•
regular_600_cell () → Polytope

Create exact regular 600-cell in Q(sqrt{5}).

##### Returns
 Polytope
•
regular_simplex (d) → Polytope

Produce a regular d-simplex embedded in R^d with edge length sqrt(2).

##### Parameters
 Int d the dimension
##### Options
 Bool group
##### Returns
 Polytope

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->RAYS_ACTION->GENERATORS; 1 0 2 3 3 0 1 2 • rhombicosidodecahedron () → Polytope Create exact rhombicosidodecahedron in Q(sqrt{5}). An Archimedean solid. ##### Returns  Polytope • rhombicuboctahedron () → Polytope Create rhombicuboctahedron. An Archimedean solid. ##### Returns  Polytope • root_system (type) → VectorConfiguration Produce the root systems of the Coxeter arrangement of a given type The roots lie at infinity to facilitate reflecting in them. ##### Parameters  String type the type of the Coxeter arrangement, for example A4 or E8 ##### Returns  VectorConfiguration • simple_roots_type_A (index) → SparseMatrix Produce 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}. ##### Parameters  Int index of the arrangement (3, 4, etc) ##### Returns  SparseMatrix • simple_roots_type_B (index) → SparseMatrix Produce 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}. ##### Parameters  Int index of the arrangement (3, 4, etc) ##### Returns  SparseMatrix • simple_roots_type_C (index) → SparseMatrix Produce 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}. ##### Parameters  Int index of the arrangement (3, 4, etc) ##### Returns  SparseMatrix • simple_roots_type_D (index) → SparseMatrix Produce 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}. ##### Parameters  Int index of the arrangement (3, 4, etc) ##### Returns  SparseMatrix • simple_roots_type_E6 () → SparseMatrix Produce 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 () → SparseMatrix Produce 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 () → SparseMatrix Produce 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 () → SparseMatrix Produce 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 () → SparseMatrix Produce 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 () → SparseMatrix Produce 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 () → SparseMatrix Produce 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 Create regular tetrahedron. A Platonic solid. ##### Returns  Polytope • truncated_cube () → Polytope Create truncated cube. An Archimedean solid. ##### Returns  Polytope • truncated_cuboctahedron () → Polytope Create 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 Create exact truncated dodecahedron in Q(sqrt{5}). An Archimedean solid. ##### Returns  Polytope • truncated_icosahedron () → Polytope Create exact truncated icosahedron in Q(sqrt{5}). An Archimedean solid. Also known as the soccer ball. ##### Returns  Polytope • truncated_icosidodecahedron () → Polytope Create exact truncated icosidodecahedron in Q(sqrt{5}). An Archimedean solid. ##### Returns  Polytope • truncated_octahedron () → Polytope Create truncated octahedron. An Archimedean solid. Also known as the 3-permutahedron. ##### Returns  Polytope • wythoff (type, rings) → Polytope Produce the orbit polytope of a point under a Coxeter arrangement with exact coordinates, possibly in a qudratic extension field of the rationals ##### Parameters  String type single letter followed by rank representing the type of the arrangement Set rings indices of the hyperplanes corresponding to simple roots of the arrangement that the initial point should NOT lie on. You may specify just an integer or a perl array ref like [0,1] or [0..2]. ##### Options  Bool lattice Should the vertices of the orbit polytope be chosen to lie on the corresponding root lattice? default 0, which means that the vertices will instead be chosen to lie as symmetrically as possible. ##### Returns  Polytope • ### Quotient spaces 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. ##### Parameters  Polytope P , centrally symmetric Example: • >$P = cube(3);> cs_quotient($P);> print$P->CS_PERMUTATION; 7 6 5 4 3 2 1 0 The zeroth vertex gets identified with the seventh, the first with the sixth etc.
•
cylinder_2 () → Polytope

Return a 2-dimensional cylinder obtained by identifying two opposite sides of a square.

