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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:
uses:
Objects
AffineLattice
:
a lattice that is displaced from the origin, i.e., a set of the form x + L, where x is a nonzero vector and L a (linear) latticeGroebnerBasis
:
The Groebner basis of the homogeneous toric ideal associated to the polytope, the term order is given in matrix form.LinearProgram
:
A linear program specified by a linear or abstract objective functionMixedIntegerLinearProgram
:
A mixed integer linear program specified by a linear or abstract objective functionPointConfiguration
:
ThePOINTS
of an object of type PointConfiguration encode a not necessarily convex finite point set. The difference to a parentVectorConfiguration
is that the points have homogeneous coordinates, i.e. they will be normalized to have first coordinate 1 without warning.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 ofVERTICES_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 fromCone
.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 seeJoswig: Polytope Propagation on Graphs.
Chapter 6 in Pachter/Sturmfels: Algebraic Statistics for Computational Biology, Cambridge 2005.
QuotientSpace
:
A topological quotient space obtained from aPolytope
by identifying faces. This object will sit inside the polytope.SchlegelDiagram
:
A Schlegel diagram of a polytope.SlackIdeal
:
UNDOCUMENTEDVectorConfiguration
:
An object of type VectorConfiguration deals with properties of row vectors, assembled into an n x d matrix calledVECTORS
. The entries of these row vectors are interpreted as nonhomogeneous coordinates. In particular, the coordinates of a VECTOR will *NOT* be normalized to have a leading 1.Visual::Cone
:
Visualization of a Cone as a graph (if 1d), or as a solid object (if 2d or 3d)Visual::Gale
:
A gale diagram prepared for drawing.Visual::PointConfiguration
:
Visualization of the point configuration.Visual::Polytope
:
Visualization of a polytope as a graph (if 1d), or as a solid object (if 2d or 3d), or as a Schlegel diagram (4d).Visual::PolytopeGraph
:
Visualization of the graph of a polyhedron.Visual::PolytopeLattice
:
Visualization of theHASSE_DIAGRAM
of a polyhedron as a multilayer graph..Visual::SchlegelDiagram
:
Visualization of the Schlegel diagram of a polytope.VoronoiPolyhedron
:
For a finite set ofSITES
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.
Functions
Combinatorics
Combinatorial functions.

circuits2matrix(Set<Pair<Set<Int>,Set<Int>>> co)
Convert
CIRCUITS
orCOCIRCUITS
to a 0/+1/1 matrix, with one row for each circuit/cocircuit, and as many columns as there are VECTORs/POINTS. Parameters:
 Returns:

cocircuit_equation_of_ridge(Cone C, Set rho)
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

cocircuit_equations(Cone C, Array<Set> interior_ridge_simplices, Array<Set> interior_simplices)
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

codegree<Scalar>(Cone 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.

codegree<Scalar>(PointConfiguration 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:

contraction(VectorConfiguration C, Int v)
Contract a vector configuration C along a specified vector v.
 Parameters:
Int
v
: index of the vector to contract

degree<Scalar>(PointConfiguration 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 3cube, type
> print degree(cube(3)); 3

deletion(VectorConfiguration C, Int v)
Delete a specified vector v from a vector configuration C.
 Parameters:
Int
v
: index of the vector to delete
Comparing
Functions based on graph isomorphisms.

congruent(Polytope P1, Polytope P2)
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 polytopePolytope
P2
: the second polytope Returns:
Scalar
 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(Polytope P1, Polytope P2)
 Parameters:
Polytope
P1
: the first polytopePolytope
P2
: the second polytope Options:
Bool
verbose
: Prints information on the difference between P1 and P2 if they are not equal. Returns:
 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,simplex(2),verbose => 1); Inequality 1 1 1 not satisfied by point 1 1 1. false

find_facet_vertex_permutations(Cone P1, Cone P2)
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.

included_polyhedra(Polytope P1, Polytope P2)
 Parameters:
Polytope
P1
: the first polytopePolytope
P2
: the second polytope Options:
Bool
verbose
: Prints information on the difference between P1 and P2 if none is included in the other. Returns:
 Example:
> print included_polyhedra(simplex(3),cube(3)); true
To see in what way the two polytopes differ, try this:

isomorphic(Cone P1, Cone P2)
Check whether the face lattices of two cones or polytopes are isomorphic. The problem is reduced to graph isomorphism of the vertexfacet incidence graphs.
 Parameters:
Cone
P1
: the first cone/polytopeCone
P2
: the second cone/polytope Returns:
 Example:
The following compares the standard 2cube 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(Polytope P1, Polytope P2)
Tests whether two smooth lattice polytopes are lattice equivalent by comparing lattice distances between vertices and facets.
Consistency check
These functions are for checking the consistency of some properties.

check_inc(Matrix points, Matrix hyperplanes, String sign, Bool verbose)
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:
 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(IncidenceMatrix VIF)
Try to check whether a given vertexfacet incidence matrix VIF defines a polytope. Note that a successful certification by check_poly is not sufficient to determine whether an incidence matrix actually defines a polytope. Think of it as a plausibility check.
 Parameters:
IncidenceMatrix
VIF
 Options:
Bool
dual
: transposes the incidence matrixBool
verbose
: prints information about the check. Returns:

validate_moebius_strip(Polytope P)
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:

validate_moebius_strip_quads(Polytope P)
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.
Coordinate conversions
The following functions allow for the conversion of the coordinate type of cones and polytopes.

affine_float_coords(Polytope P)
Dehomogenize the vertex coordinates and convert them to Float

convert_to<Coord>(Cone c)
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)

convert_to<Coord>(Polytope P)
provide a Polytope object with desired coordinate type
Finite metric spaces
Tight spans and their connections to polyhedral geometry

tight_span_envelope(SubdivisionOfPoints sd)
Computes the envelope for a given subdivision of points.
 Parameters:
 Options:
Bool
extended
: If True, the envelope of the extended tight span is computed. Returns:
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(Polytope P)
Compute the Steiner points of all faces of a polyhedron P using a randomized approximation of the angles. P must be
BOUNDED
.

center_distance(Polytope p)
Compute the mean or median distance of the
VERTICES
to theVERTEX_BARYCENTER
.

circuit_completions(Matrix L, Matrix R)
Given two matrices L (n x d) and R (m x d) such that (L/R) has rank r, select all (r+1n)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:
 Example:
Divide the vertex set of the 3cube 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(Cone P, Vector q)
Return the vertices of the face of P whose normal cone contains a point q.

containing_outer_cone(Polytope P, Vector q)
Return the vertices of the face of P whose outer cone contains a point q.

dihedral_angle(Vector<Scalar> H1, Vector<Scalar> H2)
Compute the dihedral angle between two (oriented) affine or linear hyperplanes.
 Parameters:
Vector<Scalar>
H1
: : first hyperplaneVector<Scalar>
H2
: : second hyperplane Options:
Bool
deg
: output in degrees rather than radians, default is falseBool
cone
: hyperplanes seen as linear hyperplanes, default is false Returns:
 Example:
> $H1 = new Vector(1,5,5); > $H2 = new Vector(1,5,5); > print dihedral_angle($H1,$H2,deg=>1); 90

induced_lattice_basis(Polytope p)
Returns a basis of the affine lattice spanned by the vertices

integer_points_bbox(Polytope<Scalar> P)
Enumerate all integer points in the given polytope by searching a bounding box.

minimal_ball(Polytope P_in)
Finds the minimal Ball which contains a given Polytope P_in with semidefinit programming. The ball is given by its center c and the square of it radius r.

minimal_vertex_angle(Polytope P)
Computes the minimal angle between two vertices of the input polytope P.

normaliz_compute(Cone C)
Compute degree one elements, Hilbert basis or Hilbert series of a cone C with libnormaliz Hilbert series and Hilbert hvector depend on the given grading and will not work unless C is
HOMOGENEOUS
or aMONOID_GRADING
is set Parameters:
Cone
C
 Options:
Bool
from_facets
: supply facets instead of rays to normalizBool
degree_one_generators
: compute the generators of degree one, i.e. lattice points of the polytopeBool
hilbert_basis
: compute Hilbert basis of the cone CBool
h_star_vector
: compute Hilbert hvector of the cone CBool
hilbert_series
: compute Hilbert series of the monoidBool
ehrhart_quasi_polynomial
: compute Ehrhart quasi polynomial of a polytopeBool
facets
: compute support hyperplanes (=FACETS,LINEAR_SPAN)Bool
rays
: compute extreme rays (=RAYS)Bool
dual_algorithm
: use the dual algorithm by PottierBool
skip_long
: do not try to use long coordinates firstBool
verbose
: libnormaliz debug output Returns:
 from extension:

occluding_cone(Cone P, Set F)
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

optimal_contains(Polytope P_in, Polytope P_out)
Finds for a given inner Polytope P_in and a given outer Polytope P_out a maximal a scalar s and a vector t, such that P_in scaled with s and shifted by t is a subset of P_out.

print_face_lattice(IncidenceMatrix VIF, Bool dual)
Write the face lattice of a vertexfacet 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(Vector q, Cone P)
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 PCone
P
: the polytope/cone from which q is to be separated Options:
Bool
strong
: Test for strong separability. default: true Returns:
 Example:
> $q = cube(2)>VERTICES>row(0); > print separable(cube(2), $q, strong=>0); true

simple_polytope_vertices_rs(Polytope<Scalar> P, Vector<Scalar> min_vertex)
Use reverse search method to find the vertices of a polyhedron. While applying this method, also collect the directed graph of cost optimization with respect to a (optionally) provided objective. If no objective is provided, one will be selected that cuts of
ONE_VERTEX
The input polytope must beSIMPLE
andPOINTED
, these properties are not checked by the algorithm.

steiner_point(Polytope P)
Compute the Steiner point of a polyhedron P using a randomized approximation of the angles.

violations(Cone P, Vector q)
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 qInt
violating_criterion
: has the options: +1 (positive values violate; this is the default), 0 (*non*zero values violate), 1 (negative values violate) Returns:
 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_face_indices(Cone P, Vector q)
Return the indices (in the HASSE_DIAGRAM) of all faces that are visible from a point q.

