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Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2

BigObject Polytope<Scalar>

from application polytope

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

Type Parameters:

Scalar: inherited from Scalar

derived from:
Specializations:

Polytope::Lattice: A polytope all of whose vertex coordinates are integral.

Polytope<Float>: A pointed polyhedron with float coordinates realized in Rd. It mainly exists for visualization. Convex hull and related algorithms use floating-point arithmetics. Due to numerical errors inherent to this kind of computations, the resulting combinatorial description can be arbitrarily far away from the truth, or even not correspond to any valid polytope. You have been warned. None of the standard construction clients produces objects of this type. If you want to get one, create it with the explicit constructor or convert_to.

Polytope<Rational>: A rational polyhedron realized in Q^d

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

Example:

To construct a polytope as the convex hull of three points in the plane use

> $p=new Polytope(POINTS=>[[1,0,0],[1,1,0],[1,0,1]]); > print$p->N_FACETS
3

Note that homogeneous coordinates are used throughout.

Example:

Many standard constructions are available directly. For instance, to get a regular 120-cell (which is 4-dimensional) use:

> $c=regular_120_cell(); > print$c->VOLUME;
1575+705r5

This is the exact volume 1575+705*\sqrt{5}. polymake has limited support for polytopes with non-rational coordinates.

Input property

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

EQUATIONS

Equations that hold for all points of the polyhedron. A vector (A0, A1, …, Ad) describes the hyperplane of all points (1, x1, …, xd) such that A0 + A1 x1 + … + Ad xd = 0. All vectors in this section must be non-zero. Input section only. Ask for AFFINE_HULL if you want to see an irredundant description of the affine span.

Type:

INEQUALITIES

Inequalities that describe half-spaces such that the polyhedron is their intersection. Redundancies are allowed. Dual to POINTS. A vector (A0, A1, …, Ad) defines the (closed affine) half-space of points (1, x1, …, xd) such that A0 + A1 x1 + … + Ad xd >= 0. Input section only. Ask for FACETS and AFFINE_HULL if you want to compute an H-representation from a V-representation.

Type:

POINTS

Points such that the polyhedron is their convex hull. Redundancies are allowed. The vector (x0, x1, … xd) represents a point in d-space given in homogeneous coordinates. Affine points are identified by x0 > 0. Points with x0 = 0 can be interpreted as rays. polymake automatically normalizes each coordinate vector, dividing them by the first non-zero element. The clients and rule subroutines can always assume that x0 is either 0 or 1. All vectors in this section must be non-zero. Dual to INEQUALITIES. Input section only. Ask for VERTICES if you want to compute a V-representation from an H-representation. Alias for property INPUT_RAYS.

Type:
Example:

Given some (homogeneous) points in 3-space we first construct a matrix containing them. Assume we don't know wether these are all vertices of their convex hull or not. To safely produce a polytope from these points, we set the input to the matrix representing them. In the following the points under consideration are the vertices of the 3-simplex together with their barycenter, which will be no vertex:

> $M = new Matrix([[1,0,0,0],[1,1,0,0],[1,0,1,0],[1,0,0,1],[1,1/4,1/4,1/4]]); >$p = new Polytope(POINTS=>$M); > print$p->VERTICES;
1 0 0 0
1 1 0 0
1 0 1 0
1 0 0 1

Combinatorics

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

BALANCE

Maximal dimension in which all facets are balanced.

Type:
Example:

The following full dimensional polytope given by 10 specific vertices on the 8-dimensional sphere is 3-neighborly. Hence the dual polytope is 3-balanced, where we first center and then polarize it.

> $p = rand_sphere(8,10,seed=>8866463); >$q = polarize(center($p)); > print$q->BALANCE;
3

BALANCED

Dual to NEIGHBORLY.

Type:
Example:

Since cyclic polytopes generated by vertices on the moment curve are neighborly, their dual polytopes are balanced. The following checks this for the 4-dimensional case by centering the cyclic polytope and then polarizing it:

> $p = cyclic(4,6); >$q = polarize(center($p)); > print$q->BALANCED;
true

CD_INDEX_COEFFICIENTS

Coefficients of the cd-index.

Type:

COCUBICAL

Dual to CUBICAL.

Type:
Example:

Since the cross-polytope is dual to a cube of same dimension, it is cocubical. The following checks this for the 3-dimensional case:

> print cross(3)->COCUBICAL;
true

COCUBICALITY

Dual to CUBICALITY.

Type:
Example:

After stacking a facet of the 3-dimensional cube, its cubicality is lowered to 2. Hence its dual polytope has cocubicality 2 as well. The following produces such a stacked cube and asks for its cocubicality after polarization:

> $p = stack(cube(3),5); > print polarize($p)->COCUBICALITY;
2

COMPLEXITY

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

Type:

CUBICAL

True if all facets are cubes.

Type:
Example:

A k-dimensional cube has k-1-dimensional cubes as facets and is therefore cubical. The following checks if this holds for the 3-dimensional case:

> print cube(3)->CUBICAL;
true
Example:

This checks if a zonotope generated by 4 random points on the 3-dimensional sphere is cubical, which is always the case.