##### Returns
 Polytope

Example:
• To obtain a topological space homeomorphic to a cylinder, type> $p = cylinder_2();> print$p->QUOTIENT_SPACE->IDENTIFICATION_ACTION->GENERATORS; 2 3 0 1> print $p->QUOTIENT_SPACE->IDENTIFICATION_ACTION->ORBITS; {0 2} {1 3} Thus, vertices 0,2 and vertices 1,3 are identified.> print$p->QUOTIENT_SPACE->FACES; {{0} {1}} {{0 1} {0 2} {1 3}} {{0 1 2 3}} Thus, after identification two vertices, three edges, and one two-dimensional face remain:> print $p->QUOTIENT_SPACE->F_VECTOR; 2 3 1 • davis_manifold () → Polytope Return the 4-dimensional hyperbolic manifold obtained by Michael Davis ##### Returns  Polytope [Proceedings of the AMS, Vol. 93, No. 2 (Feb., 1985), pp. 325-328] by identifying opposite faces of the 120-cell Example: • The Davis manifold is the centrally symmetric quotient of the regular 210-cell:>$d = davis_manifold();> print $d->F_VECTOR; 600 1200 720 120> print$d->QUOTIENT_SPACE->F_VECTOR; 300 600 360 60 1
•
quarter_turn_manifold () → Polytope

Return 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.