visible_facet_indices(Cone P, Vector q)
Return the indices of all facets that are visible from a point q.

zonotope_tiling_lattice(Polytope P)
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:
 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(GeometricSimplicialComplex c, Int facet_index, Rational conv_eps)
Computes the inequalities and the linear objective for an LP to lift a simplicial dball embedded starshaped in R^{d}.

core_point_algo(Polytope p, Rational optLPvalue)
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).

core_point_algo_Rote(Polytope p, Rational optLPvalue)
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).

find_transitive_lp_sol(Matrix Inequalities)
Algorithm to solve symmetric linear programs (LP) of the form max c^{t}x , c=(0,1,1,..,1) subject to the inequality system given by Inequalities. It is required that the symmetry group of the LP acts transitively on the coordinate directions.
 Parameters:
Matrix
Inequalities
: the inequalities describing the feasible region Returns:
 Example:
Consider the LP described by the facets of the 3cube:
> @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(Matrix points)
Compute a true inner point of a convex hull of the given set of points.

lp2poly<Scalar>(String file, Vector testvec, String prefix)
Read a linear programming problem given in LPFormat or MILPFormat (as used by cplex & Co.) and convert it to a polytope object with an added LP property or MILP property
 Type Parameters:
Scalar
: coordinate type of the resulting polytope; default isRational
. Parameters:
String
file
: filename of a linear programming problem in LPFormatVector
testvec
: If present, after reading in each line of the LP it is checked whether testvec fulfills itString
prefix
: If testvec is present, all variables in the LP file are assumed to have the form $prefix$i Options:
Bool
create_lp
: Create an LP property regardless of the format of the given file. If the file has MILPFormat, the created LP property will have an attachment INTEGER_VARIABLES Returns:
Polytope<Scalar>

poly2lp(Polytope P, LinearProgram LP, Bool maximize, String file)
Convert a polymake description of a polyhedron to LP format (as used by CPLEX and other linear problem solvers) and write it to standard output or to a file. If LP comes with an attachment 'INTEGER_VARIABLES' (of type Array<Bool>), the output will contain an additional section 'GENERAL', allowing for IP computations in CPLEX. If the polytope is not FEASIBLE, the function will throw a runtime error.
 Parameters:
Polytope
P
LinearProgram
LP
: default value: P→LPBool
maximize
: produces a maximization problem; default value: 0 (minimize)String
file
: default value: standard output

poly2porta(Polytope<Rational> p, String file)
take a rational polytope and write a porta input file (.ieq or .poi)

porta2poly(String file)
Read an .ieq or .poi file (porta input) or .poi.ieq or .ieq.poi (porta output) and convert it to a polytope object
 Parameters:
String
file
: filename of a porta file (.ieq or .poi) Returns:

print_constraints(Cone<Scalar> C)
Write the
FACETS
/INEQUALITIES
and theLINEAR_SPAN
/EQUATIONS
(if present) of a polytope P or cone C in a readable way.COORDINATE_LABELS
are adopted if present. Parameters:
Cone<Scalar>
C
: the given polytope or cone Options:
 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

rand_aof(Polytope P, Int start)
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.

random_edge_epl(Graph<Directed> G)
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:
 Returns:

separating_hyperplane(Vector q, Matrix points)
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 pointsMatrix
points
: the points from which q is to be separated Returns:
 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(Polytope p1, Polytope p2)
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 hyperplanePolytope
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:

totally_dual_integral(Matrix inequalities)
Checks weather a given system of inequalities is totally dual integral or not. The inequalities should describe a full dimensional polyhedron

vertex_colors(Polytope P, LinearProgram LP)
Calculate RGBcolorvalues 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
 Options:
RGB
min
: the minimal RGB valueRGB
max
: the maximal RGB value Returns:
 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(Polytope P, String 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
 Example:
For the 0/1 2cube 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.
 Example:
For the 0/1 3cube, 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(Polytope 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 2dimensional 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(Polytope P, String 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 2dimensional 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(Polytope P, Set<Int> isotypic_components, String 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
String
outfile_name
: . Setting this to '' (as is the default) prints the LP to stdout. Example:
For the 3cube, 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 3cube, there are 4 simplices in the same symmetry class as {0,1,2,4}, and one in the class of {0,3,5,6}.
Producing a cone
Various constructions of cones.

inner_cone(Cone p, Set<Int> F)
Computes the inner cone of p at a face F (or a vertex v).
 Parameters:
Cone
p
 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:
 Example:
To compute the inner cone at a vertex of the 3cube, do this:
> $c = inner_cone(cube(3), 1); > print $c>RAYS; 1 0 0 0 1 0 0 0 1
 Example:
To compute the inner cone along an edge of the 3cube, 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
 Example:
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(PointConfiguration p, Set<Int> F)
Computes the normal cone of p at a face F (or vertex v). By default this is the inner normal cone.
 Parameters:
 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:
 Example:
To compute the outer normal cone of a doubled 2cube, 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(Cone p, Set<Int> F)
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
 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:
 Example:
To compute the outer normal cone at a vertex of the 3cube, do this:
> $c = normal_cone(cube(3), 0, outer=>1); > print $c>RAYS; 1 0 0 0 1 0 0 0 1
 Example:
To compute the outer normal cone along an edge of the 3cube, do this:
> print normal_cone(cube(3), [0,1], outer=>1)>RAYS; 0 1 0 0 0 1
 Example:
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(Polytope<Scalar> P)
retuns the recession cone (tail cone, characteristic cone) of a polytope

subcone(Cone C)
Make a subcone from a cone.
 Parameters:
Cone
C
: the input cone Options:
Bool
no_labels
: Do not createRAY_LABELS
. default: 0 Returns:
Producing a point configuration
Constructing a point configuration, either from scratch or from existing objects.

minkowski_sum(PointConfiguration P1, PointConfiguration P2)
Produces the Minkowski sum of P1 and P2.
 Parameters:
 Returns:
 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(Scalar lambda, PointConfiguration P1, Scalar mu, PointConfiguration P2)
Produces the polytope lambda*P1+mu*P2, where * and + are scalar multiplication and Minkowski addition, respectively.
 Parameters:
Scalar
lambda
Scalar
mu
 Returns:
 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>(GraphAdjacency<Directed> G, EdgeMap<Directed,Scalar> Arc_Bounds, Int source, Int sink)
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

flow_polytope<Scalar>(Graph<Directed> G, Array<Scalar> Arc_Bounds, Int source, Int sink)
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

fractional_matching_polytope(Graph G)
Matching polytope of an undirected graph.
 Parameters:
Graph
G
 Returns:

tutte_lifting(Graph G)
Let G be a 3connected 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 R^{3}.
 Parameters:
Graph
G
 Returns:

weighted_digraph_polyhedron(Matrix encoding)
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:
Producing a polytope from other objects
Polytope constructions which take other big objects as input.

billera_lee(Vector<Integer> H)
Produces a simplicial polytope whose Hvector is the given input vector. The Gvector coming from the given vector must be an Msequence. 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: 10.1016/00973165(81)900583
 Parameters:
 Returns:
 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(Polytope P, Scalar z, Scalar z_prime)
Make a bipyramid over a pointed polyhedron. The bipyramid is the convex hull of the input polyhedron P and two points (v, z), (v, z_prime) on both sides of the affine span of P. For bounded polyhedra, the apex projections v to the affine span of P coincide with the vertex barycenter of P.
 Parameters:
Polytope
P
Scalar
z
: distance between the vertex barycenter and the first apex, default value is 1.Scalar
z_prime
: distance between the vertex barycenter and the second apex, default value is z. Options:
Bool
no_coordinates
: : don't compute the coordinates, purely combinatorial description is produced.Bool
no_labels
: Do not copyVERTEX_LABELS
from the original polytope. default: 0 label the new vertices with “Apex” and “Apex'”. Returns:
 Example:
Here's a way to construct the 3dimensional cross polytope:
> $p = bipyramid(bipyramid(cube(1))); > print equal_polyhedra($p,cross(3)); true

blending(Polytope P1, Int v1, Polytope P2, Int v2)
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 vertexPolytope
P2
Int
v2
: the index of the second vertex Options:
Bool
no_labels
: Do not copyVERTEX_LABELS
from the original polytopes. default: 0 Returns:
 Example:
The following gives the smallest
EVEN
3polytope 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(Polytope P_0, Polytope P_1, Scalar t_0, Scalar t_1)
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 polytopePolytope
P_1
: the second polytopeScalar
t_0
: the extra coordinate for the vertices of P_0Scalar
t_1
: the extra coordinate for the vertices of P_1 Options:
Bool
no_labels
: Do not copyVERTEX_LABELS
from the original polytope. default: 0 Returns:

cayley_embedding(Array<Polytope> A)
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 ith polytope is embedded to vt_i e_i, where t_i is the ith entry of the optional array t.
 Parameters:
 Options:
Array<Scalar>
factors
: array of scaling factors for the Cayley embedding; defaults to the all1 vectorBool
no_labels
: Do not copyVERTEX_LABELS
from the original polytope. default: 0 Returns:

cayley_polytope(Array<Polytope> P_Array)
Construct the cayley polytope of a set of pointed lattice polytopes contained in P_Array which is the convex hull of P_{1}×e_{1}, …, P_{k}×e_{k} where e_{1}, …,e_{k} are the standard unit vectors in R^{k}. In this representation the last k coordinates always add up to 1. The option proj projects onto the complement of the last coordinate.
 Parameters:
 Options:
Bool
proj
 Returns:

cell_from_subdivision(Polytope P, Int cell)
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 copyVERTEX_LABELS
from the original polytope. default: 0 Returns:
 Example:
First we create a nice subdivision for our favourite 2polytope, 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

cells_from_subdivision(Polytope<Scalar> P, Set<Int> cells)
Extract the given cells of the subdivision of a polyhedron and create a new polyhedron that has as vertices the vertices of the cells.
 Parameters:
Polytope<Scalar>
P
 Options:
Bool
no_labels
: Do not copyVERTEX_LABELS
from the original polytope. default: 0 Returns:
Polytope<Scalar>
 Example:
First we create a nice subdivision for a small polytope:
> $p = new Polytope(VERTICES=>[[1,0,0],[1,0,1],[1,1,0],[1,1,1],[1,3/2,1/2]]); > $p>POLYTOPAL_SUBDIVISION(MAXIMAL_CELLS=>[[0,1,3],[1,2,3],[2,3,4]]);
Then we create the polytope that has as vertices the vertices from cell 1 and 2, while keeping their labels.
> $c = cells_from_subdivision($p,[1,2]); > 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

conv(Array<Polytope> P_Array)
Construct a new polyhedron as the convex hull of the polyhedra given in P_Array.
 Parameters:
 Returns:
 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(Polytope P, Bool maximize)
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.