> print zonotope(rand_sphere(3,4)->VERTICES)->CUBICAL;
true

CUBICALITY

Maximal dimension in which all facets are cubes.

Type:
Example:

We will modify the 3-dimensional cube in two different ways. While stacking some facets (in this case facets 4 and 5) preserves the cubicality up to dimension 2, truncating an arbitrary vertex reduces the cubicality to 1.

> print stack(cube(3),[4,5])->CUBICALITY;
2
> print truncation(cube(3),5)->CUBICALITY;
1

CUBICAL_H_VECTOR

Cubical h-vector. Defined for cubical polytopes.

Type:

DUAL_BOUNDED_H_VECTOR

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

Type:

DUAL_GRAPH
Type:
Properties of DUAL_GRAPH:
DIHEDRAL_ANGLES

Dihedral angles (in radians) between the two facets corresponding to each edge of the dual graph, i.e. the ridges of the polytope.

Type:

DUAL_H_VECTOR

dual h-vector, defined via recursion on the face lattice of a polytope. Coincides for simple polytopes with the combinatorial definition of the h-vector via abstract objective functions.

Type:

EDGE_ORIENTABLE

True if there exists an edge-orientation (see EDGE_ORIENTATION for a definition). The polytope is required to be 2-cubical.

Type:
Example:

The following checks a 3-dimensional cube for edge orientability:

> $p = cube(3); > print$p->EDGE_ORIENTABLE;
true
Example:

A 3-dimensinal cube with one stacked facet is still 2-cubical. Therefore we can check for edge orientability:

> $p = stack(cube(3),5); > print$p->EDGE_ORIENTABLE;
true

EDGE_ORIENTATION

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

Type:
Example:

The following prints a list of oriented edges of a 2-dimensional cube such that opposing edges have the same orientation:

> $p = cube(2); > print$p->EDGE_ORIENTATION;
0 2
1 3
0 1
2 3

EXCESS_VERTEX_DEGREE

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

Type:
Example:

The excess vertex degree of an egyptian pyramid is one.

> print pyramid(cube(2))->EXCESS_VERTEX_DEGREE;
1

F2_VECTOR

fik is the number of incident pairs of i-faces and k-faces; the main diagonal contains the F_VECTOR.

Type:
Example:

The following prints the f2-vector of a 3-dimensional cube:

> print cube(3)->F2_VECTOR;
8 24 24
24 12 24
24 24 6

FACETS_THRU_VERTICES

transposed VERTICES_IN_FACETS Notice that this is a temporary property; it will not be stored in any file. Alias for property FACETS_THRU_RAYS.

Type:

FACE_SIMPLICITY

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

Type:

FATNESS

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

Type:

FOLDABLE_MAX_SIGNATURE_UPPER_BOUND

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

Type:

F_VECTOR

fk is the number of k-faces.

Type:
Example:

This prints the f-vector of a 3-dimensional cube. The first entry represents the vertices.

> print cube(3)->F_VECTOR;
8 12 6
Example:

This prints the f-vector of the 3-dimensional cross-polytope. Since the cube and the cross polytope of equal dimension are dual, their f-vectors are the same up to reversion.

> print cross(3)->F_VECTOR;
6 12 8
Example:

After truncating the first standard basis vector of the 3-dimensional cross-polytope the f-vector changes. Only segments of the incident edges of the cut off vertex remain and the intersection of these with the new hyperplane generate four new vertices. These also constitute four new edges and a new facet.

> print truncation(cross(3),4)->F_VECTOR;
9 16 9

GRAPH
Type:
Properties of GRAPH:
EDGE_DIRECTIONS

Difference of the vertices for each edge (only defined up to signs).

Type:
SQUARED_EDGE_LENGTHS

Squared Euclidean length of each edge

Type:
LATTICE_ACCUMULATED_EDGE_LENGTHS

a map associating to each edge length of the polytope the number of edges with this length the lattice edge length of an edge is one less than the number of lattice points on that edge

Type:
LATTICE_EDGE_LENGTHS

the lattice lengths of the edges of the polytope i.e. for each edge one less than the number of lattice points on that edge

Type:

G_VECTOR

(Toric) g-vector, defined via the (generalized) h-vector as gi = hi - hi-1.

Type:

HASSE_DIAGRAM
Type:

H_VECTOR

h-vector, defined via recursion on the face lattice of a polytope. Coincides for simplicial polytopes with the combinatorial definition of the h-vector via shellings

Type:

MINIMAL_NON_FACES

Minimal non-faces of a SIMPLICIAL polytope.

Type:

MOEBIUS_STRIP_EDGES

Ordered list of edges of a Moebius strip with parallel interior edges. Consists of k lines of the form (vi wi), for i=1, …, k. The Moebius strip in question is given by the quadrangles (vi, wi, wi+1,vi+1), for i=1, …, k-1, and the quadrangle (v1, w1, vk, wk). Validity can be verified with the client validate_moebius_strip. The polytope is required to be 2-cubical.