##### Returns
 Polytope

Example:
• To obtain a topological space homeomorphic to the quarter turn manifold, type> $p = quarter_turn_manifold();> print$p->QUOTIENT_SPACE->IDENTIFICATION_ACTION->GENERATORS; 5 7 4 6 2 0 3 1 6 2 1 5 7 3 0 4 To see which vertices are identified, type> print $p->QUOTIENT_SPACE->IDENTIFICATION_ACTION->ORBITS; {0 3 5 6} {1 2 4 7} Thus, two classes of vertices remain, with 0 and 1 being representatives. To see the faces remaining after identification, type> print$p->QUOTIENT_SPACE->FACES; {{0} {1}} {{0 1} {0 2} {0 4} {0 7}} {{0 1 2 3} {0 1 4 5} {0 1 6 7}} {{0 1 2 3 4 5 6 7}}> print $p->QUOTIENT_SPACE->F_VECTOR; 2 4 3 1 • write_quotient_space_simplexity_ilp () outputs a linear program whose optimal value is a lower bound for the number of simplices necessary to triangulate the polytope in such a way that its symmetries respect the triangulation of the boundary. • ### Symmetry These functions capture information of the object that is concerned with the action of permutation groups. • cocircuit_equations_support_reps (points, gens, rirs, rmis) → Int write the indices of the representatives of the support of the cocircuit equations to a file ##### Parameters  Matrix points Array> gens the generators of the action of the symmetry group Array rirs representatives of interior ridge simplices Array rmis representatives of maximal interior simplices ##### Options  String filename where large output should go to. 'filename=>"-"' writes to stdout. ##### Returns  Int • combinatorial_symmetries (p) → group::PermutationAction Compute the combinatorial symmetries (i.e., automorphisms of the face lattice) of a given polytope p. They are stored in terms of a GROUP.VERTICES_ACTION and a GROUP.FACETS_ACTION property in p, and the GROUP.VERTICES_ACTION is also returned. ##### Parameters  Polytope p ##### Returns  group::PermutationAction the action of the combinatorial symmetry group on the vertices Example: • To get the vertex symmetry group of the square and print its generators, type the following:> print combinatorial_symmetries(cube(2))->GENERATORS; 2 3 0 1 1 0 2 3>$p = cube(2); combinatorial_symmetries($p);> print$p->GROUP->VERTICES_ACTION->GENERATORS; 0 2 1 3 1 0 3 2> print $p->GROUP->FACETS_ACTION->GENERATORS; 2 3 0 1 1 0 2 3 • combinatorial_symmetrized_cocircuit_equations (P, comps) calculate a sparse representation of the cocircuit equations corresponding to a direct sum of isotypic components ##### Parameters  Cone P Set comps the list of indices of the isotypic components to project to; default [0], which amounts to summing all cocircuit equations corresponding to the orbit of each ridge. • combinatorial_symmetrized_cocircuit_equations (P, rirs, rmis, comps) → Array<Pair<SetType, HashMap<SetType,Rational>>> calculate the projection of the cocircuit equations to a direct sum of isotypic components and represent it combinatorially ##### Parameters  Cone P Array rirs representatives of interior ridge simplices Array rmis representatives of maximal interior simplices Set comps the list of indices of the isotypic components to project to; default [0], which amounts to summing all cocircuit equations corresponding to the orbit of each ridge. ##### Options  String filename where large output should go to. 'filename=>"-"' writes to stdout. ##### Returns  Array>> indexed_cocircuit_equations a list of interior ridge simplices together with the corresponding sparsely represented cocircuit equation • isotypic_configuration (P, i) → polytope::PointConfiguration<Float> Given a polytope that has a matrix group acting on it, return the projections of the vertices to the i-th isotypic component C_i. If the input is a group with a permutation action a, regard a as acting on the unit basis vectors of the ambient space and return the projection of the unit basis vectors to the i-th isotypic component. ##### Parameters  Polytope P a polytope with a matrix action, or a group::Group g with a permutation action Int i the index of the desired isotypic component ##### Returns  polytope::PointConfiguration Example: • Consider the symmetry group of the cyclic polytope c(4,10) in the Carathéodory realization.>$p = cyclic_caratheodory(4,10,group=>1); For i=4, we obtain a 10-gon:> print isotypic_configuration($p,4)->POINTS; 1 1 0 1 0.8090169944 0.5877852523 1 0.3090169944 0.9510565163 1 -0.3090169944 0.9510565163 1 -0.8090169944 0.5877852523 1 -1 0 1 -0.8090169944 -0.5877852523 1 -0.3090169944 -0.9510565163 1 0.3090169944 -0.9510565163 1 0.8090169944 -0.5877852523 Similarly, for i=5 we get two copies of a pentagon. • lattice_automorphisms_smooth_polytope (P) → Array<Array<Int>> Returns a generating set for the lattice automorphism group of a smooth polytope P by comparing lattice distances between vertices and facets. ##### Parameters  Polytope P the given polytope ##### Returns  Array> the generating set for the lattice automorphism group Example: • > print lattice_automorphisms_smooth_polytope(cube(2)); 2 3 0 1 1 0 3 2 0 2 1 3 • linear_symmetries () wrapper function to store the symmetry group in the parent object Contained in extension sympol. • linear_symmetries (m) → group::Group Use sympol to compute the linear symmetries of - the rows of a rational matrix m, or - the RAYS|VERTICES, FACETS, or POINTS of a rational cone or polytope C (whatever is available, in this order), or - the VECTORS|POINTS of a rational vector or point configuration P. Except for matrices, the action of the symmetry group is stored inside the parent object. In the case of cones, sympol might compute only a subset of the linear symmetry group. Sympol, and therefore this function, can only handle rational objects. Contained in extension sympol. ##### Parameters  Matrix m | Cone C | VectorConfiguration P ##### Returns  group::Group the linear symmetry group, together with a PERMUTATION_ACTION, VERTEX_ACTION, FACETS_ACTION, or VECTOR_ACTION Example: • >$ls = linear_symmetries(cube(2)->VERTICES);> print $ls->PERMUTATION_ACTION->GENERATORS; 0 2 1 3 3 1 2 0 2 3 0 1> print linear_symmetries(cube(3)->VERTICES)->PERMUTATION_ACTION->GENERATORS; 0 4 2 6 1 5 3 7 0 1 4 5 2 3 6 7 7 6 5 4 3 2 1 0 2 6 0 4 3 7 1 5> print linear_symmetries(cube(3))->FACETS_ACTION->GENERATORS; 1 0 2 3 4 5 0 1 3 2 4 5 2 3 0 1 4 5 0 1 2 3 5 4 0 1 4 5 2 3 • nestedOPGraph (gen_point, points, lattice_points, group, verbose) → ARRAY Constructs the NOP-graph of an orbit polytope. It is used by the rule for the NOP_GRAPH. ##### Parameters  Vector gen_point the generating point Matrix points the vertices of the orbit polytope Matrix lattice_points the lattice points of the orbit polytope group::Group group the generating group Bool verbose print out additional information ##### Returns  ARRAY ($Graph, $lp_reps,$minInStartOrbit, \@core_point_reps, $CPindices) • orbit_polytope (input_point, a) → Polytope Constructs the orbit polytope of a given point input_point with respect to a given group action a. ##### Parameters  Vector input_point the basis point of the orbit polytope group::PermutationAction a the action of a permutation group on the coordinates of the ambient space ##### Returns  Polytope the orbit polytope of input_point w.r.t. the action a Example: • The orbit polytope of a set of points A in affine d-space is the convex hull of the images of A under the action of a group G on the affine space. polymake implements several variations of this concept. The most basic one is the convex hull of the orbit of a single point under a set of coordinate permutations. For example, consider the cyclic group C_6 that acts on 6-dimensional space by cyclically permuting the coordinates. This action is represented in polymake by group::cyclic_group(6)->PERMUTATION_ACTION. To compute the convex hull of cyclic shifts of the vector v = [1,6,0,5,-5,0,-5] in homogeneous coordinates, type>$p = orbit_polytope(new Vector([1,6,0,5,-5,0,-5]), group::cyclic_group(6)->PERMUTATION_ACTION); After this assignment, the orbit polytope is still in implicit form, and the only properties that are defined reside in GROUP->COORDINATE_ACTION:> print $p->GROUP->COORDINATE_ACTION->properties(); type: PermutationAction<Int, Rational> as Polytope<Rational>::GROUP::COORDINATE_ACTION  GENERATORS 1 2 3 4 5 0   INPUT_RAYS_GENERATORS 1 6 0 5 -5 0 -5 To calculate the vertices of the orbit polytope explicitly, say> print$p->VERTICES; 1 -5 0 -5 6 0 5 1 -5 6 0 5 -5 0 1 0 -5 6 0 5 -5 1 0 5 -5 0 -5 6 1 5 -5 0 -5 6 0 1 6 0 5 -5 0 -5
•
orbit_polytope (input_points, a) → Polytope