edge_middle(Polytope P)
Produce the convex hull of all edge middle points of some polytope P. The polytope must be
BOUNDED
. Parameters:
Polytope
P
 Returns:

face(Cone P, Set S)
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 copyVERTEX_LABELS
from the original polytope. default: 0 Returns:
 Example:
To create a cone from the vertices of the zeroth facet of the 3cube, type this:
> $p = face(cube(3),[0,1]);

facet(Cone P, Int facet)
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 copyVERTEX_LABELS
from the original polytope. default: 0 Returns:
 Example:
To create a cone from the vertices of the zeroth facet of the 3cube, type this:
> $p = facet(cube(3),0);

facet_to_infinity(Polytope P, Int i)
Make an affine transformation such that the ith facet is transformed to infinity

free_sum(Polytope P1, Polytope P2)
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:
 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(Polytope P)
Decompose a given polytope into the free sum of smaller ones
 Parameters:
Polytope
P
 Returns:

free_sum_decomposition_indices(Polytope P)
Decompose a given polytope into the free sum of smaller ones, and return the vertex indices of the summands

gc_closure(Polytope P)
Produces the gomorychvatal closure of a full dimensional polyhedron
 Parameters:
Polytope
P
 Returns:

integer_hull(Polytope P)
Produces the integer hull of a polyhedron

intersection(Cone C …)
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.

join_polytopes(Polytope P1, Polytope P2)
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:
 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(Polytope P, Vector v, Vector v_prime, Rational z, Rational z_prime)
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 apexVector
v_prime
: basis for the second apex If v_prime is omitted, v will be used for both apices. If both v and v_prime are omitted, it tries to find two vertices which don't lie in a common facet. If no such vertices can be found or P is a simplex, it uses an interior lattice point as both v and v_prime.Rational
z
: height for the first apex, default value is 1Rational
z_prime
: height for the second apex, default value is z Options:
Bool
no_labels
: Do not copyVERTEX_LABELS
from the original polytope. default: 0 label the new vertices with “Apex” and “Apex'”. Returns:
 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(Polytope P, Rational z, Vector v)
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 copyVERTEX_LABELS
from the original polytope. default: 0 label the new top vertex with “Apex”. Returns:
 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(Cone P)
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:

make_totally_dual_integral(Polytope P)
Produces a polyhedron with an totally dual integral inequality formulation of a full dimensional polyhedron
 Parameters:
Polytope
P
 Returns:

mapping_polytope(Polytope P1, Polytope P2)
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 R^{p} to R^{q}, that map P1 into P2. Mapping polytopes are also called Hompolytopes; cf. Bogart, Contois & Gubeladze, doi:10.1007/s0020901210535. The label of a new facet corresponding to v_{1} and h_{1} will have the form “v_{1}*h_{1}”.
 Parameters:
Polytope
P1
Polytope
P2
 Options:
Bool
no_labels
: Do not assignFACET_LABELS
. default: 0 Returns:
 Example:
Let us look at the mapping polytope of the unit interval and the standard unimodular triangle.
> $I=simplex(1); $T=simplex(2); $Hom_IT=mapping_polytope($I,$T);
The dimension equals (dim I + 1) * dim T.
> print $Hom_IT>DIM 4
> print rows_labeled($Hom_IT>FACETS,$Hom_IT>FACET_LABELS); v0*F0:1 1 0 1 0 v0*F1:0 1 0 0 0 v0*F2:0 0 0 1 0 v1*F0:1 1 1 1 1 v1*F1:0 1 1 0 0 v1*F2:0 0 0 1 1
> print $Hom_IT>N_VERTICES; 9
This is how to turn, e.g., the first vertex into a linear map.
> $v=$Hom_IT>VERTICES>[0]; > $M=new Matrix([$v>slice([1..2]),$v>slice([3..4])]); > print $I>VERTICES * transpose($M); 0 0 0 1
The above are coordinates in R^2, without the homogenization commonly used in polymake.

minkowski_sum(Polytope P1, Polytope P2)
Produces the Minkowski sum of P1 and P2.

minkowski_sum(Scalar lambda, Polytope P1, Scalar mu, Polytope P2)
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:
 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(Array<Polytope> summands)
Computes the (
VERTICES
of the) Minkowski sum of a list of polytopes using the algorithm by Fukuda described inKomei Fukuda, From the zonotope construction to the Minkowski addition of convex polytopes, J. Symbolic Comput., 38(4):12611272, 2004.
 Parameters:
 Returns:
 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(Polytope P, Array<Int> int_coords)
Produces the mixed integer hull of a polyhedron
 Parameters:
Polytope
P
 Returns:

pointed_part(Polytope P)
Produces the pointed part of a polyhedron

prism(Polytope P, Scalar z1, Scalar z2)
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 polytopeScalar
z1
: the left endpoint of the interval; default value: 1Scalar
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 0Bool
no_coordinates
: only combinatorial information is handledBool
no_labels
: Do not copyVERTEX_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:
 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(Polytope P1, Polytope P2)
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 handledBool
no_vertices
: do not compute verticesBool
no_facets
: do not compute facetsBool
no_labels
: Do not copyVERTEX_LABELS
from the original polytopes, if present. the label of a new vertex corresponding to v_{1} ⊕ v_{2} will have the form LABEL_1*LABEL_2. default: 0Bool
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:
 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);

project_full(Cone P)
Orthogonally project a polyhedron to a coordinate subspace such that redundant columns are omitted, i.e., the projection becomes fulldimensional 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 FourierMotzkin 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 FourierMotzkin eliminationBool
no_labels
: Do not copyVERTEX_LABELS
to the projection. default: 0 Returns:

projection(Cone P, Array<Int> indices)
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 FourierMotzkin 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
revert
: inverts the coordinate listBool
nofm
: suppresses FourierMotzkin elimination Returns:
 Example:
project the 3cube 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(Array<Cone> P_Array)
Construct a new polyhedron that projects to a given array of polyhedra. If the n polyhedra are d_1, d_2, …, d_ndimensional and all have m vertices, the resulting polyhedron is (d_1+…+d_n)dimensional, has m vertices, and the projection to the ith d_i coordinates gives the ith input polyhedron.
 Parameters:
 Returns:
 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

pyramid(Polytope P, Scalar z)
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 0Bool
no_coordinates
: don't compute new coordinates, produce purely combinatorial description.Bool
no_labels
: Do not copyVERTEX_LABELS
from the original polytope. default: 0 label the new top vertex with “Apex”. Returns:
 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(Polytope P, Int n)
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 beBOUNDED
.

rand_vert(Matrix V, Int n)
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 0/1polytopes.

spherize(Polytope P)

stack(Polytope P, Set<Int> stack_facets)
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
 Options:
Rational
lift
: controls the exact coordinates of the new vertices; rational number between 0 and 1; default value: 1/2Bool
no_coordinates
: produces a pure combinatorial description (in contrast to lift)Bool
no_labels
: Do not copyVERTEX_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:
 Example:
To generate a cubical polytope by stacking all facets of the 3cube to height 1/4, do this:
> $p = stack(cube(3),All,lift=>1/4);

stellar_all_faces(Polytope P, Int d)
Perform a stellar subdivision of all proper faces, starting with the facets. Parameter d specifies the lowest dimension of the faces to be divided. It can also be negative, then treated as the codimension. Default is 1, that is, the edges of the polytope.

stellar_indep_faces(Polytope P, Array<Set<Int>> in_faces)
Perform a stellar subdivision of the faces in_faces of a polyhedron P. The faces must have the following property: The open vertex stars of any pair of faces must be disjoint.
 Parameters:
Polytope
P
: , must be bounded Returns:

tensor(Polytope P1, Polytope P2)
Construct a new polytope as the convex hull of the tensor products of the vertices of two polytopes P1 and P2. Unbounded polyhedra are not allowed. Does depend on the vertex coordinates of the input.

