BigObject PolyhedralFan<Scalar>

from application fan

A polyhedral fan. The current restriction is that each cone in the fan has to be pointed. This will be relaxed later. If a fan is specified via INPUT_RAYS and INPUT_CONES each input cone must list all the input rays incident. Once non-trivial linealities are allowed the following will apply: The RAYS always lie in a linear subspace which is complementary to the LINEALITY_SPACE.

Type Parameters:

Scalar: numeric data type used for the coordinates, must be an ordered field. Default is Rational.

Specializations:

PolyhedralFan<Rational>: Unnamed full specialization of PolyhedralFan

Example:

A typical example is the normal fan of a convex polytope.

 > $f=normal_fan(polytope::cube(3));
 > print $f->F_VECTOR;
 6 12 8

Permutations:

ConesPerm:

permuting the MAXIMAL_CONES

RaysPerm:

permuting the RAYS

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

---

INPUT_CONES

Maybe redundant list of not necessarily maximal cones. Indices refer to INPUT_RAYS. Each cone must list all rays of INPUT_RAYS it contains. Any incident cones will automatically be added. The cones are allowed to contain lineality. Cones which do not have any rays correspond to the trivial cone (contains only the origin). An empty fan does not have any cones. Input section only. Ask for MAXIMAL_CONES if you want to know the maximal cones (indexed by RAYS).

Type:

IncidenceMatrix<NonSymmetric>

Example:

In 3-space, we construct the fan containing the closure of the all-positive octant and the (x⇐0, y>=0, z=0) quadrant with the following input (note that we do not have to state the combinatorics for cones contained in a given cone of higher dimension, e.g. the the (x>=0, z>=0) quadrant in the y=0 plane):

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0,0],[0,1,0],[0,0,1],[-1,0,0]],INPUT_CONES=>[[0,1,2],[1,3]]);
 > print rows_labeled($f->RAYS);
 0:1 0 0
 1:0 1 0
 2:0 0 1
 3:-1 0 0

 > print $f->CONES;
 <{1}
 {3}
 {2}
 {0}
 >
 <{1 3}
 {1 2}
 {0 2}
 {0 1}
 >
 <{0 1 2}
 >

---

INPUT_CONES_REPS

Maybe redundant list of not necessarily maximal cones, one from each orbit. All vectors in the input must be non-zero. Indices refer to INPUT_RAYS.

Type:

IncidenceMatrix<NonSymmetric>

---

INPUT_LINEALITY

Vectors whose linear span defines a subset of the lineality space of the fan; redundancies are allowed. Input section only. Ask for LINEALITY_SPACE if you want to know the lineality space.

Type:

Matrix<Scalar,NonSymmetric>

Example:

In 3-space, we can “stretch” the (x>=0, y>=0) quadrant in the z=0 hyperplane to obtain the joint of the two octants where x and y are non-negative by adding linearity in z-direction:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0,0],[0,1,0]],INPUT_CONES=>[[0,1]],INPUT_LINEALITY=>[[0,0,1]]);

---

INPUT_RAYS

Rays from which the cones are formed. May be redundant. All vectors in the input must be non-zero. You also need to provide INPUT_CONES to define a fan completely. Input section only. Ask for RAYS if you want a list of non-redundant rays.

Type:

Matrix<Scalar,NonSymmetric>

Example:

When constructing the fan which displays the division of the plane into quadrants, we can use redundant rays such that the indices given in INPUT_CONES suggest that two rays form the border of our fan, but due to our choice in INPUT_RAYS, they actually are the same ray adjacent to two distinct 2-dimensional cones:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0],[0,1],[-1,0],[0,-1],[2,0]],INPUT_CONES=>[[0,1],[1,2],[2,3],[3,4]]);
 > print rows_labeled($f->RAYS);
 0:1 0
 1:0 1
 2:-1 0
 3:0 -1

 > print $f->CONES;
 <{0}
 {1}
 {2}
 {3}
 >
 <{0 1}
 {1 2}
 {2 3}
 {0 3}
 >

---

INPUT_RAYS_REPS

One Ray from each orbit. May be redundant.

Type:

Matrix<Scalar,NonSymmetric>

---

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

---

COMBINATORIAL_DIM

Combinatorial dimension of the fan.

Type:

Int

Example:

The normal fan of the 6-cube has combinatorial dimension 5:

 > print normal_fan(cube(6))->COMBINATORIAL_DIM;
 5

---

COMPLETE

The polyhedral fan is complete if its suport is the whole space. Due to undecidability issues this is checked heuristically only. See the remarks on SPHERE for details. Note that in the case of a polyhedral complex, this refers to the underlying fan, so should always be false.

Type:

Bool

Example:

The normal fan of a polytope always is complete; here we see this for the 8-cube:

 > print normal_fan(cube(8))->COMPLETE;
 true

---

CONES

List of all cones of the fan of each dimension. Indices refer to RAYS.

Type:

Array<IncidenceMatrix<NonSymmetric>>

Example:

A fan constructed with a 2-dimensional cone in INPUT_CONES also contains the incident 1-dimensional cones.