Type:

Unordered list of quads which forms a Moebius strip with parallel interior edges. Each line lists the vertices of a quadrangle in cyclic order. Validity can be verified with the client validate_moebius_strip_quads. The polytope is required to be 2-cubical.

Type:

NEIGHBORLINESS

Maximal dimension in which all facets are neighborly.

Type:
Example:

This determines that the full dimensional polytope given by 10 specific vertices on the 8-dimensional sphere is 3-neighborly, i.e. all 3-dimensional faces are tetrahedra. Hence the polytope is not neighborly.

> print rand_sphere(8,10,seed=>8866463)->NEIGHBORLINESS;
3

NEIGHBORLY

True if the polytope is neighborly.

Type:
Example:

This checks the 4-dimensional cyclic polytope with 6 points on the moment curve for neighborliness, i.e. if it is ⌊dim/2⌋ neighborly:

> print cyclic(4,6)->NEIGHBORLY;
true

N_VERTEX_FACET_INC

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

Type:

N_VERTICES

Number of VERTICES. Alias for property N_RAYS.

Type:
Example:

The following prints the number of vertices of a 3-dimensional cube:

> print cube(3)->N_VERTICES;
8
Example:

The following prints the number of vertices of the convex hull of 10 specific points lying in the unit square [0,1]^2:

> print rand_box(2,10,1,seed=>4583572)->N_VERTICES;
4

SELF_DUAL

True if the polytope is self-dual.

Type:
Example:

The following checks if the centered square with side length 2 is self dual:

> print cube(2)->SELF_DUAL;
true
Example:

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

> print johnson_solid(8)->SELF_DUAL;
true

SIMPLE

True if the polytope is simple. Dual to SIMPLICIAL.

Type:
Example:

This determines if a 3-dimensional cube is simple or not:

> print cube(3)->SIMPLE;
true

SIMPLEXITY_LOWER_BOUND

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

Type:

SIMPLICIAL

True if the polytope is simplicial.

Type:
Example:

A polytope with random vertices uniformly distributed on the unit sphere is simplicial. The following checks this property and prints the result for 8 points in dimension 3:

> print rand_sphere(3,8)->SIMPLICIAL;
true

SIMPLICIALITY

Maximal dimension in which all faces are simplices.

Type:
Example:

The 3-dimensional cross-polytope is simplicial, i.e. its simplicity is 2. After truncating an arbitrary vertex the simplicity is reduced to 1.

> print cross(3)->SIMPLICIALITY;
2
> print truncation(cross(3),4)->SIMPLICIALITY;
1

SIMPLICITY

Maximal dimension in which all dual faces are simplices.

Type:
Example:

This checks the 3-dimensional cube for simplicity. Since the cube is dual to the cross-polytope of equal dimension and all its faces are simplices, the result is 2.

> print cube(3)->SIMPLICITY;
2

SUBRIDGE_SIZES

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

Type:

TWO_FACE_SIZES

Lists for each occurring size (= number of incident vertices or edges) of a 2-face how many there are.

Type:
Example:

This prints the number of facets spanned by 3,4 or 5 vertices a truncated 3-dimensional cube has.

> $p = truncation(cube(3),5); > print$p->TWO_FACE_SIZES;
{(3 1) (4 3) (5 3)}

VERTEX_SIZES

Number of incident facets for each vertex. Alias for property RAY_SIZES.

Type:
Example:

The following prints the number of incident facets for each vertex of the elongated pentagonal pyramid (Johnson solid 9)

> print johnson_solid(9)->VERTEX_SIZES;
5 4 4 4 4 4 3 3 3 3 3

VERTICES_IN_FACETS

Vertex-facet incidence matrix, with rows corresponding to facets and columns to vertices. Vertices and facets are numbered from 0 to N_VERTICES-1 rsp. N_FACETS-1, according to their order in VERTICES rsp. FACETS. This property is at the core of all combinatorial properties. It has the following semantics: (1) The combinatorics of an unbounded and pointed polyhedron is defined to be the combinatorics of the projective closure. (2) The combiantorics of an unbounded polyhedron which is not pointed is defined to be the combinatorics of the quotient modulo the lineality space. Therefore: VERTICES_IN_FACETS and each other property which is grouped under “Combinatorics” always refers to some polytope. Alias for property RAYS_IN_FACETS.

Type:
Example:

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

> print n_gon(5)->VERTICES_IN_FACETS;
{1 2}
{2 3}
{3 4}
{0 4}
{0 1}
Example:

The following prints the Vertex_facet incidence matrix of the standard 3-simplex together with the facet numbers:

> print rows_numbered(simplex(3)->VERTICES_IN_FACETS);
0:1 2 3
1:0 2 3
2:0 1 3
3:0 1 2

Geometry

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

AFFINE_HULL

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

Type:

BOUNDED

True if and only if LINEALITY_SPACE trivial and FAR_FACE is trivial.

Type:
Example:

A pyramid over a square is bounded. Removing the base square yields an unbounded pointed polyhedron (the vertices with first entry equal to zero correspond to rays).