truncation(Polytope P, Set<Int> trunc_vertices)
Cut off one or more vertices of a polyhedron. The exact location of the cutting hyperplane(s) can be controlled by the option cutoff, a rational number between 0 and 1. When cutoff=0, the hyperplane would go through the chosen vertex, thus cutting off nothing. When cutoff=1, the hyperplane touches the nearest neighbor vertex of a polyhedron. Alternatively, the option no_coordinates can be specified to produce a pure combinatorial description of the resulting polytope, which corresponds to the cutoff factor 1/2.
 Parameters:
Polytope
P
 Options:
Scalar
cutoff
: controls the exact location of the cutting hyperplane(s); rational number between 0 and 1; default value: 1/2Bool
no_coordinates
: produces a pure combinatorial description (in contrast to cutoff)Bool
no_labels
: Do not copyVERTEX_LABELS
from the original polytope. default: 0 New vertices get labels of the form 'LABEL1LABEL2', where LABEL1 is the original label of the truncated vertex, and LABEL2 is the original label of its neighbor. Returns:
 Example:
To truncate the second vertex of the square at 1/4, try this:
> $p = truncation(cube(2),2,cutoff=>1/4); > print $p>VERTICES; 1 1 1 1 1 1 1 1 1 1 1 1/2 1 1/2 1

unirand(Polytope P, Int n)
Produce a polytope with n random points that are uniformly distributed within the given polytope P. P must be bounded and fulldimensional.
 Parameters:
Polytope
P
Int
n
: the number of random points Options:
Bool
boundary
: forces the points to lie on the boundary of the given polytopeInt
seed
: controls the outcome of the random number generator; fixing a seed number guarantees the same outcome. Returns:
 Example:
This produces a polytope as the convex hull of 5 random points in the square with the origin as its center and side length 2:
> $p = unirand(cube(2),5);
 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(Polytope p, Int n)
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 copyVERTEX_LABELS
from the original polytope. default: 0 by default, the labels are produced from the corresponding neighbor vertices of the original polytope. Returns:
 Example:
This produces a vertex figure of one vertex of a 3dimensional cube with the origin as its center and side length 2. The result is a 2simplex.
> $p = vertex_figure(cube(3),5); > print $p>VERTICES; 1 1 1 0 1 0 1 1 1 1 0 1

wedge(Polytope P, Int facet, Rational z, Rational z_prime)
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 boundedInt
facet
: the `cutting edge'.Rational
z
: default value is 0.Rational
z_prime
: default value is z, or 1 if z==0. Options:
Bool
no_coordinates
: don't compute coordinates, pure combinatorial description is produced.Bool
no_labels
: Do not copyVERTEX_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:
 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(Polytope P1, Polytope P2)
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 productBool
no_labels
: Do not copyVERTEX_LABELS
from the original polytopes. default: 0 the label of a new vertex corresponding to v_{1} ⊕ v_{2} will have the form LABEL_1*LABEL_2. Returns:
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(Int d)
Produce a ddimensional associahedron (or Stasheff polytope). We use the facet description given in Ziegler's book on polytopes, section 9.2.

binary_markov_graph(Array<Bool> observation)
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:
 Returns:

binary_markov_graph(String observation)
 Parameters:
String
observation
: encoded as a string of “0” and “1”.

birkhoff(Int n, Bool even)
Constructs the Birkhoff polytope of dimension n^{2}. It is the polytope of nxn stochastic matrices (encoded as n^{2} row vectors), thus matrices with nonnegative entries whose row and column entries sum up to one. Its vertices are the permutation matrices.

cyclic(Int d, Int n)
Produce a ddimensional cyclic polytope with n points. Prototypical example of a neighborly polytope. Combinatorics completely known due to Gale's evenness criterion. Coordinates are chosen on the (spherical) moment curve at integer steps from start, or 0 if unspecified. If spherical is true the vertices lie on the sphere with center (1/2,0,…,0) and radius 1/2. In this case (the necessarily positive) parameter start defaults to 1.
 Parameters:
Int
d
: the dimensionInt
n
: the number of points Options:
Int
start
: defaults to 0 (or to 1 if spherical)Bool
spherical
: defaults to false Returns:
 Example:
To create the 2dimensional 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(Int d, Int n)
Produce a ddimensional 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
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<Rational>, else Polytope<Float> so that the matrices corresponding to the symmetry action may be approximated Returns:

delpezzo(Int d, Scalar scale)
Produce a ddimensional delPezzo polytope, which is the convex hull of the cross polytope together with the allones and minus allones vector. All coordinates are +/ scale or 0.

dwarfed_product_polygons(Int d, Int s)
Produce a ddimensional dwarfed product of polygons of size s.

explicit_zonotope(Matrix zones)
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:
 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(Int d)
Produce a Fano dsimplex.

fractional_knapsack(Vector<Rational> b)
Produce a knapsack polytope defined by one linear inequality (and nonnegativity constraints).
 Parameters:
 Returns:

goldfarb(Int d, Scalar e, Scalar g)
Produces a ddimensional 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 KleeMinty cubes.
 Parameters:
Int
d
: the dimensionScalar
e
Scalar
g
 Returns:

goldfarb_sit(Int d, Scalar eps, Scalar delta)
Produces a ddimensional variation of the KleeMinty cube if eps<1/2 and delta⇐1/2. This KleeMinty cube is scaled in direction x_(di) 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 dimensionScalar
eps
Scalar
delta
 Returns:

hypersimplex(Int k, Int d)
Produce the hypersimplex $ Δ(k,d) $, that is the the convex hull of all 0/1vector in $ R^d $ with exactly k 1s. Note that the output is never fulldimensional.
 Parameters:
Int
k
: number of 1sInt
d
: ambient dimension Options:
Bool
group
Bool
no_vertices
: do not compute verticesBool
no_facets
: do not compute facetsBool
no_vif
: do not compute vertices in facets Returns:
 Example:
This creates the hypersimplex in dimension 4 with vertices with exactly two 1entries and computes its symmetry group:
> $h = hypersimplex(2,4,group=>1);

hypertruncated_cube<Scalar>(Int d, Scalar k, Scalar lambda)
Produce a ddimensional hypertruncated cube. With symmetric linear objective function (0,1,1,…,1).

k_cyclic(Int n, Vector s)
Produce a (rounded) 2*kdimensional kcyclic 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, floatingpoint, or rational numbers. The coordinates of the ith 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 kcyclic polytopes are not simplicial. Since the components are rounded, this function might output a polytope which is not a kcyclic polytope! More information can be found in the following references:
P. Schuchert: “MatroidPolytope und Einbettungen kombinatorischer Mannigfaltigkeiten”,
PhD thesis, TU Darmstadt, 1995.
Z. Smilansky: “Bicyclic 4polytopes”,
Isr. J. Math. 70, 1990, 8292

klee_minty_cube(Int d, Scalar e)
Produces a ddimensional KleeMintycube if e<1/2. Uses the
goldfarb
client with g=0. Parameters:
Int
d
: the dimensionScalar
e
 Returns:

lecture_hall_simplex(Int d)
Produce a lecture hall dsimplex.

long_and_winding(Int r)
Produce polytope in dimension 2r with 3r+2 facets such that the total curvature of the central path is at least Omega(2^r); see Allamigeon, Benchimol, Gaubert and Joswig, SIAM J. Appl. Algebra Geom. (2018). See also
perturbed_long_and_winding
. Parameters:
Int
r
: defining parameter Options:
Rational
eval_ratio
: parameter for evaluating the puiseux rational functionsInt
eval_exp
: to evaluate at eval_ratio^eval_exp, default: 1Float
eval_float
: parameter for evaluating the puiseux rational functions Returns:
 Example:
This yields a 4polytope over the field of Puiseux fractions.
> $p = long_and_winding(2);
 Example:
This yields a rational 4polytope with the same combinatorics.
> $p = long_and_winding(2,eval_ratio=>2);

max_GC_rank(Int d)
Produce a ddimensional polytope of maximal GomoryChvatal 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:

multiplex(Int d, Int n)
Produce a combinatorial description of a multiplex with parameters d and n. This yields a selfdual ddimensional polytope with n+1 vertices. They are introduced by
T. Bisztriczky,
On a class of generalized simplices, Mathematika 43:27285, 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.

n_gon(Int n, Rational r, Rational alpha_0)
Produce a regular ngon. All vertices lie on a circle of radius r. The radius defaults to 1.
 Parameters:
Int
n
: the number of verticesRational
r
: the radius (defaults to 1)Rational
alpha_0
: the initial angle divided by pi (defaults to 0) Options:
Bool
group
 Returns:
 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

neighborly_cubical(Int d, Int n)
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. 24:315344 (2000).

newton(Polynomial p)
Produce the Newton polytope of a polynomial p.
 Parameters:
 Returns:
 Example:
Create the newton polytope of 1+x^2+y like so:

perles_irrational_8_polytope()
Create an 8dimensional polytope without rational realizations due to Perles
 Returns:

permutahedron(Int d)
Produce a ddimensional permutahedron. The vertices correspond to the elements of the symmetric group of degree d+1.

perturbed_long_and_winding(Int r)
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 functionsInt
eval_exp
: to evaluate at eval_ratio^eval_exp, default: 1Float
eval_float
: parameter for evaluating the puiseux rational functions Returns:
 Example:
This yields a simple 4polytope over the field of Puiseux fractions.
> $p = perturbed_long_and_winding(2);

pile(Vector<Int> sizes)
Produce a (d+1)dimensional polytope from a pile of cubes. Start with a ddimensional pile of cubes. Take a generic convex function to lift this polytopal complex to the boundary of a (d+1)polytope.
 Parameters:
 Returns:

pseudo_delpezzo(Int d, Scalar scale)
Produce a ddimensional delPezzo polytope, which is the convex hull of the cross polytope together with the allones vector. All coordinates are +/ scale or 0.

rand01(Int d, Int n)
Produce a ddimensional 0/1polytope with n random vertices. Uniform distribution.

rand_box(Int d, Int n, Int b)
Computes the convex hull of n points sampled uniformly at random from the integer points in the cube [0,b]^{d}.

rand_cyclic(Int d, Int n)
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.

rand_metric<Scalar>(Int n)
Produce an npoint metric with random distances. The values are uniformily distributed in [1,2].

rand_metric_int<Scalar>(Int n)
Produce an npoint metric with random distances. The values are uniformily distributed in [1,2].

rand_normal(Int d, Int n)
Produce a rational ddimensional polytope from n random points approximately normally distributed in the unit ball.

rand_sphere<Num>(Int d, Int n)
Produce a rational ddimensional polytope with n random vertices approximately uniformly distributed on the unit sphere.
 Type Parameters:
Num
: can be AccurateFloat (the default) or Rational WithAccurateFloat
the distribution should be closer to uniform, but the vertices will not exactly be on the sphere. WithRational
the vertices are guaranteed to be on the unit sphere, at the expense of both uniformity and logheight of points. Parameters:
Int
d
: the dimension of sphereInt
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:

rss_associahedron(Int l)
Produce a polytope of constrained expansions in dimension l according to
Rote, Santos, and Streinu: Expansive motions and the polytope of pointed pseudotriangulations.
Discrete and computational geometry, 699–736, Algorithms Combin., 25, Springer, Berlin, 2003.
 Parameters:
Int
l
: ambient dimension Returns:

signed_permutahedron(Int d)
Produce a ddimensional signed permutahedron.

