BigObject Polytope

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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.

derived from [' Cone']

Specializations:

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

Polytope<Float>: A pointed polyhedron with float coordinates realized in R^d.

It mainly exists for visualization.

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

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

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

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

Examples:

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.

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

H_VECTOR

Type: 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

CUBICAL

Type: Bool
True if all facets are cubes.
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
```

MINIMAL_NON_FACES

Type: Array
Minimal non-faces of a SIMPLICIAL polytope.

GRAPH

Type: Graph
Properties of GRAPH:
- EDGE_DIRECTIONS
  - Type: EdgeMap
  - Difference of the vertices for each edge (only defined up to signs).
- SQUARED_EDGE_LENGTHS
  - Type: EdgeMap
  - Squared Euclidean length of each edge
- LATTICE_ACCUMULATED_EDGE_LENGTHS
  - Type: Map
  - a map associating to each edge length of the polytope the number of edges with this length
    the lattice edge length of an edge is one less than the number of lattice points on that edge
- LATTICE_EDGE_LENGTHS
  - Type: EdgeMap
  - 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

COMPLEXITY

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

G_VECTOR

Type: Vector
(Toric) g-vector, defined via the (generalized) h-vector as g_i = h_i - h_i-1.

COCUBICAL

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

DUAL_GRAPH

Type: Graph
Properties of DUAL_GRAPH:
- DIHEDRAL_ANGLES
  - Type: EdgeMap
  - Dihedral angles (in radians) between the two facets corresponding to
    each edge of the dual graph, i.e. the ridges of the polytope.

EDGE_ORIENTATION

Type: Matrix
List of all edges with orientation, such that for each 2-face the opposite edges point in the same direction.
Each line is of the form (u v), which indicates that the edge {u,v} is oriented from u to v.
The polytope is required to be 2-cubical.
Example:
The following prints a list of oriented edges of a 2-dimensional cube such that opposing edges have the same orientation:
```
 >  $p = cube(2);
 print $p->EDGE_ORIENTATION;
 0 2
 1 3
 0 1
 2 3
```

F2_VECTOR

Type: Matrix
f_ik is the number of incident pairs of i-faces and k-faces; the main diagonal contains the F_VECTOR.
Example:
The following prints the f2-vector of a 3-dimensional cube:
```
 >  print cube(3)->F2_VECTOR;
 8 24 24
 24 12 24
 24 24 6
```

MOEBIUS_STRIP_EDGES

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

EDGE_ORIENTABLE

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

N_VERTICES

Type: Int
Number of VERTICES.
Alias for property N_RAYS.
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
```

SIMPLICIALITY

Type: Int
Maximal dimension in which all faces are simplices.
Example:
The 3-dimensional cross-polytope is simplicial, i.e. its simplicity is 2. After truncating an arbitrary vertex
the simplicity is reduced to 1.
```
 >  print cross(3)->SIMPLICIALITY;
 2
```
```
 >  print truncation(cross(3),4)->SIMPLICIALITY;
 1
```

FOLDABLE_MAX_SIGNATURE_UPPER_BOUND

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

SIMPLE

Type: Bool
True if the polytope is simple. Dual to SIMPLICIAL.
Example:
This determines if a 3-dimensional cube is simple or not:
```
 >  print cube(3)->SIMPLE;
 true
```