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

CENTERED_ZONOTOPE

is the zonotope calculated from ZONOTOPE_INPUT_POINTS or ZONOTOPE_INPUT_VECTORS to be centered at the origin? The zonotope is always calculated as the Minkowski sum of all segments conv {x,v}, where

• v ranges over the ZONOTOPE_INPUT_POINTS or ZONOTOPE_INPUT_VECTORS, and
• x = -v if CENTERED_ZONOTOPE = 1,
• x = 0 if CENTERED_ZONOTOPE = 0.

Input section only.

Type:

CENTRALLY_SYMMETRIC

True if P = -P.

Type:
Example:

A centered 3-cube is centrally symmetric. By stacking a single facet (5), this property is lost. We can recover it by stacking the opposing facet (4) as well.

> $p = cube(3); > print$p->CENTRALLY_SYMMETRIC;
true
> print stack($p,5)->CENTRALLY_SYMMETRIC; false > print stack($p,new Set<Int>(4,5))->CENTRALLY_SYMMETRIC;
true

CENTROID

Centroid (center of mass) of the polytope.

Type:
Vector<Scalar>

CONE_AMBIENT_DIM

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

Type:

CONE_DIM

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

Type:
Example:

This prints the cone dimension of a 3-cube. Since the dimension of its affine closure is 3, the result is 4.

> print cube(3)->CONE_DIM;
4

CS_PERMUTATION

The permutation induced by the central symmetry, if present.

Type:

FACETS_THRU_POINTS

similar to FACETS_THRU_VERTICES, but with POINTS instead of VERTICES Notice that this is a temporary property; it will not be stored in any file. Alias for property FACETS_THRU_INPUT_RAYS.

Type:

FAR_HYPERPLANE

Valid strict inequality for all affine points of the polyhedron.

Type:
Vector<Scalar>

FEASIBLE

True if the polyhedron is not empty.

Type:

GALE_TRANSFORM

Coordinates of the Gale transform.

Type:

INEQUALITIES_THRU_VERTICES

transposed VERTICES_IN_INEQUALITIES Alias for property INEQUALITIES_THRU_RAYS.

Type:

LATTICE

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

Type:

MINIMAL_VERTEX_ANGLE

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

Type:

MINKOWSKI_CONE

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

Type:

N_01POINTS

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

Type:
depends on extension:

N_POINTS

Number of POINTS. Alias for property N_INPUT_RAYS.

Type:

ONE_VERTEX

A vertex of a pointed polyhedron. Alias for property ONE_RAY.

Type:
Vector<Scalar>
Example:

This prints the first vertex of the 3-cube (corresponding to the first row in the vertex matrix).

> print cube(3)->ONE_VERTEX;
1 -1 -1 -1

POINTED

True if the polyhedron does not contain an affine line.

Type:
Example:

A square does not contain an affine line and is therefore pointed. Removing one facet does not change this, although it is no longer bounded. After removing two opposing facets, it contains infinitely many affine lines orthogonal to the removed facets.

> $p = cube(2); > print$p->POINTED;
true
> print facet_to_infinity($p,0)->POINTED; true > print new Polytope(INEQUALITIES=>$p->FACETS->minor([0,1],All))->POINTED;
false

POINTS_IN_FACETS

Similar to VERTICES_IN_FACETS, but with columns corresponding to POINTS instead of VERTICES. This property is a byproduct of convex hull computation algorithms. It is discarded as soon as VERTICES_IN_FACETS is computed. Alias for property INPUT_RAYS_IN_FACETS.

Type:

QUOTIENT_SPACE

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

Type:

SLACK_IDEAL

The slack ideal of the polytope as described in > João Gouveia, Antonio Macchia, Rekha R. Thomas, Amy Wiebe: > The Slack Realization Space of a Polytope > (https://arxiv.org/abs/1708.04739) Since saturation is computationally expensive, a non-saturated version of the ideal is stored in the property NON_SATURATED. Asking for GENERATORS will do the saturation.

Type:
depends on extension:
Example:

> $c = cube(2,1,0); >$I = $c->SLACK_IDEAL; >$I->properties;
type: SlackIdeal

NON_SATURATED
type: Ideal

> $I->NON_SATURATED->properties; type: Ideal N_VARIABLES 8 GENERATORS x_0*x_3*x_5*x_6 - x_1*x_2*x_4*x_7 > print$I->GENERATORS;
x_0*x_3*x_5*x_6 - x_1*x_2*x_4*x_7

SLACK_MATRIX

The slack matrix of the polytope. The (i,j)-th entry is the value of the j-th facet on the i-th vertex. See > João Gouveia, Antonio Macchia, Rekha R. Thomas, Amy Wiebe: > The Slack Realization Space of a Polytope > (https://arxiv.org/abs/1708.04739)

Type:

SPECIAL_FACETS

The following is defined for CENTERED polytopes only: A facet is special if the cone over that facet with the origin as the apex contains the VERTEX_BARYCENTER. Motivated by Obro's work on Fano polytopes.

Type:
Set<Int>

SPLITS

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

Type:

SPLIT_COMPATIBILITY_GRAPH

Two SPLITS are compatible if the defining hyperplanes do not intersect in the interior of the polytope. This defines a graph.