simplex(Int d, Scalar scale)
Produce the standard dsimplex. Combinatorially equivalent to a regular polytope corresponding to the Coxeter group of type A_{d1}. Optionally, the simplex can be scaled by the parameter scale.
 Parameters:
Int
d
: the dimensionScalar
scale
: default value: 1 Options:
Bool
group
 Returns:
 Example:
To print the vertices (in homogeneous coordinates) of the standard 2simplex, i.e. a rightangled 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.
 Example:
To create a 3simplex and also calculate its symmetry group, type this:
> simplex(3, group=>1);

stable_set(Graph G)
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:

transportation(Vector r, Vector c)
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.

zonotope(Matrix<Scalar> M)
Create a zonotope from a matrix whose rows are input points or vectors. This method merely defines a Polytope object with the property
ZONOTOPE_INPUT_POINTS
. Parameters:
Matrix<Scalar>
M
: input points or vectors Options:
Bool
rows_are_points
: true if M are points instead of vectors; default trueBool
centered
: true if output should be centered; default true Returns:
Polytope<Scalar>
 Example:
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
 Example:
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(Matrix M)
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:
 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(VectorConfiguration P1, VectorConfiguration P2)
Construct the free sum of two vector configurations.

project_full<Scalar>(VectorConfiguration P)
Orthogonally project a vector configuration to a coordinate subspace such that redundant columns are omitted, i.e., the projection becomes fulldimensional without changing the combinatorial type.
 Type Parameters:
Scalar
: coordinate type Parameters:
 Options:
Bool
no_labels
: Do not copyVERTEX_LABELS
to the projection. default: 0 Returns:

project_out<Scalar>(VectorConfiguration V, Matrix B)
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:
Matrix
B
: a matrix whose rows contain the basis of the space to be projected out Returns:

project_out<Scalar>(Cone C, Matrix B)
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.

project_to<Scalar>(VectorConfiguration V, Matrix B)
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 nonzero 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.

project_to<Scalar>(Cone C, Matrix B)
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.

projection<Scalar>(VectorConfiguration P, Array<Int> indices)
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:
 Options:
Bool
revert
: inverts the coordinate list Returns:

projection_preimage<Scalar>(Array<VectorConfiguration> P_Array)
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_ndimensional and all have m vectors, the resulting vector configuration is (d_1+…+d_n)dimensional, has m vectors, and the projection to the ith d_i coordinates gives the ith input vector configuration.
 Type Parameters:
Scalar
: coordinate type Parameters:
Array<VectorConfiguration>
P_Array
 Returns:
Producing other objects
Functions producing big objects which are not contained in application polytope.

coxeter_group(String type)
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:

crosscut_complex(Polytope p)
Produce the crosscut complex of the boundary of a polytope.
 Parameters:
Polytope
p
 Options:
Bool
geometric_realization
: create aGeometricSimplicialComplex
; default is true Returns:
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 4polytopes.

archimedean_solid(String s)
Create Archimedean solid of the given name. Some polytopes are realized with floating point numbers and thus not exact; Vertexfacetincidences are correct in all cases.
 Parameters:
String
s
: the name of the desired Archimedean solid Returns:
 Example:
To show the mirror image of the snub cube use:
> scale(archimedean_solid('snub_cube'),1)>VISUAL;

catalan_solid(String s)
Create Catalan solid of the given name. Some polytopes are realized with floating point numbers and thus not exact; Vertexfacetincidences are correct in all cases.
 Parameters:
String
s
: the name of the desired Catalan solid Returns:

cross<Scalar>(Int d, Scalar scale)
Produce a ddimensional cross polytope. Regular polytope corresponding to the Coxeter group of type B_{d1} = C_{d1}. All coordinates are +/ scale or 0.
 Type Parameters:
Scalar
: Coordinate type of the resulting polytope. Unless specified explicitly, deduced from the type of bound values, defaults to Rational. Parameters:
Int
d
: the dimensionScalar
scale
: the absolute value of each nonzero vertex coordinate. Needs to be positive. The default value is 1. Options:
Bool
group
: add a symmetry group description to the resulting polytopeBool
character_table
: add the character table to the symmetry group description, if 0<d<7; default 1 Returns:
Polytope<Scalar>
 Example:
To create the 3dimensional 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>(Int d, Scalar x_up, Scalar x_low)
Produce a ddimensional cube. Regular polytope corresponding to the Coxeter group of type B_{d1} = C_{d1}. The bounding hyperplanes are x_{i} ⇐ x_up and x_{i} >= x_low.
 Type Parameters:
Scalar
: Coordinate type of the resulting polytope. Unless specified explicitly, deduced from the type of bound values, defaults to Rational. Parameters:
Int
d
: the dimensionScalar
x_up
: upper bound in each dimensionScalar
x_low
: lower bound in each dimension Options:
Bool
group
: add a symmetry group description to the resulting polytopeBool
character_table
: add the character table to the symmetry group description, if 0<d<7; default 1 Returns:
Polytope<Scalar>
 Example:
This yields a +/1 cube of dimension 3 and stores it in the variable $c.
> $c = cube(3);
 Example:
This stores a standard unit cube of dimension 3 in the variable $c.
> $c = cube(3,0);
 Example:
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()
Create cuboctahedron. An Archimedean solid.
 Returns:

dodecahedron()
Create exact regular dodecahedron in Q(sqrt{5}). A Platonic solid.
 Returns:

icosahedron()
Create exact regular icosahedron in Q(sqrt{5}). A Platonic solid.
 Returns:

icosidodecahedron()
Create exact icosidodecahedron in Q(sqrt{5}). An Archimedean solid.
 Returns:

johnson_solid(Int n)
Create Johnson solid number n, where 1 ⇐ n ⇐ 92. A Johnson solid is a 3polytope 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:

johnson_solid(String s)
Create Johnson solid with the given name. A Johnson solid is a 3polytope 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 polytope Returns:

octahedron()
Produce a regular octahedron, which is the same as the 3dimensional cross polytope.
 Returns:

platonic_solid(String s)
Create Platonic solid of the given name.
 Parameters:
String
s
: the name of the desired Platonic solid Returns:

reduced(Rational t, Rational x, Rational s, Rational h, Rational r)
Produce a 3dimensional 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

regular_120_cell()
Create exact regular 120cell in Q(sqrt{5}).
 Returns:

regular_24_cell()
Create regular 24cell.
 Returns:

regular_600_cell()
Create exact regular 600cell in Q(sqrt{5}).
 Returns:

regular_simplex(Int d)
Produce a regular dsimplex embedded in R^d with edge length sqrt(2).
 Parameters:
Int
d
: the dimension Options:
Bool
group
 Returns:
 Example:
To print the vertices (in homogeneous coordinates) of the regular 2simplex, i.e. an equilateral triangle, type this:
> print regular_simplex(2)>VERTICES; 1 1 0 1 0 1 1 1/21/2r3 1/21/2r3
The polytopes cordinate type is
QuadraticExtension<Rational>
, thus numbers that can be represented as a + b*sqrt© with Rational numbers a, b and c. The last row vectors entries thus represent the number $ 1/2 * ( 1  sqrt(3) ) $. Example:
To store a regular 3simplex 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()
Create exact rhombicosidodecahedron in Q(sqrt{5}). An Archimedean solid.
 Returns:

rhombicuboctahedron()
Create rhombicuboctahedron. An Archimedean solid.
 Returns:

root_system(String type)
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:

simple_roots_type_A(Int index)
Produce the simple roots of the Coxeter arrangement of type A Indices are taken w.r.t. the Dynkin diagram 0 — 1 — … — n1 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:

simple_roots_type_B(Int index)
Produce the simple roots of the Coxeter arrangement of type B Indices are taken w.r.t. the Dynkin diagram 0 — 1 — … — n2 –(4)–> n1 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:

simple_roots_type_C(Int index)
Produce the simple roots of the Coxeter arrangement of type C Indices are taken w.r.t. the Dynkin diagram 0 — 1 — … — n2 ←(4)– n1 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:

simple_roots_type_D(Int index)
Produce the simple roots of the Coxeter arrangement of type D Indices are taken w.r.t. the Dynkin diagram n2 / 0  1  …  n3 # n1 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:

simple_roots_type_E6()
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:

simple_roots_type_E7()
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:

simple_roots_type_E8()
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:

simple_roots_type_F4()
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:

simple_roots_type_G2()
Produce the simple roots of the Coxeter arrangement of type G2 Indices are taken w.r.t. the Dynkin diagram 0 ←(6)– 1
 Returns:

simple_roots_type_H3()
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:

simple_roots_type_H4()
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:

tetrahedron()
Create regular tetrahedron. A Platonic solid.
 Returns:

truncated_cube()
Create truncated cube. An Archimedean solid.
 Returns:

truncated_cuboctahedron()
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 2faces.
 Returns:

truncated_dodecahedron()
Create exact truncated dodecahedron in Q(sqrt{5}). An Archimedean solid.
 Returns:

truncated_icosahedron()
Create exact truncated icosahedron in Q(sqrt{5}). An Archimedean solid. Also known as the soccer ball.
 Returns:

truncated_icosidodecahedron()
Create exact truncated icosidodecahedron in Q(sqrt{5}). An Archimedean solid.
 Returns:

truncated_octahedron()
Create truncated octahedron. An Archimedean solid. Also known as the 3permutahedron.
 Returns:

wythoff(String type, Set rings)
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 arrangementSet
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:
Quotient spaces
Topologic cell complexes defined as quotients over polytopes modulo a discrete group.

cs_quotient(Polytope P)
For a centrally symmetric polytope, divide out the central symmetry, i.e, identify diametrically opposite faces.