SELF_DUAL

Type: Bool
True if the polytope is self-dual.
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
```

COCUBICALITY

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

EXCESS_VERTEX_DEGREE

Type: Int
Measures the deviation of the cone from being simple in terms of the GRAPH.
Alias for property EXCESS_RAY_DEGREE.
Example:
The excess vertex degree of an egyptian pyramid is one.
```
 >  print pyramid(cube(2))->EXCESS_VERTEX_DEGREE;
 1
```

CD_INDEX_COEFFICIENTS

Type: Vector
Coefficients of the cd-index.

FATNESS

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

TWO_FACE_SIZES

Type: Map
Lists for each occurring size (= number of incident vertices or edges) of a 2-face how many there are.
Example:
This prints the number of facets spanned by 3,4 or 5 vertices a truncated 3-dimensional cube has.
```
 >  $p = truncation(cube(3),5);
 print $p->TWO_FACE_SIZES;
 {(3 1) (4 3) (5 3)}
```

BALANCE

Type: Int
Maximal dimension in which all facets are balanced.
Example:
The following full dimensional polytope given by 10 specific vertices on the 8-dimensional sphere is 3-neighborly. Hence the dual polytope is
3-balanced, where we first center and then polarize it.
```
 >  $p = rand_sphere(8,10,seed=>8866463);
 $q = polarize(center($p));
 print $q->BALANCE;
 3
```

SIMPLICITY

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

DUAL_BOUNDED_H_VECTOR

Type: 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.

SUBRIDGE_SIZES

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

SIMPLEXITY_LOWER_BOUND

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

VERTICES_IN_FACETS

Type: IncidenceMatrix
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.
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
```