Type:

STEINER_POINT

Steiner point of the whole polytope.

Type:
Vector<Scalar>

STEINER_POINTS

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

Type:

TILING_LATTICE

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

Type:
AffineLattice<Scalar>

VALID_POINT

Some point belonging to the polyhedron.

Type:
Vector<Scalar>
Example:

This stores a (homogeneous) point belonging to the 3-cube as a vector and prints its coordinates:

> $v = cube(3)->VALID_POINT; > print$v;
1 -1 -1 -1

VERTEX_BARYCENTER

The center of gravity of the vertices of a bounded polytope.

Type:
Vector<Scalar>
Example:

This prints the vertex barycenter of the standard 3-simplex:

> print simplex(3)->VERTEX_BARYCENTER;
1 1/4 1/4 1/4

VERTEX_NORMALS

The i-th row is the normal vector of a hyperplane separating the i-th vertex from the others. This property is a by-product of redundant point elimination algorithm. All vectors in this section must be non-zero. Alias for property RAY_SEPARATORS.

Type:
Example:

This prints a matrix in which each row represents a normal vector of a hyperplane seperating one vertex of a centered square with side length 2 from the other ones. The first and the last hyperplanes as well as the second and third hyperplanes are the same up to orientation.

> print cube(2)->VERTEX_NORMALS;
0 1/2 1/2
0 -1/2 1/2
0 1/2 -1/2
0 -1/2 -1/2

VERTICES

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

Type:
Example:

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

> print simplex(2)->VERTICES;
(3) (0 1)
1 1 0
1 0 1
Example:

If we know some points to be vertices of their convex hull, we can store them as rows in a Matrix and construct a new polytope with it. The following produces a 3-dimensioanl pyramid over the standard 2-simplex with the specified vertices:

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

VERTICES_IN_INEQUALITIES

Similar to VERTICES_IN_FACETS, but with rows corresponding to INEQUALITIES instead of FACETS. This property is a byproduct of convex hull computation algorithms. It is discarded as soon as VERTICES_IN_FACETS is computed. Alias for property RAYS_IN_INEQUALITIES.

Type:

VERTICES_IN_RIDGES

Alias for property RAYS_IN_RIDGES.

Type:

WEAKLY_CENTERED

True if (1, 0, 0, …) is contained (possibly in the boundary).

Type:
Example:

The cube [0,1]^3 is only weakly centered, since the origin is on the boundary.