Constructs the orbit polytope of a given set of points input_points with respect to a given group action a.

##### Parameters
 Matrix input_points the basis points of the orbit polytope group::PermutationAction a the action of a permutation group on the coordinates of the ambient space
##### Returns
 Polytope the orbit polytope of input_points w.r.t. the action a

Example:
• To find the orbit of more than one point under a PermutationAction on the coordinates, say> $p = orbit_polytope(new Matrix([ [1,6,0,5,-5,0,-5], [1,1,2,3,4,5,6] ]), new group::PermutationAction(GENERATORS=>[ [1,2,3,4,5,0] ]));> print$p->VERTICES; 1 -5 0 -5 6 0 5 1 -5 6 0 5 -5 0 1 0 -5 6 0 5 -5 1 0 5 -5 0 -5 6 1 5 -5 0 -5 6 0 1 6 0 5 -5 0 -5 1 1 2 3 4 5 6 1 2 3 4 5 6 1 1 3 4 5 6 1 2 1 4 5 6 1 2 3 1 5 6 1 2 3 4 1 6 1 2 3 4 5
•
orbit_polytope (input_point, g) → Polytope

Constructs the orbit polytope of a given point input_point with respect to a given group action a.

##### Parameters
 Vector input_point the basis point of the orbit polytope group::Group g a group with a PERMUTATION_ACTION that acts on the coordinates of the ambient space
##### Returns
 Polytope the orbit polytope of input_point w.r.t. the action of g

Example:
• As a convenience function, you can also directly specify a group::Group that contains a PERMUTATION_ACTION:> $p = orbit_polytope(new Vector([1,6,0,5,-5,0,-5]), group::cyclic_group(6)); Up to now, the orbit polytope is still in implicit form. To calculate the vertices explicitly, say> print$p->VERTICES; 1 -5 0 -5 6 0 5 1 -5 6 0 5 -5 0 1 0 -5 6 0 5 -5 1 0 5 -5 0 -5 6 1 5 -5 0 -5 6 0 1 6 0 5 -5 0 -5
•
orbit_polytope (input_points, g) → Polytope

Constructs the orbit polytope of a given set of points input_points with respect to a given group action a.

##### Parameters
 Matrix input_points the basis points of the orbit polytope group::Group g a group with a PERMUTATION_ACTION that acts on the coordinates of the ambient space
##### Returns
 Polytope the orbit polytope of input_points w.r.t. the action of g

Example:
• As a convenience function, you can also directly specify a group::Group that contains a PERMUTATION_ACTION:> $p = orbit_polytope(new Matrix([ [1,6,0,5,-5,0,-5], [1,1,2,3,4,5,6] ]), group::cyclic_group(6));> print$p->VERTICES; 1 -5 0 -5 6 0 5 1 -5 6 0 5 -5 0 1 0 -5 6 0 5 -5 1 0 5 -5 0 -5 6 1 5 -5 0 -5 6 0 1 6 0 5 -5 0 -5 1 1 2 3 4 5 6 1 2 3 4 5 6 1 1 3 4 5 6 1 2 1 4 5 6 1 2 3 1 5 6 1 2 3 4 1 6 1 2 3 4 5
•
orbit_polytope (input_points, gens) → Polytope

Constructs the orbit polytope of a given set of points input_points with respect to a given set of generators gens.