F_VECTOR

Type: Vector
f_k is the number of k-faces.
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
```

SIMPLICIAL

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

ALTSHULER_DET

Type: Integer
Let M be the vertex-facet incidence matrix, then the Altshuler determinant is
defined as max{det(M &lowast; M^T), det(M^T &lowast; M)}.
Example:
This prints the Altshuler determinant of the built-in pentagonal pyramid (Johnson solid 2):
```
 >  print johnson_solid("pentagonal_pyramid")->ALTSHULER_DET;
 25
```

HASSE_DIAGRAM

Type: Lattice

FACETS_THRU_VERTICES

Type: IncidenceMatrix
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.

FACE_SIMPLICITY

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

NEIGHBORLINESS

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

VERTEX_SIZES

Type: Array
Number of incident facets for each vertex.
Alias for property RAY_SIZES.
Example:
The following prints the number of incident facets for each vertex of the elongated pentagonal pyramid (Johnson solid 9)
```
 >  print johnson_solid(9)->VERTEX_SIZES;
 5 4 4 4 4 4 3 3 3 3 3
```

DUAL_H_VECTOR

Type: 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.

N_VERTEX_FACET_INC

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

NEIGHBORLY

Type: Bool
True if the polytope is neighborly.
Example:
This checks the 4-dimensional cyclic polytope with 6 points on the moment curve for neighborliness, i.e. if it is &lfloor;dim/2&rfloor; neighborly:
```
 >  print cyclic(4,6)->NEIGHBORLY;
 true
```

BALANCED

Type: Bool
Dual to NEIGHBORLY.
Example:
Since cyclic polytopes generated by vertices on the moment curve are neighborly, their dual polytopes are balanced. The following checks this
for the 4-dimensional case by centering the cyclic polytope and then polarizing it:
```
 >  $p = cyclic(4,6);
 $q = polarize(center($p));
 print $q->BALANCED;
 true
```

CUBICAL_H_VECTOR

Type: Vector
Cubical h-vector. Defined for cubical polytopes.

CUBICALITY

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

MOEBIUS_STRIP_QUADS

Type: Matrix
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.

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

CENTRALLY_SYMMETRIC

Type: Bool
True if P = -P.

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

 >  $p = cube(3);
 print $p->CENTRALLY_SYMMETRIC;
 true

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

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

ONE_VERTEX

Type: Vector
A vertex of a pointed polyhedron.
Alias for property ONE_RAY.
Example:
This prints the first vertex of the 3-cube (corresponding to the first row in the vertex matrix).
```
 >  print cube(3)->ONE_VERTEX;
 1 -1 -1 -1
```

INEQUALITIES_THRU_VERTICES

Type: IncidenceMatrix
transposed VERTICES_IN_INEQUALITIES
Alias for property INEQUALITIES_THRU_RAYS.

VERTICES

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

STEINER_POINTS

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

POINTS_IN_FACETS

Type: IncidenceMatrix
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.

SPLIT_COMPATIBILITY_GRAPH

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

N_POINTS

Type: Int
Number of POINTS.
Alias for property N_INPUT_RAYS.

QUOTIENT_SPACE

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

FACETS_THRU_POINTS

Type: IncidenceMatrix
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.

FEASIBLE

Type: Bool
True if the polyhedron is not empty.

GALE_TRANSFORM

Type: Matrix
Coordinates of the Gale transform.

VERTEX_BARYCENTER

Type: Vector
The center of gravity of the vertices of a bounded polytope.
Example:
This prints the vertex barycenter of the standard 3-simplex:
```
 >  print simplex(3)->VERTEX_BARYCENTER;
 1 1/4 1/4 1/4
```

CONE_DIM

Type: Int
One more than the dimension of the affine hull of the polyhedron
= one more than the dimension of the polyhedron.
= dimension of the homogenization of the polyhedron
If the polytope is given purely combinatorially, this is the dimension of a minimal embedding space
Example:
This prints the cone dimension of a 3-cube. Since the dimension of its affine closure is 3, the result is 4.
```
 >  print cube(3)->CONE_DIM;
 4
```

SPLITS

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

LATTICE

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

CS_PERMUTATION

Type: Array
The permutation induced by the central symmetry, if present.

CENTERED_ZONOTOPE

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

MINKOWSKI_CONE

Type: 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

BOUNDED

Type: Bool
True if and only if LINEALITY_SPACE trivial and FAR_FACE is trivial.
Example:
A pyramid over a square is bounded. Removing the base square yields an unbounded pointed polyhedron
(the vertices with first entry equal to zero correspond to rays).
```
 >  $p = pyramid(cube(2));
 print $p->BOUNDED;
 true
```
```
 >  $q = facet_to_infinity($p,4);
 print $q->BOUNDED;
 false
```

ZONOTOPE_INPUT_POINTS

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

AFFINE_HULL

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

VERTICES_IN_RIDGES

Type: IncidenceMatrix
Alias for property RAYS_IN_RIDGES.

CONE_AMBIENT_DIM

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

VERTEX_NORMALS

Type: Matrix
The i-th row is the normal vector of a hyperplane separating the i-th vertex from the others.
This property is a by-product of redundant point elimination algorithm.
All vectors in this section must be non-zero.
Alias for property RAY_SEPARATORS.
Example:
This prints a matrix in which each row represents a normal vector of a hyperplane seperating one vertex of a centered square
with side length 2 from the other ones. The first and the last hyperplanes as well as the second and third hyperplanes are the same
up to orientation.
```
 >  print cube(2)->VERTEX_NORMALS;
 0 1/2 1/2
 0 -1/2 1/2
 0 1/2 -1/2
 0 -1/2 -1/2
```

STEINER_POINT

Type: Vector
Steiner point of the whole polytope.

TILING_LATTICE

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

VERTICES_IN_INEQUALITIES

Type: IncidenceMatrix
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.

VALID_POINT

Type: Vector
Some point belonging to the polyhedron.
Example:
This stores a (homogeneous) point belonging to the 3-cube as a vector and prints its coordinates:
```
 >  $v = cube(3)->VALID_POINT;
 print $v;
 1 -1 -1 -1
```

WEAKLY_CENTERED

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

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

 >  $p = cube(3,0,0);
 print $p->WEAKLY_CENTERED;
 true

 >  print $p->CENTERED;
 false

FAR_HYPERPLANE

Type: Vector
Valid strict inequality for all affine points of the polyhedron.

CENTROID

Type: Vector
Centroid (center of mass) of the polytope.

POINTED

Type: Bool
True if the polyhedron does not contain an affine line.
Example:
A square does not contain an affine line and is therefore pointed. Removing one facet does not change this, although
it is no longer bounded. After removing two opposing facets, it contains infinitely many affine lines orthogonal to the
removed facets.
```
 >  $p = cube(2);
 print $p->POINTED;
 true
```
```
 >  print facet_to_infinity($p,0)->POINTED;
 true
```
```
 >  print new Polytope(INEQUALITIES=>$p->FACETS->minor([0,1],All))->POINTED;
 false
```

CENTERED

Type: Bool
True if (1, 0, 0, …) is in the relative interior.
If full-dimensional then polar to BOUNDED.

Example:
The cube [0,1]^3 is not centered, since the origin is on the boundary. By a small translation we can make it centered:

 >  $p = cube(3,0,0);
 print $p->CENTERED;
 false

 >  $t = new Vector([-1/2,-1/2,-1/2]);
 print translate($p,$t)->CENTERED;
 true

MINIMAL_VERTEX_ANGLE

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

SPECIAL_FACETS

Type: Set
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.

N_01POINTS

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

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

INEQUALITIES

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

POINTS

Type: Matrix
Points such that the polyhedron is their convex hull.
Redundancies are allowed.
The vector (x₀, x₁, … x_d) 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.
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
```