> $p = cube(3,0,0); > print$p->WEAKLY_CENTERED;
true
> print join("\n",@{$p->EHRHART_QUASI_POLYNOMIAL}); 2/9*x^2 + 2/3*x + 1 2/9*x^2 + 2/9*x + 5/9 2/9*x^2 -2/9*x + 5/9 FACET_VERTEX_LATTICE_DISTANCES The entry (i,j) equals the lattice distance of vertex j from facet i. Type: FACET_WIDTH The maximal integral width of the polytope with respect to the facet normals. Type: FACET_WIDTHS The integral width of the polytope with respect to each facet normal. Type: GORENSTEIN The polytope is Gorenstein if a dilation of the polytope is REFLEXIVE up to translation. Type: GORENSTEIN_INDEX If the polytope is GORENSTEIN then this is the multiple such that the polytope is REFLEXIVE. Type: GORENSTEIN_VECTOR If the polytope is GORENSTEIN, then this is the unique interior lattice point in the multiple of the polytope that is REFLEXIVE. Type: GROEBNER_BASIS The Groebner basis for the toric ideal associated to the lattice points in the polytope using any term order. Type: LATTICE_BASIS VERTICES are interpreted as coefficient vectors for this basis given in affine form assumed to the the standard basis if not explicitely specified. Type: LATTICE_CODEGREE COMBINATORIAL_DIM+1-LATTICE_DEGREE or the smallest integer k such that k*P has an interior lattice point. Type: LATTICE_DEGREE The degree of the h*-polynomial or Ehrhart polynomial. Type: LATTICE_EMPTY True if the polytope contains no lattice points other than the vertices. Type: LATTICE_VOLUME The normalized volume of the polytope. Type: LATTICE_WIDTH The minimal integral width of the polytope. Type: LATTICE_WIDTH_DIRECTION One direction which realizes LATTICE_WIDTH of the polytope. Type: NORMAL The polytope is normal if the Hilbert basis of the cone spanned by P x {1} is at height 1. Equivalently points in integral dilates of P are postive integral sums of lattice points of P. Type: depends on extension: POLAR_SMOOTH The lattice polytope is polar to smooth if it is REFLEXIVE and the polar of the polytope (wrt to its interior point) is a SMOOTH lattice polytope. Type: REFLEXIVE True if the polytope and its dual have integral vertices. Type: SMOOTH The polytope is smooth if the associated projective variety is smooth; the determinant of the edge directions is +/-1 at every vertex. Type: SPANNING The polytope is spanning if the lattice points generate the lattice Type: TERMINAL The polytope is terminal if there is exactly one interior lattice point and all other lattice points are vertices. Type: VERY_AMPLE The polytope is very ample if the Hilbert Basis of the cone spanned by the edge-directions of any vertex lies inside the polytope. Type: depends on extension: Lattice points in polytopes These properties capture information that depends on the lattice structure of the polytope. polymake always works with the integer lattice. BOUNDARY_LATTICE_POINTS The lattice points on the boundary of the polytope, including the vertices. Type: INTERIOR_LATTICE_POINTS The lattice points strictly in the interior of the polytope Type: LATTICE_POINTS_GENERATORS The lattice points generators in the polytope. The output consists of three matrices [P,R,L], where P are lattice points which are contained in the polytope R are rays and L is the lineality. Together they form a description of all lattice points. Every lattice point can be described as p + lambda*R + mu*L where p is a row in P and lambda has only non-negative integral coordinates and mu has arbitrary integral coordinates. Type: depends on extension: N_BOUNDARY_LATTICE_POINTS The number of BOUNDARY_LATTICE_POINTS Type: N_INTERIOR_LATTICE_POINTS The number of INTERIOR_LATTICE_POINTS Type: N_LATTICE_POINTS The number of LATTICE_POINTS Type: Matroid properties Properties which belong to the corresponding (oriented) matroid CHIROTOPE Chirotope corresponding to the VERTICES. TOPCOM format. Type: depends on extension: CIRCUITS Circuits in VECTORS Type: depends on extension: COCIRCUITS Cocircuits in VECTORS Type: depends on extension: Optimization These properties provide tools from linear, integer and dicrete optimization. In particular, linear programs are defined here. LP Linear program applied to the polytope Type: LinearProgram<Scalar> MILP Mixed integer linear program applied to the polytope Type: Symmetry These properties capture information of the object that is concerned with the action of permutation groups. GROUP derived from: Type: Methods of GROUP: REPRESENTATIVE_INEQUALITIES UNDOCUMENTED Properties of GROUP: COORDINATE_ACTION Type: Properties of COORDINATE_ACTION: CP_INDICES The row indices of all core points among the REPRESENTATIVE_CERTIFIERS. Type: Set<Int> NOP_GRAPH The NOP-graph of POINTS_GENERATORS with respect to the GROUP. The nodes of the NOP-graph correspond to the REPRESENTATIVE_CERTIFIERS, which represent the different orbit polytopes contained in the given orbit polytope. Type: N_POINTS_GENERATORS Alias for property N_INPUT_RAYS_GENERATORS. Type: N_REPRESENTATIVE_CERTIFIERS The number of REPRESENTATIVE_CERTIFIERS. Type: N_REPRESENTATIVE_CORE_POINTS Type: N_VERTICES_GENERATORS Alias for property N_RAYS_GENERATORS. Type: POINTS_GENERATORS Alias for property INPUT_RAYS_GENERATORS. Type: REPRESENTATIVE_CERTIFIERS A matrix of representatives of all certifiers for POINTS_GENERATORS with respect to the GROUP. A certifier is an integer point in the given orbit polytope. Note that the representative certifiers must be in the same order as the corresponding nodes in the NOP_GRAPH. Further, the CP_INDICES refer to row indices of this property. Type: REPRESENTATIVE_CORE_POINTS A matrix of representatives of all core points in the given orbit polytope. A core point is an integer point whose orbit polytope is lattice-free (i.e. does not contain integer points besides its vertices). Type: VERTICES_GENERATORS Alias for property RAYS_GENERATORS. Type: MATRIX_ACTION derived from: Type: Properties of MATRIX_ACTION: VERTICES_ORBITS Alias for property VECTORS_ORBITS. Type: POINTS_ACTION Alias for property INPUT_RAYS_ACTION. Type: REPRESENTATIVE_VERTICES Alias for property REPRESENTATIVE_RAYS. Type: SYMMETRIC_FACETS Type: SYMMETRIC_RAYS Type: VERTICES_ACTION Type: Properties of VERTICES_ACTION: SYMMETRIZED_COCIRCUIT_EQUATIONS The cocircuit equations, projected to a certain direct sum of isotypic components Type: Triangulation and volume Everything in this group is defined for BOUNDED polytopes only. MAHLER_VOLUME Mahler volume (or volume product) of the polytope. Defined as the volume of the polytope and the volume of its polar (for BOUNDED, CENTERED and FULL_DIM polytopes only). Often studied for centrally symmetric convex bodies, where the regular cubes are conjectured to be the global minimiers. Type: Scalar Example: The following prints the Mahler volume of the centered 2-cube: > print cube(2)->MAHLER_VOLUME; 8 POLYTOPAL_SUBDIVISION Type: Properties of POLYTOPAL_SUBDIVISION: REFINED_SPLITS The splits that are coarsenings of the subdivision. If the subdivision is regular these form the unique split decomposition of the corresponding weight function. Type: Set<Int> RELATIVE_VOLUME The k-dimensional Euclidean volume of a k-dimensional rational polytope embedded in R^n. This value is obtained by summing the square roots of the entries in SQUARED_RELATIVE_VOLUMES using the function naive_sum_of_square_roots. Since this latter function does not try very hard to compute the real value, you may have to resort to a computer algebra package. The value is encoded as a map collecting the coefficients of various roots encountered in the sum. For example, {(3 1/2),(5 7)} represents sqrt{3}/2 + 7 sqrt{5}. If the output is not satisfactory, please use a symbolic algebra package. Type: Example: The following prints the 2-dimensional volume of a centered square with side length 2 embedded in the 3-space (the result is 4): >$M = new Matrix([1,-1,1,0],[1,-1,-1,0],[1,1,-1,0],[1,1,1,0]);
> $p = new Polytope<Rational>(VERTICES=>$M);
> print \$p->RELATIVE_VOLUME;
{(1 4)}