cylinder_2()
Return a 2dimensional cylinder obtained by identifying two opposite sides of a square.
 Returns:
 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 twodimensional face remain:
> print $p>QUOTIENT_SPACE>F_VECTOR; 2 3 1

davis_manifold()
Return the 4dimensional hyperbolic manifold obtained by Michael Davis

quarter_turn_manifold()
Return the 3dimensional Euclidean manifold obtained by identifying opposite faces of a 3dimensional cube by a quarter turn. After identification, two classes of vertices remain.
 Returns:
 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(Matrix<Scalar> points, Array<Array<Int>> gens, Array<SetType> rirs, Array<SetType> rmis)
write the indices of the representatives of the support of the cocircuit equations to a file

combinatorial_symmetries(Polytope p)
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:
 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(Cone P, Set<Int> comps)
calculate a sparse representation of the cocircuit equations corresponding to a direct sum of isotypic components
 Parameters:
Cone
P

combinatorial_symmetrized_cocircuit_equations(Cone P, Array<SetType> rirs, Array<SetType> rmis, Set<Int> comps)
calculate the projection of the cocircuit equations to a direct sum of isotypic components and represent it combinatorially

isotypic_configuration(Polytope P, Int i)
Given a polytope that has a matrix group acting on it, return the projections of the vertices to the ith 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 ith isotypic component.
 Parameters:
Polytope
P
: a polytope with a matrix action, or a group::Group g with a permutation actionInt
i
: the index of the desired isotypic component Returns:
 Example:
Consider the symmetry group of the cyclic polytope c(4,10) in the Carathéodory realization.
> $p = cyclic_caratheodory(4,10,group=>1);
For i=4, we obtain a 10gon:
> print isotypic_configuration($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(Polytope P)
Returns a generating set for the lattice automorphism group of a smooth polytope P by comparing lattice distances between vertices and facets.

linear_symmetries
wrapper function to store the symmetry group in the parent object
 from extension:

linear_symmetries(Matrix m)
Use sympol to compute the linear symmetries of
 the rows of a rational matrix m, or
 the RAYSVERTICES, FACETS, or POINTS of a rational cone or polytope C (whatever is available, in this order), or
 the VECTORSPOINTS 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.
 Parameters:
Matrix
m
:  Cone C  VectorConfiguration P Returns:
 from extension:
 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(Vector gen_point, Matrix points, Matrix lattice_points, Group group, Bool verbose)
Constructs the NOPgraph of an orbit polytope. It is used by the rule for the
NOP_GRAPH
.

orbit_polytope(Vector input_point, PermutationAction a)
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 polytopePermutationAction
a
: the action of a permutation group on the coordinates of the ambient space Returns:
 Example:
The orbit polytope of a set of points A in affine dspace 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 6dimensional 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(Matrix input_points, PermutationAction a)
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 polytopePermutationAction
a
: the action of a permutation group on the coordinates of the ambient space Returns:
 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(Vector input_point, Group g)
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 polytopeGroup
g
: a group with a PERMUTATION_ACTION that acts on the coordinates of the ambient space Returns:
 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(Matrix input_points, Group g)
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 polytopeGroup
g
: a group with a PERMUTATION_ACTION that acts on the coordinates of the ambient space Returns:
 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(Matrix input_points, Array<Array<Int>> gens)
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 Returns:
 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>(Vector input_point, MatrixActionOnVectors a)
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 polytopeMatrixActionOnVectors
a
: the action of a matrix group on the coordinates of the ambient space Returns:
 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>(Matrix<Scalar> input_points, MatrixActionOnVectors<Scalar> a)
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<Scalar>
input_points
: the generating points of the orbit polytopeMatrixActionOnVectors<Scalar>
a
: the action of a matrix group on the coordinates of the ambient space Returns:
 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(Polytope p)
Projects a symmetric polytope in R^{4} cap H_{1,k} to R^{3}. (See also the polymake extension 'tropmat' by S. Horn.)
 Parameters:
Polytope
p
: the symmetric polytope to be projected Returns:

representation_conversion_up_to_symmetry(Cone c)
Computes the dual description of a polytope up to its linear symmetry group.
 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 VString
method
: specifies sympol's method of ray computation via 'lrs' (default), 'cdd', 'beneath_beyond', 'ppl' Returns:
 from extension:

symmetrized_cocircuit_equations<Scalar>(Cone P, Set<Int> comps)
calculate the projection of the cocircuit equations to a direct sum of isotypic components

truncated_orbit_polytope(Polytope P, Scalar eps)
Gives an implicit representation of the allvertex 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 polytopeScalar
eps
: scaled distance by which the vertices of the orbit polytope are to be cut off Returns:
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(Polytope p)
Transform to a fulldimensional 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:
 Example:
Consider a line segment embedded in 2space 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.
 Example:
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(Polytope P)
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:
 Example:
Observe the transformation of a simple unbounded 2polyhedron:
> $P = new Polytope(VERTICES=>[[1,0,0],[0,1,1],[0,0,1]]); > print bound($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(Polytope P)
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:
 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(Polytope P, Int v)
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.

orthonormal_col_basis(Matrix M)
Return an orthonormal column basis of the input matrix.
 Parameters:
Matrix
M
: the input matrix Returns:

orthonormal_row_basis(Matrix M)
Return an orthonormal row basis of the input matrix.
 Parameters:
Matrix
M
: the input matrix Returns:

polarize(Cone C)
This method takes either a polytope (1.) or a cone (2.) as input. 1. Given a bounded, centered, not necessarily fulldimensional 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 fulldimensional, the output will contain lineality orthogonal to the affine span of P. In particular, polarize() of a pointed polytope will always produce a fulldimensional 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:
 Example:
To save the polar of the square in the variable $p and then print its vertices, do this:
> $p = polarize(cube(2)); > print $p>VERTICES; 1 1 0 1 1 0 1 0 1 1 0 1
 Example:
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

porta_dual
Dual transformation via porta. Computes vertices and lineality space from inequalities and equations.

porta_primal
Primal transformation via porta. Computes facets and affine hull from vertices or points.

revert(Polytope P)
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:
 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(Polytope P, Scalar factor, Bool store)
Scale a polyhedron P by a given scaling parameter factor.
 Parameters:
Polytope
P
: the polyhedron to be scaledScalar
factor
: the scaling factorBool
store
: stores the reverse transformation as an attachment (REVERSE_TRANSFORMATION); default value: 1. Returns:
 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(Polytope P, Matrix trans, Bool store)
Transform a polyhedron P according to the linear transformation trans.
 Parameters:
Polytope
P
: the polyhedron to be transformedMatrix
trans
: the transformation matrixBool
store
: stores the reverse transformation as an attachment (REVERSE_TRANSFORMATION); default value: 1. Returns:
 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(Polytope P, Vector trans, Bool store)
Translate a polyhedron P by a given translation vector trans.
 Parameters:
Polytope
P
: the polyhedron to be translatedVector
trans
: the translation vectorBool
store
: stores the reverse transformation as an attachment (REVERSE_TRANSFORMATION); default value: 1. Returns:
 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(Polytope p)
Transform to a fulldimensional polytope while preserving the lattice spanned by vertices induced lattice of new vertices = Z^dim
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(Cone c)
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 generateVERTEX_LABELS
from the faces of the original cone. default: 0Bool
geometric_realization
: create aGeometricSimplicialComplex
; default is true Returns:

chirotope(Polytope P)
Compute the chirotope of a polytope using TOPCOM.
 Parameters:
Polytope
P
 Returns:

chirotope(VectorConfiguration P)
Compute the chirotope of a point or vector configuration using TOPCOM.
 Parameters:
 Returns:

chirotope(Polytope P)
Compute the chirotope of a polytope using polymake's native implementation.
 Parameters:
Polytope
P
 Returns:

chirotope(VectorConfiguration P)
Compute the chirotope of a point or vector configuration using polymake's native implementation.
 Parameters:
 Returns:

coherency_index(Polytope p1, Polytope p2, Matrix points, Vector w1, Vector w2)
DOC_FIXME: Incomprehensible description! Computes the coherency index of w1 w.r.t. w2

coherency_index(Matrix points, Vector w1, Vector w2)
DOC_FIXME: Incomprehensible description! Computes the coherency index of w1 w.r.t. w2

coherency_index(Polytope p1, Polytope p2)
DOC_FIXME: Erroneous description! w1 is not a parameter here! Computes the coherency index of p1 w.r.t. p2

common_refinement(Matrix points, IncidenceMatrix sub1, IncidenceMatrix sub2, Int dim)
Computes the common refinement of two subdivisions of points. It is assumed that there exists a common refinement of the two subdivisions.
 Parameters:
Matrix
points
IncidenceMatrix
sub1
: first subdivisionIncidenceMatrix
sub2
: second subdivisionInt
dim
: dimension of the point configuration Returns:
 Example:
A simple 2dimensional set of points:
> $points = new Matrix<Rational>([[1,0,0],[1,1,0],[1,0,1],[1,1,1],[1,2,1]]);
Two different subdivisions…
> $sub1 = new IncidenceMatrix([[0,1,2],[1,2,3,4]]); > $sub2 = new IncidenceMatrix([[1,3,4],[0,1,2,3]]);
…and their common refinement:
> print common_refinement($points,$sub1,$sub2,2); {0 1 2} {1 3 4} {1 2 3}

common_refinement(Polytope p1, Polytope p2)
Computes the common refinement of two subdivisions of the same polytope p1, p2. It is assumed that there exists a common refinement of the two subdivisions. It is not checked if p1 and p2 are indeed the same!