EQUATIONS

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

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

SPANNING

Type: Bool
The polytope is spanning if the lattice points generate the lattice

FACET_WIDTHS

Type: Vector
The integral width of the polytope with respect to each facet normal.

COMPRESSED

Type: Bool
True if the FACET_WIDTH is one.

GORENSTEIN_INDEX

Type: Integer
If the polytope is GORENSTEIN then this is the multiple such that the polytope is REFLEXIVE.

CANONICAL

Type: Bool
The polytope is canonical if there is exactly one interior lattice point.

LATTICE_EMPTY

Type: Bool
True if the polytope contains no lattice points other than the vertices.

POLAR_SMOOTH

Type: Bool
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.

REFLEXIVE

Type: Bool
True if the polytope and its dual have integral vertices.

EHRHART_POLYNOMIAL

Type: UniPolynomial
The Ehrhart polynomial.

LATTICE_VOLUME

Type: Integer
The normalized volume of the polytope.

EHRHART_QUASI_POLYNOMIAL

Type: Array
The Ehrhart quasi-polynomial of a rational polytope.
Coefficients are periodic functions of integral period.

Example:
To obtain the Ehrhart quasi-polynomial of a scaled 2-dimensional cross polytope write:

 >  $p=scale(cross(2),1/3);
 print join("\n",@{$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

NORMAL

Type: Bool
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.

FACET_VERTEX_LATTICE_DISTANCES

Type: Matrix
The entry (i,j) equals the lattice distance of vertex j from facet i.

SMOOTH

Type: Bool
The polytope is smooth if the associated projective variety is smooth; the determinant of the edge directions is +/-1 at every vertex.

GROEBNER_BASIS

Type: GroebnerBasis
The Groebner basis for the toric ideal associated to the lattice points in the polytope using any term order.

LATTICE_CODEGREE

Type: Int
COMBINATORIAL_DIM+1-LATTICE_DEGREE or the smallest integer k such that k*P has an interior lattice point.

LATTICE_WIDTH_DIRECTION

Type: Vector
One direction which realizes LATTICE_WIDTH of the polytope.

TERMINAL

Type: Bool
The polytope is terminal if there is exactly one interior lattice point and all other lattice points are vertices.

LATTICE_BASIS

Type: Matrix
VERTICES are interpreted as coefficient vectors for this basis
given in affine form
assumed to the the standard basis if not explicitely specified.

LATTICE_WIDTH

Type: Integer
The minimal integral width of the polytope.

GORENSTEIN

Type: Bool
The polytope is Gorenstein if a dilation of the polytope is REFLEXIVE up to translation.

GORENSTEIN_VECTOR

Type: Vector
If the polytope is GORENSTEIN, then this is the unique interior lattice point
in the multiple of the polytope that is REFLEXIVE.

LATTICE_DEGREE

Type: Int
The degree of the h*-polynomial or Ehrhart polynomial.

VERY_AMPLE

Type: Bool
The polytope is very ample if the Hilbert Basis of the cone spanned by the edge-directions of any vertex lies inside the polytope.

FACET_WIDTH

Type: Integer
The maximal integral width of the polytope with respect to the facet normals.

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

N_LATTICE_POINTS

Type: Integer
The number of LATTICE_POINTS

BOUNDARY_LATTICE_POINTS

Type: Matrix
The lattice points on the boundary of the polytope, including the vertices.