SQUARED_RELATIVE_VOLUMES

Array of the squared relative k-dimensional volumes of the simplices in a triangulation of a d-dimensional polytope.

Type:
Array<Scalar>

TRIANGULATION
derived from:
Type:
Properties of TRIANGULATION:
GKZ_VECTOR

GKZ-vector

See Chapter 7 in Gelfand, Kapranov, and Zelevinsky:
Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994
Type:
Vector<Scalar>

VOLUME

Volume of the polytope.

Type:
Scalar
Example:

The following prints the volume of the centered 3-dimensional cube with side length 2:

> print cube(3)->VOLUME;
8

Unbounded polyhedra

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

BOUNDED_COMPLEX
Type:
Properties of BOUNDED_COMPLEX:
GRAPH
derived from:
Type:
Properties of GRAPH:
EDGE_COLORS

Each edge indicates the maximal dimension of a bounded face containing it. Mainly used for visualization purposes.

Type:
EDGE_DIRECTIONS

Difference of the vertices for each edge (only defined up to signs).

Type:
EDGE_LENGTHS

The length of each edge measured in the maximum metric.

Type:
TOTAL_LENGTH

Sum of all EDGE_LENGTHS.

Type:
Scalar
VERTEX_MAP

For every row of VERTICES this indicates the corresponding row in the VERTICES of the parent polytope.

Type:

FAR_FACE

Indices of vertices that are rays.

Type:
Set<Int>

N_BOUNDED_VERTICES

Number of bounded vertices (non-rays).

Type:

SIMPLE_POLYHEDRON

True if each bounded vertex of a (possibly unbounded) d-polyhedron has vertex degree d in the GRAPH. The vertex degrees of the vertices on the FAR_FACE do not matter.

Type:

TOWARDS_FAR_FACE

A linear objective function for which each unbounded edge is increasing; only defined for unbounded polyhedra.

Type:
Vector<Scalar>

UNBOUNDED_FACETS

Indices of facets that are unbounded.

Type:
Set<Int>

Visualization

These properties are for visualization.

FACET_LABELS

Unique names assigned to the FACETS, analogous to VERTEX_LABELS.

Type:

FTV_CYCLIC_NORMAL

Reordered transposed VERTICES_IN_FACETS. Dual to VIF_CYCLIC_NORMAL. Alias for property FTR_CYCLIC_NORMAL.

Type:

GALE_VERTICES

Coordinates of points for an affine Gale diagram.

Type:

INEQUALITY_LABELS

Unique names assigned to the INEQUALITIES, analogous to VERTEX_LABELS.

Type:

NEIGHBOR_VERTICES_CYCLIC_NORMAL

Reordered GRAPH. Dual to NEIGHBOR_FACETS_CYCLIC_NORMAL. Alias for property NEIGHBOR_RAYS_CYCLIC_NORMAL.

Type:

POINT_LABELS

Unique names assigned to the POINTS, analogous to VERTEX_LABELS. Alias for property INPUT_RAY_LABELS.

Type:

SCHLEGEL_DIAGRAM

Holds one special projection (the Schlegel diagram) of the polytope.

Type:
SchlegelDiagram<Scalar>

VERTEX_LABELS

Unique names assigned to the VERTICES. If specified, they are shown by visualization tools instead of vertex indices. For a polytope build from scratch, you should create this property by yourself, either manually in a text editor, or with a client program. If you build a polytope with a construction function taking some other input polytope(s), the labels are created the labels automatically except if you call the function with a no_labels option. The exact format of the abels is dependent on the construction, and is described in the corresponding help topic. Alias for property RAY_LABELS.

Type:

VIF_CYCLIC_NORMAL

Reordered VERTICES_IN_FACETS for 2d and 3d-polytopes. Vertices are listed in the order of their appearance when traversing the facet border counterclockwise seen from outside of the polytope. For a 2d-polytope (which is a closed polygon), lists all vertices in the border traversing order. Alias for property RIF_CYCLIC_NORMAL.

Type:

Combinatorics

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

CD_INDEX()

Prettily print the cd-index given in CD_INDEX_COEFFICIENTS

Returns:

Geometry

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

AMBIENT_DIM()

returns the dimension of the ambient space of the polytope

Returns:

DIM()

returns the dimension of the polytope

Returns:

INNER_DESCRIPTION()

Returns the inner description of a Polytope: [V,L] where V are the vertices and L is the lineality space

Returns:
Array<Matrix<Scalar>>

MINKOWSKI_CONE_COEFF(Vector<Rational> coeff)

returns the Minkowski summand of a polytope P given by a coefficient vector to the rays of the MINKOWSKI_CONE.