##### Parameters
 Matrix input_points the basis point of the orbit polytope Array> gens the generators of a permutation group that acts on the coordinates of the ambient space
##### Returns
 Polytope the orbit polytope of input_points w.r.t. the coordinate action generated by gens

Example:
• This is a variation where several points are given as the row of a matrix, and the permutation action on coordinates is given by explicitly listing the generators. In this example, the matrix has just one row, and there is just one generator.> print orbit_polytope(new Matrix([ [1,6,0,5,-5,0,-5] ]), [ [1,2,3,4,5,0] ])->VERTICES; 1 -5 0 -5 6 0 5 1 -5 6 0 5 -5 0 1 0 -5 6 0 5 -5 1 0 5 -5 0 -5 6 1 5 -5 0 -5 6 0 1 6 0 5 -5 0 -5
•
orbit_polytope <Scalar> (input_point, a) → Polytope

Constructs the orbit polytope of a given point input_point with respect to a given matrix group action a.

##### Type Parameters
 Scalar S the underlying number type
##### Parameters
 Vector input_point the generating point of the orbit polytope group::MatrixActionOnVectors a the action of a matrix group on the coordinates of the ambient space
##### Returns
 Polytope the orbit polytope of input_point w.r.t. the action a

Example:
• polymake also supports orbit polytopes under the action of a group by matrices. To find the orbit of a point in the plane under the symmetry group of the square, say> $p = orbit_polytope(new Vector([1,2,1]), cube(2, group=>1)->GROUP->MATRIX_ACTION);> print$p->VERTICES; 1 -2 -1 1 -2 1 1 -1 -2 1 -1 2 1 1 -2 1 1 2 1 2 -1 1 2 1
•
orbit_polytope <Scalar> (input_points, a) → Polytope

Constructs the orbit polytope of a given set of points input_points with respect to a given matrix group action a.

##### Type Parameters
 Scalar S the underlying number type
##### Parameters
 Matrix input_points the generating points of the orbit polytope group::MatrixActionOnVectors a the action of a matrix group on the coordinates of the ambient space
##### Returns
 Polytope the orbit polytope of the input_points w.r.t. the action a

Example:
• To find the orbit of more than one point in the plane under the symmetry group of the square, say> $p = orbit_polytope(new Matrix([ [1,2,1], [1,5/2,0] ]), cube(2, group=>1)->GROUP->MATRIX_ACTION);> print$p->VERTICES; 1 -2 -1 1 -2 1 1 -1 -2 1 -1 2 1 1 -2 1 1 2 1 2 -1 1 2 1 1 -5/2 0 1 0 -5/2 1 0 5/2 1 5/2 0
•
ortho_project (p) → Polytope

Projects a symmetric polytope in R4 cap H1,k to R3. (See also the polymake extension 'tropmat' by S. Horn.)

##### Parameters
 Polytope p the symmetric polytope to be projected
##### Returns
 Polytope the image of p in R3
•
representation_conversion_up_to_symmetry (c) → Pair

Computes the dual description of a polytope up to its linear symmetry group.

Contained in extension sympol.
##### Parameters
 Cone c the cone (or polytope) whose dual description is to be computed, equipped with a GROUP
##### Options
 Bool v_to_h 1 (default) if converting V to H, false if converting H to V String method specifies sympol's method of ray computation via 'lrs' (default), 'cdd', 'beneath_beyond', 'ppl'
##### Returns
 Pair (Matrix vertices/inequalities, Matrix lineality/equations)
•
symmetrized_cocircuit_equations <Scalar> (P, comps)

calculate the projection of the cocircuit equations to a direct sum of isotypic components

##### Type Parameters
 Scalar the underlying data type
##### Parameters
 Cone P or PointConfiguration Set comps the list of indices of the isotypic components to project to; default [0], which amounts to summing all cocircuit equations corresponding to the orbit of each ridge.
##### Options
 Bool reduce_rows Should we return only linearly independent equations? default: 1
•
truncated_orbit_polytope (P, eps) → Polytope

Gives an implicit representation of the all-vertex truncation of an orbit polytope P, in which all vertices are cut off by hyperplanes at distance eps. The input polytope P must have a GROUP.COORDINATE_ACTION. The output is a polytope with a GROUP.COORDINATE_ACTION equipped with INEQUALITIES_GENERATORS.