LATTICE_POINTS_GENERATORS

Type: Array
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.

N_BOUNDARY_LATTICE_POINTS

Type: Integer
The number of BOUNDARY_LATTICE_POINTS

N_INTERIOR_LATTICE_POINTS

Type: Integer
The number of INTERIOR_LATTICE_POINTS

INTERIOR_LATTICE_POINTS

Type: Matrix
The lattice points strictly in the interior of the polytope

Properties which belong to the corresponding (oriented) matroid

CHIROTOPE

Type: Text
Chirotope corresponding to the VERTICES. TOPCOM format.

CIRCUITS

Type: Set
Circuits in VECTORS

COCIRCUITS

Type: Set
Cocircuits in VECTORS

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

LP

Type: LinearProgram
Linear program applied to the polytope

MILP

Type: MixedIntegerLinearProgram
Mixed integer linear program applied to the polytope

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

GROUP

Type: Group
Methods of GROUP:
- REPRESENTATIVE_INEQUALITIES
  - UNDOCUMENTED
Properties of GROUP:
- MATRIX_ACTION
  - Type: MatrixActionOnVectors
  - Properties of MATRIX_ACTION:
  - VERTICES_ORBITS
    - Type: Array
    - Alias for property VECTORS_ORBITS.
- POINTS_ACTION
  - Type: PermutationAction
  - Alias for property INPUT_RAYS_ACTION.
- COORDINATE_ACTION
  - Type: PermutationAction
  - Properties of COORDINATE_ACTION:
  - POINTS_GENERATORS
    - Type: Matrix
    - Alias for property INPUT_RAYS_GENERATORS.
  - N_POINTS_GENERATORS
    - Type: Int
    - Alias for property N_INPUT_RAYS_GENERATORS.
  - REPRESENTATIVE_CERTIFIERS
    - Type: Matrix
    - 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.
  - CP_INDICES
    - Type: Set
    - The row indices of all core points among
      the REPRESENTATIVE_CERTIFIERS.
  - N_REPRESENTATIVE_CERTIFIERS
    - Type: Int
    - The number of REPRESENTATIVE_CERTIFIERS.
  - VERTICES_GENERATORS
    - Type: Matrix
    - Alias for property RAYS_GENERATORS.
  - REPRESENTATIVE_CORE_POINTS
    - Type: Matrix
    - 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).
  - NOP_GRAPH
    - Type: 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.
  - N_VERTICES_GENERATORS
    - Type: Int
    - Alias for property N_RAYS_GENERATORS.
  - N_REPRESENTATIVE_CORE_POINTS
    - Type: Int
    - The number of REPRESENTATIVE_CORE_POINTS.
- REPRESENTATIVE_VERTICES
  - Type: Matrix
  - Alias for property REPRESENTATIVE_RAYS.
- SYMMETRIC_RAYS
  - Type: Matrix
- VERTICES_ACTION
  - Type: PermutationAction
  - Properties of VERTICES_ACTION:
  - SYMMETRIZED_COCIRCUIT_EQUATIONS
    - Type: SymmetrizedCocircuitEquations
    - The cocircuit equations, projected to a certain direct sum of isotypic components
- SYMMETRIC_FACETS
  - Type: Matrix

Everything in this group is defined for BOUNDED polytopes only.

SQUARED_RELATIVE_VOLUMES

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

MAHLER_VOLUME

Type:
Mahler volume (or volume product) of the polytope.
Defined as the volume of the polytope and the volume of its polar (for BOUNDED, CENTERED and FULL_DIM polytopes only).
Often studied for centrally symmetric convex bodies, where the regular cubes are conjectured to be the global minimiers.
Example:
The following prints the Mahler volume of the centered 2-cube:
```
 >  print cube(2)->MAHLER_VOLUME;
 8
```

VOLUME

Type:
Volume of the polytope.
Example:
The following prints the volume of the centered 3-dimensional cube with side length 2:
```
 >  print cube(3)->VOLUME;
 8
```