delaunay_triangulation(VoronoiPolyhedron V)
Compute the Delaunay triangulation of the given
SITES
of a VoronoiPolyhedron V. If the sites are not in general position, the nontriangular facets of the Delaunay subdivision are triangulated (by applying the beneathbeyond algorithm). Parameters:
 Returns:
 Example:
> $VD = new VoronoiPolyhedron(SITES=>[[1,1,1],[1,0,1],[1,1,1],[1,1,1],[1,0,1],[1,1,1]]); > $D = delaunay_triangulation($VD); > print $D; {0 1 3} {1 3 4} {1 2 4} {2 4 5}

fiber_polytope(PointConfiguration pc, PointConfiguration pc)
Computes the fiber polytope of a projection of point configurations P→Q via the GKZ secondary configuration.
 Parameters:
PointConfiguration
pc
: (or Polytope) source point configuration or polytopePointConfiguration
pc
: target point configuration Returns:

fiber_polytope(PointConfiguration pc, Polytope pc)
Computes the fiber polytope of a projection of point configurations P→Q via the GKZ secondary configuration.
 Parameters:
PointConfiguration
pc
: (or Polytope) source point configuration or polytopePolytope
pc
: target polytope Returns:

fiber_polytope(PointConfiguration P, Matrix pi)
Computes the fiber polytope of a projection of point configurations P pi→ Q via the GKZ secondary configuration.
 Parameters:
PointConfiguration
P
: (or Polytope) source point configuration or polytopeMatrix
pi
: the projection acting on P Returns:

foldable_max_signature_ilp(Int d, Matrix points, Rational volume, SparseMatrix cocircuit_equations)
Set up an ILP whose MAXIMAL_VALUE is the maximal signature of a foldable triangulation of a polytope, point configuration or quotient manifold
 Parameters:
Int
d
: the dimension of the input polytope, point configuration or quotient manifoldMatrix
points
: the input points or verticesRational
volume
: the volume of the convex hullSparseMatrix
cocircuit_equations
: the matrix of cocircuit equations Returns:

foldable_max_signature_upper_bound(Int d, Matrix points, Rational volume, SparseMatrix cocircuit_equations)
Calculate the LP relaxation upper bound to the maximal signature of a foldable triangulation of polytope, point configuration or quotient manifold
 Parameters:
Int
d
: the dimension of the input polytope, point configuration or quotient manifoldMatrix
points
: the input points or verticesRational
volume
: the volume of the convex hullSparseMatrix
cocircuit_equations
: the matrix of cocircuit equations Returns:

interior_and_boundary_ridges(Polytope P)
Find the (d1)dimensional simplices in the interior and in the boundary of a ddimensional polytope or cone

is_regular(Matrix points, Array<Set<Int>> subdiv)
For a given subdivision subdiv of points tests if the subdivision is regular and if yes computes a weight vector inducing this subdivsion. The output is a pair of Bool and the weight vector. Options can be used to ensure properties of the resulting vector. The default is having 0 on all vertices of the first face of subdiv.
 Parameters:
Matrix
points
: in homogeneous coordinates Options:
Matrix<Scalar>
equations
: system of linear equation the cone is cut with.Int
lift_face_to_zero
: gives only lifting functions lifting all vertices of the designated face to 0 Returns:
 Example:
A regular subdivision of the square, with the first cell lifted to zero:
> $points = cube(2)>VERTICES; > print is_regular($points,[[0,1,3],[1,2,3]],lift_to_zero=>[0,1,3]); 1 <0 0 1 0>

is_subdivision(Matrix points, Array<Set<Int>> faces)
 Parameters:
Matrix
points
 Options:
 Example:
Two potential subdivisions of the square without inner points:
> $points = cube(2)>VERTICES; > print is_subdivision($points,[[0,1,3],[1,2,3]],interior_points=>[ ]); true
> print is_subdivision($points,[[0,1,2],[1,2]],interior_points=>[ ]); false
 Example:
Three points in RR^1
> $points = new Matrix([[1,0],[1,1],[1,2]]); > print is_subdivision($points, [[0,2]]); true
> print is_subdivision($points, [[0,1]]); false

iterated_barycentric_subdivision(Cone c, Int n)
Create a simplicial complex as an iterated barycentric subdivision of a given cone or polytope.
 Parameters:
Cone
c
: input cone or polytopeInt
n
: how many times to subdivide Options:
Bool
no_labels
: Do not generateVERTEX_LABELS
from the faces of the original cone. default: 0Bool
geometric_realization
: create aGeometricSimplicialComplex
; default is false Returns:

max_interior_simplices(Polytope P)
Find the maximal interior simplices of a polytope P. Symmetries of P are NOT taken into account.

max_interior_simplices(PointConfiguration P)
find the maximal interior simplices of a point configuration Symmetries of the configuration are NOT taken into account.
 Parameters:
PointConfiguration
P
: the input point configuration Returns:
 Example:
To calculate the maximal interior simplices of a point configuration, type
> $p=new PointConfiguration(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,1,1],[1,1/2,1/2]]); > print max_interior_simplices($p); {0 1 2} {0 1 3} {0 1 4} {0 2 3} {0 2 4} {1 2 3} {1 3 4} {2 3 4}

mixed_volume(Polytope<Scalar> P1, Polytope<Scalar> P2, Polytope<Scalar> Pn)
Produces the mixed volume of polytopes P_{1},P_{2},…,P_{n}.

n_fine_triangulations(Matrix M, Bool optimization)
Calculates the number of fine triangulations of a planar point configuration. This can be space intensive. Victor Alvarez, Raimund Seidel: A Simple Aggregative Algorithm for Counting Triangulations of Planar Point Sets and Related Problems. In Proc. of the 29th Symposium on Computational Geometry (SoCG '13), pages 18, Rio de Janeiro, Brazil, 2013

placing_triangulation(Matrix Points)
Compute the placing triangulation of the given point set using the beneathbeyond algorithm.
 Parameters:
Matrix
Points
: the given point set Options:
Bool
non_redundant
: whether it's already known that Points are nonredundant Returns:
 Example:
To compute the placing triangulation of the square (of whose vertices we know that they're nonredundant), do this:
> $t = placing_triangulation(cube(2)>VERTICES, non_redundant=>1); > print $t; {0 1 2} {1 2 3}

points2metric(Matrix points)
Define a metric by restricting the Euclidean distance function to a given set of points. Due to floating point computations (sqrt is used) the metric defined may not be exact. If the option max or l1 is set to true the maxnorm or l1norm is used instead (with exact computation).

poly2metric(Polytope P)
Define a metric by restricting the Euclidean distance function to the vertex set of a given polytope P. Due to floating point computations (sqrt is used) the metric defined may not be exact. If the option max or l1 is set to true the maxnorm or l1norm is used instead (with exact computation).

positive_circuits(Polytope or, Set<Int> S)
returns all sets of points that form a circuit with the given list of points
 Parameters:
Polytope
or
: PointConfiguration P Returns:

quotient_space_simplexity_ilp(Int d, Matrix V, Scalar volume, SparseMatrix cocircuit_equations)
Set up an LP whose MINIMAL_VALUE is a lower bound for the minimal number of simplices needed to triangulate a polytope, point configuration or quotient manifold
 Parameters:
Int
d
: the dimension of the input polytope, point configuration or quotient manifoldMatrix
V
: the input points or verticesScalar
volume
: the volume of the convex hullSparseMatrix
cocircuit_equations
: the matrix of cocircuit equations Options:
String
filename
: a name for a file in .lp format to store the linear program Returns:

quotient_space_simplexity_lower_bound(Int d, Matrix V, Scalar volume, SparseMatrix cocircuit_equations)
Calculate a lower bound for the minimal number of simplices needed to triangulate a polytope, point configuration or quotient manifold
 Parameters:
Int
d
: the dimension of the input polytope, point configuration or quotient manifoldMatrix
V
: the input points or verticesScalar
volume
: the volume of the convex hullSparseMatrix
cocircuit_equations
: the matrix of cocircuit equations Returns:

regular_subdivision(Matrix points, Vector weights)
Compute a regular subdivision of the polytope obtained by lifting points to weights and taking the lower complex of the resulting polytope. If the weight is generic the output is a triangulation.

regularity_lp(Matrix points, Array<Set<Int>> subdiv)
For a given subdivision subdiv of points determines a LinearProgram to decide whether the subdivision is regular. The output a Polytope with an attached LP. Options can be used to ensure properties of the resulting LP. The default is having 0 on all vertices of the first face of subdiv.
 Parameters:
Matrix
points
: in homogeneous coordinates Options:
Matrix<Scalar>
equations
: system of linear equation the cone is cut with.Int
lift_face_to_zero
: gives only lifting functions lifting all vertices of the designated face to 0Scalar
epsilon
: minimum distance from all inequalities Returns:
Polytope<Scalar>

secondary_polytope(PointConfiguration pc)
Computes the GKZ secondary configuration of a point configuration via its chirotope.
 Parameters:
PointConfiguration
pc
: input point configuration Returns:

secondary_polytope(Polytope pc)
Computes the GKZ secondary configuration of a point configuration via its chirotope.
 Parameters:
Polytope
pc
: input polytope Returns:

simplexity_ilp(Int d, Matrix points, Array<Set> MIS, Scalar volume, SparseMatrix cocircuit_equations)
Set up an ILP whose MINIMAL_VALUE is the minimal number of simplices needed to triangulate a polytope, point configuration or quotient manifold
 Parameters:
Int
d
: the dimension of the input polytope, point configuration or quotient manifoldMatrix
points
: the input points or verticesScalar
volume
: the volume of the convex hullSparseMatrix
cocircuit_equations
: the matrix of cocircuit equations Returns:

simplexity_ilp_with_angles(Int d, Matrix V, Matrix F, IncidenceMatrix VIF, IncidenceMatrix VIR, Array<Array<Int>> gens, Array<Set> MIS, Scalar volume, SparseMatrix cocircuit_equations)
Set up an ILP whose MINIMAL_VALUE is the minimal number of simplices needed to triangulate a polytope, point configuration or quotient manifold
 Parameters:
Int
d
: the dimension of the input polytope, point configuration or quotient manifoldMatrix
V
: the input points or verticesMatrix
F
: the facets of the input polytopeIncidenceMatrix
VIF
: the verticesinfacets incidence matrixIncidenceMatrix
VIR
: the verticesinridges incidence matrixScalar
volume
: the volume of the convex hullSparseMatrix
cocircuit_equations
: the matrix of cocircuit equations Returns:

simplexity_lower_bound(Int d, Matrix points, Scalar volume, SparseMatrix cocircuit_equations)
Calculate the LP relaxation lower bound for the minimal number of simplices needed to triangulate a polytope, point configuration or quotient manifold
 Parameters:
Int
d
: the dimension of the input polytope, point configuration or quotient manifoldMatrix
points
: the input points or verticesScalar
volume
: the volume of the convex hullSparseMatrix
cocircuit_equations
: the matrix of cocircuit equations Returns:

split_compatibility_graph(Matrix splits, Polytope P)
DOC_FIXME: Incomprehensible description! Computes the compatibility graph among the splits of a polytope P.