##### Parameters
 Polytope P the input polytope Scalar eps scaled distance by which the vertices of the orbit polytope are to be cut off
##### Returns
 Polytope the truncated orbit polytope
•

### Transformations

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) → Polytope

Transform 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 $P->VERTICES; 1 0 1 1 To get the transformation, do the following:>$M = $P->get_attachment('REVERSE_LATTICE_PROJECTION');> print$M; 1 0 0 0 1 2> print $P->VERTICES *$M; 1 0 0 1 1 2
•
bound (P) → Polytope

Make 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.

##### Parameters
 Polytope P a positive polyhedron
##### Returns
 Polytope

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) → Polytope

Make 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).

##### Parameters
 Polytope P
##### Returns
 Polytope

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 $P->contains_in_interior($origin); false To create a translate that contains the origin, do this:> $PC = center($P);> print $PC->contains_in_interior($origin); true 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; true
•
orthantify (P, v) → Polytope

Make 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
•
orthonormal_col_basis (M) → Matrix<Float>

Return an orthonormal column basis of the input matrix.

##### Parameters
 Matrix M the input matrix
##### Returns
 Matrix
•
orthonormal_row_basis (M) → Matrix<Float>

Return an orthonormal row basis of the input matrix.

##### Parameters
 Matrix M the input matrix
##### Returns
 Matrix
•
polarize (C) → Cone

This method takes either a polytope (1.) or a cone (2.) as input. 1. Given a bounded, centered, not necessarily full-dimensional polytope P, produce its polar with respect to the standard Euclidean scalar product. 2. Given a cone C produce its dual with respect to the standard Euclidean scalar product, i.e. all the vectors that evaluate positively on C. 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.

##### Parameters
 Cone C
##### Options
 Bool no_coordinates only combinatorial information is handled
##### Returns
 Cone

Examples:
• To save the polar of the square in the variable $p and then print its vertices, do this:>$p = polarize(cube(2));> print $p->VERTICES; 1 1 0 1 -1 0 1 0 1 1 0 -1 • To dualize the cone over a hexagon and print its rays, do this:>$c = new Cone(INPUT_RAYS=>[[1,0,0],[1,1,0],[1,2,1],[1,2,2],[1,1,2],[1,0,1]]);> $cd = polarize($c);> print $cd->RAYS; 1 -1 1 0 0 1 0 1 0 1 1 -1 1 0 -1/2 1 -1/2 0 • revert (P) → Polytope Apply a reverse transformation to a given polyhedron P. All transformation clients keep track of the polytope's history. They write or update the attachment REVERSE_TRANSFORMATION. Applying revert to the transformed polytope reconstructs the original polytope. ##### Parameters  Polytope P a (transformed) polytope ##### Returns  Polytope the original polytope Example: • The following translates the square and then reverts the transformation:>$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) → Polytope

Scale a polyhedron P by a given scaling parameter factor.

##### Parameters
 Polytope P the polyhedron to be scaled Scalar factor the scaling factor Bool store stores the reverse transformation as an attachment (REVERSE_TRANSFORMATION); default value: 1.
##### Returns
 Polytope

Example:
• To scale 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) → Polytope

Transform a polyhedron P according to the linear transformation trans.

##### Parameters
 Polytope P the polyhedron to be transformed Matrix trans the transformation matrix Bool 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 $p->get_attachment('REVERSE_TRANSFORMATION'); 1 -23 -23 0 1 0 0 0 1 Check out the revert function to learn how to undo the transformation. It might be more comfortable to use the translate function to achieve the same result. • translate (P, trans, store) → Polytope Translate a polyhedron P by a given translation vector trans. ##### Parameters  Polytope P the polyhedron to be translated Vector trans the translation vector Bool store stores the reverse transformation as an attachment (REVERSE_TRANSFORMATION); default value: 1. ##### Returns  Polytope Example: • This translates the square by (23,23) and stores the transformation:>$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) → Polytope

Transform to a full-dimensional polytope while preserving the lattice spanned by vertices induced lattice of new vertices = Z^dim

##### 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.
•

### Triangulations, subdivisions and volume

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::SimplicialComplex

Create 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 no_labels Do not generate VERTEX_LABELS from the faces of the original cone. default: 0 Bool geometric_realization create a topaz::GeometricSimplicialComplex; default is true
##### Returns
 topaz::SimplicialComplex
•
chirotope (P) → String

Compute the chirotope of a polytope using TOPCOM.

##### Parameters
 Polytope P
##### Returns
 String
•
chirotope (P) → String

Compute the chirotope of a point or vector configuration using TOPCOM.