RELATIVE_VOLUME

Type: Map
The k-dimensional Euclidean volume of a k-dimensional rational polytope
embedded in R^n.
This value is obtained by summing the square roots of the entries in SQUARED_RELATIVE_VOLUMES
using the function naive_sum_of_square_roots. Since this latter function
does not try very hard to compute the real value, you may have to resort to
a computer algebra package.
The value is encoded as a map collecting the coefficients of various roots encountered in the sum.
For example, {(3 1/2),(5 7)} represents sqrt{3}/2 + 7 sqrt{5}.
If the output is not satisfactory, please use a symbolic algebra package.

Example:
The following prints the 2-dimensional volume of a centered square with side length 2 embedded in the 3-space (the result is 4):

 >  $M = new Matrix([1,-1,1,0],[1,-1,-1,0],[1,1,-1,0],[1,1,1,0]);
 $p = new Polytope<Rational>(VERTICES=>$M);
 print $p->RELATIVE_VOLUME;
 {(1 4)}

TRIANGULATION

Type: GeometricSimplicialComplex
Properties of TRIANGULATION:
- GKZ_VECTOR
  - Type: Vector
  - GKZ-vector
    See Chapter 7 in Gelfand, Kapranov, and Zelevinsky:
    Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994

POLYTOPAL_SUBDIVISION

Type: SubdivisionOfPoints
Properties of POLYTOPAL_SUBDIVISION:
- REFINED_SPLITS
  - Type: Set
  - The splits that are coarsenings of the subdivision.
    If the subdivision is regular these form the unique split decomposition of
    the corresponding weight function.

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.

UNBOUNDED_FACETS

Type: Set
Indices of facets that are unbounded.

BOUNDED_COMPLEX

Type: PolyhedralComplex
Properties of BOUNDED_COMPLEX:
- GRAPH
  - Type: Graph
  - Properties of GRAPH:
  - TOTAL_LENGTH
    - Type:
    - Sum of all EDGE_LENGTHS.
  - EDGE_COLORS
    - Type: EdgeMap
    - Each edge indicates the maximal dimension of a bounded
      face containing it. Mainly used for visualization purposes.
  - EDGE_DIRECTIONS
    - Type: EdgeMap
    - Difference of the vertices for each edge (only defined up to signs).
  - EDGE_LENGTHS
    - Type: EdgeMap
    - The length of each edge measured in the maximum metric.
- VERTEX_MAP
  - Type: Array
  - For every row of VERTICES this indicates the corresponding row in the
    VERTICES of the parent polytope.

N_BOUNDED_VERTICES

Type: Int
Number of bounded vertices (non-rays).

SIMPLE_POLYHEDRON

Type: Bool
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.

TOWARDS_FAR_FACE

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

FAR_FACE

Type: Set
Indices of vertices that are rays.

These properties are for visualization.

VIF_CYCLIC_NORMAL

Type: Array
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.

VERTEX_LABELS

Type: Array
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.

FACET_LABELS

Type: Array
Unique names assigned to the FACETS, analogous to VERTEX_LABELS.

FTV_CYCLIC_NORMAL

Type: Array
Reordered transposed VERTICES_IN_FACETS. Dual to VIF_CYCLIC_NORMAL.
Alias for property FTR_CYCLIC_NORMAL.

INEQUALITY_LABELS

Type: Array
Unique names assigned to the INEQUALITIES, analogous to VERTEX_LABELS.

POINT_LABELS

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

NEIGHBOR_VERTICES_CYCLIC_NORMAL

Type: Array
Reordered GRAPH. Dual to NEIGHBOR_FACETS_CYCLIC_NORMAL.
Alias for property NEIGHBOR_RAYS_CYCLIC_NORMAL.

GALE_VERTICES

Type: Matrix
Coordinates of points for an affine Gale diagram.