split_polyhedron(Polytope P)
Computes the split polyhedron of a fulldimensional polyhdron P.
 Parameters:
Polytope
P
 Returns:

splits(Matrix V, Graph G, Matrix F, Int dimension)
Computes the SPLITS of a polytope. The splits are normalized by dividing by the first nonzero entry. If the polytope is not fulldimensional the first entries are set to zero unless coords are specified.

splits_in_subdivision(Matrix vertices, Array<Set<Int>> subdivision, Matrix splits)
Tests which of the splits of a polyhedron are coarsenings of the given subdivision.

staircase_weight(Int k, Int l)
Gives a weight vector for the staircase triangulation of the product of a k1 and an l1dimensional simplex.
 Parameters:
Int
k
: the number of vertices of the first simplexInt
l
: the number of vertices of the second simplex Returns:
 Example:
The following creates the staircase triangulation of the product of the 2 and the 1simplex.
> $w = staircase_weight(3,2); > $p = product(simplex(2),simplex(1)); > $p>POLYTOPAL_SUBDIVISION(WEIGHTS=>$w); > print $p>POLYTOPAL_SUBDIVISION>MAXIMAL_CELLS; {0 2 4 5} {0 2 3 5} {0 1 3 5}

symmetrized_foldable_max_signature_ilp(Int d, Matrix points, Rational volume, Array<Array<Int>> generators, SparseMatrix symmetrized_foldable_cocircuit_equations)
Set up an ILP whose MAXIMAL_VALUE is the maximal signature of a foldable triangulation of a polytope, point configuration or quotient manifold
 Parameters:
Int
d
: the dimension of the input polytope, point configuration or quotient manifoldMatrix
points
: the input points or verticesRational
volume
: the volume of the convex hullSparseMatrix
symmetrized_foldable_cocircuit_equations
: the matrix of symmetrized cocircuit equations Returns:

symmetrized_foldable_max_signature_upper_bound(Int d, Matrix points, Rational volume, SparseMatrix cocircuit_equations)
Calculate the LP relaxation upper bound to the maximal signature of a foldable triangulation of polytope, point configuration or quotient manifold
 Parameters:
Int
d
: the dimension of the input polytope, point configuration or quotient manifoldMatrix
points
: the input points or verticesRational
volume
: the volume of the convex hullSparseMatrix
cocircuit_equations
: the matrix of cocircuit equations Returns:

topcom_all_triangulations(PointConfiguration pc)
Computes all triangulations of a point configuration via its chirotope.
 Parameters:
PointConfiguration
pc
: input point configuration Returns:

topcom_fine_and_connected_triangulations(PointConfiguration pc)
Computes all fine triangulations of a point configuration that are connected by bistellar flips to a fine seed triangulation. The triangulations are computed via the chirotope. If the input point configuration or polytope has a symmetry group, only fine triangulations up to symmetry will be computed.
 Parameters:
PointConfiguration
pc
: or Polytope p input point configuration or polytope Returns:

topcom_input_format(Cone P)
This converts a polytope, cone or point configuration into a format that topcom understands
 Parameters:
Cone
P
: (or PointConfiguration) Returns:
 Example:
To convert a 2cube without symmetries into topcom format, type
> print topcom_input_format(cube(2)); [[1,1,1],[1,1,1],[1,1,1],[1,1,1]] []
If you also want the symmetry group, you can type
> print topcom_input_format(cube(2,group=>1)); [[1,1,1],[1,1,1],[1,1,1],[1,1,1]] [[1,0,3,2],[0,2,1,3]]

topcom_regular_and_connected_triangulations(PointConfiguration pc)
Computes all triangulations of a point configuration that are connected by bistellar flips to the regular triangulations. The triangulations are computed via the chirotope. If the input point configuration or polytope has a symmetry group, only triangulations up to symmetry will be computed.
 Parameters:
PointConfiguration
pc
: or Polytope p input point configuration or polytope Returns:

universal_polytope<Scalar>(PointConfiguration<Scalar> PC)
Calculate the universal polytope of a point configuration A. It is a 0/1 polytope with one vertex for every triangulation of A. Each coordinate of the ambient space corresponds to a simplex in the configuration.
 Type Parameters:
Scalar
: the underlying number type Parameters:
PointConfiguration<Scalar>
PC
: the point configuration Returns:
 Example:
To calculate the universal polytope of a point configuration, type
> $p=new PointConfiguration(POINTS=>[[1,0,0],[1,1,0],[1,0,1],[1,1,1],[1,1/2,1/2]]); > print universal_polytope($p)>VERTICES; 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1
Notice how the first vertex corresponds to the triangulation using all points, and the other ones to the triangulations that don't use the inner point.

universal_polytope(Polytope P)
Calculate the universal polytope U(P) of an input polytope P. If P has n vertices and dimension d, then U(P) is a 0/1polytope in dimension binomial(n,d+1) whose vertices correspond to the full triangulations of P. Each coordinate of a particular vertex v indicates the presence or absence of a particular simplex in the triangulation corresponding to v, and the order of the simplices (and hence the interpretation of each coordinate of v) is the one given by the property MAX_INTERIOR_SIMPLICES. Because the number of triangulations of P is typically very large, the polytope U(P) is not constructed by enumerating triangulations, but instead in the inequality description afforded by the cocircuit equations, a volume equality, and nonnegativity constraints for the coordinates.
 Parameters:
Polytope
P
: the input polytope Returns:
 Example:
Since the 2dimensional cube (i.e., the square) has just two triangulations, its universal polytope is a segment embedded in dimension binomial(4,3) = 4. The cocircuit equations read as follows:
> print universal_polytope(cube(2))>EQUATIONS; 8 4 4 4 4 (5) (2 1) (3 1) (5) (1 1) (4 1)

universal_polytope(Polytope P, Array<Set> reps, SparseMatrix cocircuit_equations)
Calculate the universal polytope of a polytope, point configuration or quotient manifold
 Parameters:
Polytope
P
: the input polytopeSparseMatrix
cocircuit_equations
: the matrix of cocircuit equations Returns:
Visualization
These functions are for visualization.

bounding_box_facets(Matrix V)
Produces boundary facets describing a box shaped polytope that contains all bounded vertices in V.
 Parameters:
Matrix
V
: vertices that should be in the box Options:
Scalar
offset
: the minimum offset between a bounding box facet and its nearest bounded vertexScalar
surplus_k
: size of the bounding box relative to the box spanned by V (added to offset)Bool
fulldim
: keeps the bounding box full dimensional even if the bounded vertices do not span the whole space and offset is zero. Useful for visualizations of Voronoi diagrams that do not have enough vertices. Default value is 0.Bool
make_cube
 Returns:

bounding_facets(Matrix H, Matrix V)
A function that turns a giving Hdescription into one that can be used as bounding facets for a given set of vertices.
 Parameters:
Matrix
H
: Hdescription of some bounded polytope PMatrix
V
: vertices of which the bounded ones will be contained in P Options:
Scalar
offset
: the minimum euclidean distance between a hyperplane and a bounded vertex. Default is 0Scalar
surplus_k
: factor multiplied with $ max(<f,v>  v in V)  min(<f,v>  v in V) $ to describe the minimum offset relative to the extents of V in f direction (added to offset)Bool
transform
: instead of simply shifting the facets. For P simplicial/(and simple?) this should produce the same as the LP and can be turned off. Default is trueBool
fulldim
: keep P full dimensional. Default is falseBool
return_nonredundant
: (shifted) hyperplanes only. If transform is true there will be no check. Regardless of this variable. Default is true Returns:

vlabels(Matrix vertices, Bool wo_zero)
Creates vertex labels for visualization from the vertices of the polytope. The parameter wo_zero decides whether the entry at position 0 (homogenizing coordinate) is omitted (1) or included (0) in the label string.“
Other
Special purpose functions.

edge_orientable(Polytope P)
Checks whether a 2cubical polytope P is edgeorientable (in the sense of Hetyei), that means that there exits an orientation of the edges such that for each 2face the opposite edges point in the same direction. It produces the certificates
EDGE_ORIENTATION
if the polytope is edgeorientable, orMOEBIUS_STRIP_EDGES
otherwise. In the latter case, the output can be checked with the clientvalidate_moebius_strip
. Parameters:
Polytope
P
: the given 2cubical polytope

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

lawrence_matrix(Matrix M)
 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:

m_sequence(Vector<Int> h)
Test if the given vector is an Msequence.
 Parameters:
 Returns:
 Example:
The hvector of a simplicial or simple polytope is an Msequence.
> print m_sequence(cyclic(7,23)>H_VECTOR); true

matroid_indices_of_hypersimplex_vertices()
For a given matroid return the bases as a subset of the vertices of the hypersimplex
 Options:
Matroid
m
: the matroid Returns:

pseudopower(Integer l, Int i)
Compute the ith pseudopower of l, commonly denoted l^<i>. See “A Proof of the Sufficiency of McMullen’s Conditions of Simplicial Convex Polytopes” by Louis Billera and Carl Lee, DOI: 10.1016/00973165(81)900583, for the definition.

wronski_center_ideal(Matrix<Int> L, Vector<Int> 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:

wronski_polynomial(Matrix<Int> M, Vector<Int> lambda, Array<Rational> coeff, Rational s)
Returns a Wronski polynomial of a
FOLDABLE
triangulation of a lattice polytope Parameters:
Rational
s
: additional Parameter in the polynomial Options:
SimplicialComplex
triangulation
: The triangulation of the pointset corresponding to the lifting function

wronski_system(Matrix<Int> M, Vector<Int> lambda, Array<Array<Rational>> coeff_array, Rational s)
Returns a Wronski system of a
FOLDABLE
triangulation of a lattice polytope Parameters:
Rational
s
: additional Parameter in the polynomial Options:
SimplicialComplex
triangulation
: The triangulation of the pointset corresponding to the lifting function