Parameters:

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

Returns:

MINKOWSKI_CONE_POINT(Vector<Rational> point)

returns the Minkowski summand of a polytope P given by a point in the MINKOWSKI_CONE.

Parameters:

Vector<Rational> point: point in the Minkowski summand cone

Returns:

OUTER_DESCRIPTION()

Returns the outer description of a Polytope: [F,A] where F are the facets and A is the affine hull

Returns:
Array<Matrix<Scalar>>

labeled_vertices(String label …)

Find the vertices by given labels.

Parameters:

String label …: vertex labels

Returns:
Set<Int>

Lattice points in cones

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

EHRHART_POLYNOMIAL_COEFF()

Vector containing the coefficients of the EHRHART_POLYNOMIAL, ordered by increasing degree of the corresponding term.

Returns:

FACET_POINT_LATTICE_DISTANCES(Vector<Rational> v)

Vector containing the distances of a given point v from all facets

Parameters:

Vector<Rational> v: point in the ambient space of the polytope

Returns:

N_LATTICE_POINTS_IN_DILATION(Int n)

The number of LATTICE_POINTS in the n-th dilation of the polytope

Parameters:

Int n: dilation factor

Returns:

POLYTOPE_IN_STD_BASIS(Polytope<Rational> P)

returns a polytope in the integer lattice basis if a LATTICE_BASIS is given

Parameters:

Polytope<Rational> P: polytope

Returns:

Lattice points in polytopes

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

LATTICE_POINTS()

Returns the lattice points in bounded Polytopes.

Returns:

Symmetry

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

VISUAL_NOP(ARRAY colors_ref, ARRAY trans_ref)

Visualizes all (nested) orbit polytopes contained in orb in one picture.

Parameters:

ARRAY colors_ref: the reference to an array of colors

ARRAY trans_ref: the reference to an array of transparency values

VISUAL_NOP_GRAPH(String filename)

Visualizes the NOP-graph of an orbit polytope. Requires 'graphviz' and a Postscript viewer. Produces a file which is to be processed with the program 'dot' from the graphviz package. If 'dot' is installed, the NOP-graph is visualized by the Postscript viewer.

Parameters:

String filename: the filename for the 'dot' file

Triangulation and volume

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

TRIANGULATION_INT_SIGNS()

the orientation of the simplices of TRIANGULATION_INT in the given order of the POINTS

Returns:

Unbounded polyhedra

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

BOUNDED_DUAL_GRAPH

Dual graph of the bounded subcomplex.

BOUNDED_FACETS()

Indices of FACETS that are bounded.

Returns:
Set<Int>

BOUNDED_GRAPH

Graph of the bounded subcomplex.

BOUNDED_HASSE_DIAGRAM

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

BOUNDED_VERTICES()

Indices of VERTICES that are no rays.

Returns:
Set<Int>

Visualization

These methods are for visualization.

GALE()

Generate the Gale diagram of a d-polyhedron with at most d+4 vertices.

Returns:

SCHLEGEL()

Create a Schlegel diagram and draw it.

Options:

Visual::Graph::decorations proj_facet: decorations for the edges of the projection face

option list schlegel_init
Returns:

VISUAL()

Visualize a polytope as a graph (if 1d), or as a solid object (if 2d or 3d), or as a Schlegel diagram (4d).

Options:
option list geometric_options
Returns:

VISUAL_BOUNDED_GRAPH()

Visualize the GRAPH of a polyhedron.

Options:

Int seed: random seed value for the string embedder

Returns:

VISUAL_DUAL()

Visualize the dual polytope as a solid 3-d object. The polytope must be BOUNDED and CENTERED.

Options:
option list geometric_options
Returns:

VISUAL_DUAL_FACE_LATTICE()

Visualize the dual face lattice of a polyhedron as a multi-layer graph.

Options:

Int seed: random seed value for the node placement

Returns:

VISUAL_DUAL_GRAPH()

Visualize the DUAL_GRAPH of a polyhedron.

Options:

Int seed: random seed value for the string embedder

Returns:

VISUAL_FACE_LATTICE()

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

Options:

Int seed: random seed value for the node placement

Returns:

VISUAL_GRAPH()

Visualize the GRAPH of a polyhedron.

Options:

Int seed: random seed value for the string embedder

Returns:

VISUAL_ORBIT_COLORED_GRAPH()

Visualizes the graph of a symmetric cone: All nodes belonging to one orbit get the same color.

Options:
Returns:

write_stl(String filename)

Take a 3-polytope and write ASCII STL output.

Parameters:

String filename

Example:

> dodecahedron()->write_stl("/tmp/dodecahedron.stl");

no category

LATTICE_HOLLOW

UNDOCUMENTED

• documentation/latest/polytope/polytope.txt