SCHLEGEL_DIAGRAM

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

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

N_EDGES

Returns: Int
The number of edges of the GRAPH

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

Returns: String
Prettily print the cd-index given in CD_INDEX_COEFFICIENTS

N_RIDGES

Returns: Int
The number of ridges (faces of codimension 2) of the polytope
equals the number of edges of the DUAL_GRAPH

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

DIM

Returns: Int
returns the dimension of the polytope

INNER_DESCRIPTION

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

MINKOWSKI_CONE_POINT( Vector < Rational > point)

Parameters:
- Vector < Rational > point : point in the Minkowski summand cone
Returns: Polytope < Rational >
returns the Minkowski summand of a polytope P given by
a point in the MINKOWSKI_CONE.

AMBIENT_DIM

Returns: Int
returns the dimension of the ambient space of the polytope

labeled_vertices( String label …)

Parameters:
- String label … : vertex labels
Returns: Set < Int >
Find the vertices by given labels.

OUTER_DESCRIPTION

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

MINKOWSKI_CONE_COEFF( Vector < Rational > coeff)

Parameters:
- Vector < Rational > coeff : coefficient vector to the rays of the Minkowski summand cone
Returns: Polytope < Rational >
returns the Minkowski summand of a polytope P given by
a coefficient vector to the rays of the MINKOWSKI_CONE.

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

FACET_POINT_LATTICE_DISTANCES( Vector < Rational > v)

Parameters:
- Vector < Rational > v : point in the ambient space of the polytope
Returns: Vector < Integer >
Vector containing the distances of a given point v from all facets

EHRHART_POLYNOMIAL_COEFF

Returns: Vector < Rational >
Vector containing the coefficients of the EHRHART_POLYNOMIAL, ordered
by increasing degree of the corresponding term.

N_LATTICE_POINTS_IN_DILATION( Int n)

Parameters:
- Int n : dilation factor
Returns: Int
The number of LATTICE_POINTS in the n-th dilation of the polytope

POLYTOPE_IN_STD_BASIS( Polytope < Rational > P)

Parameters:
- Polytope < Rational > P : polytope
Returns: Polytope < Rational >
returns a polytope in the integer lattice basis if a LATTICE_BASIS is given

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

LATTICE_POINTS

Returns: Matrix < Integer >
Returns the lattice points in bounded Polytopes.

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

TRIANGULATION_INT_SIGNS

Returns: Array < Int >
the orientation of the simplices of TRIANGULATION_INT in the given order of the POINTS

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_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_GRAPH

Graph of the bounded subcomplex.

BOUNDED_FACETS

Returns: Set < Int >
Indices of FACETS that are bounded.

BOUNDED_VERTICES

Returns: Set < Int >
Indices of VERTICES that are no rays.

BOUNDED_DUAL_GRAPH

Dual graph of the bounded subcomplex.

These methods are for visualization.

VISUAL_BOUNDED_GRAPH

Returns: Visual::PolytopeGraph
Visualize the GRAPH of a polyhedron.

VISUAL_ORBIT_COLORED_GRAPH

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

VISUAL_DUAL_FACE_LATTICE

Returns: Visual::PolytopeLattice
Visualize the dual face lattice of a polyhedron as a multi-layer graph.

SCHLEGEL

Returns: Visual::SchlegelDiagram
Create a Schlegel diagram and draw it.

VISUAL_DUAL

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

VISUAL_DUAL_GRAPH

Returns: Visual::Graph
Visualize the DUAL_GRAPH of a polyhedron.

VISUAL_FACE_LATTICE

Returns: Visual::PolytopeLattice
Visualize the HASSE_DIAGRAM of a polyhedron as a multi-layer graph.

write_stl( String filename)

Parameters:
- String filename :
Take a 3-polytope and write ASCII STL output.

Example:

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

VISUAL_GRAPH

Returns: Visual::PolytopeGraph
Visualize the GRAPH of a polyhedron.

GALE

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

VISUAL

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

BigObject Polytope

Properties

Combinatorics

Geometry

Input property

Lattice points in cones

Lattice points in polytopes

Matroid properties

Optimization

Symmetry

Triangulation and volume

Unbounded polyhedra

Visualization

Methods

Backward compatibility