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0],[0,1]],INPUT_CONES=>[[0,1]]);
 > print $f->CONES;
 <{0}
 {1}
 >
 <{0 1}
 >

---

CONES_COMBINATORIAL_DIMS

The combinatorial dimensions of the cones.

Type:

Array<Int>

---

DUAL_GRAPH

The graph whose nodes are the maximal cones which are connected if they share a common facet. Only defined if PURE.

Type:

Graph<Undirected>

Example:

The dual graph of a face fan of the 3-cube is isomorphic to the vertex-edge graph of the 3-dimensional cross polytope.

 > $c = cross(3);
 > $f = face_fan(cube(3));
 > $g1 = $f->DUAL_GRAPH;
 > $g2 = $c->GRAPH;
 > print isomorphic($g1->ADJACENCY,$g2->ADJACENCY);
 true

---

F2_VECTOR

f_ik is the number of incident pairs of i-dimensional cones and k-dimensional cones; the main diagonal contains the F_VECTOR.

Type:

Matrix<Integer,NonSymmetric>

---

F_VECTOR

f_k is the number of k-dimensional cones starting from dimension k=1. The f-vector of a polytope and the f-vector of any of its face fans coincide; for the 3-cube: > print face_fan(cube(3))→F_VECTOR; | 8 12 6

Type:

Vector<Integer>

---

GRAPH

The graph of the fan intersected with a sphere, that is, the vertices are the rays which are connected if they are contained in a common two-dimensional cone.

Type:

Graph<Undirected>

Example:

The graph of a face fan of a polytope is isomorphic to the vertex-edge graph of the polytope. Here we see this for the 3-cube:

 > $c = cube(3);
 > $f = face_fan($c);
 > $g1 = $f->GRAPH;
 > $g2 = $c->GRAPH;
 > print isomorphic($g1->ADJACENCY,$g2->ADJACENCY);
 true

---

HASSE_DIAGRAM

Type:

Lattice<BasicDecoration,Nonsequential>

Example:

To compute the Hasse diagram of the normal fan of the 2-cube and display its decoration, we can do the following. Note the artificial node on top and the empty node at the bottom. The latter represents the trivial cone consisting only of the origin.

 > $h = normal_fan(cube(2))->HASSE_DIAGRAM;

print $h→DECORATION;

Methods of HASSE_DIAGRAM:

dim()

Returns:

Int

nodes_of_dim(Int d)

Parameters:

Int d: dimension

Returns:

Set<Int>

nodes_of_dim_range(Int d1, Int d1)

Parameters:

Int d1: lower dimension

Int d1: upper dimension

Returns:

Set<Int>

---

MAXIMAL_CONES

Non redundant list of maximal cones. Indices refer to RAYS. Cones which do not have any rays correspond to the trivial cone (contains only the origin). An empty fan does not have any cones.

Type:

IncidenceMatrix<NonSymmetric>

Example:

The maximal cones of the normal fan of the 3-cube describe which faces share a common vertex:

 > print normal_fan(cube(3))->MAXIMAL_CONES;
 {0 2 4}
 {1 2 4}
 {0 3 4}
 {1 3 4}
 {0 2 5}
 {1 2 5}
 {0 3 5}
 {1 3 5}

---

MAXIMAL_CONES_COMBINATORIAL_DIMS

The combinatorial dimensions of the maximal cones.

Type:

Array<Int>

Example:

Here we construct a fan with a 3-dimensional and a 2-dimensional maximal cone, of which the latter contains lineality, and then display the corresponding combinatorial dimensions:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0,0],[0,1,0],[0,0,1],[-1,0,0],[0,-1,0]],INPUT_CONES=>[[0,1,2],[1,3,4]]);
 > print $f->MAXIMAL_CONES;
 {0 1 2}
 {3}

 > print $f->MAXIMAL_CONES_COMBINATORIAL_DIMS;
 2 0

---

MAXIMAL_CONES_INCIDENCES

Array of incidence matrices of all maximal cones. Indices refer to RAYS.

Type:

Array<IncidenceMatrix<NonSymmetric>>

Example:

Here we construct a fan with a 3-dimensional and a 2-dimensional maximal cone and then display the incident cones:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0,0],[0,1,0],[0,0,1],[-1,0,0]],INPUT_CONES=>[[0,1,2],[1,3]]);
 > print $f->MAXIMAL_CONES;
 {0 1 2}
 {1 3}

 > print $f->MAXIMAL_CONES_INCIDENCES;
 <{1 2}
 {0 2}
 {0 1}
 >
 <{1}
 {3}
 >

---

MAXIMAL_CONES_THRU_RAYS

Transposed to MAXIMAL_CONES. Notice that this is a temporary property; it will not be stored in any file.

Type:

IncidenceMatrix<NonSymmetric>

---

N_CONES

Number of CONES.

Type:

Int

---

N_MAXIMAL_CONES

Number of MAXIMAL_CONES.

Type:

Int

Example:

The number of facets of a polytope is the same as the number of maximal cones of its face fan; here we can see this for the 3-cube:

 > print face_fan(cube(3))->N_MAXIMAL_CONES;
 6

---

PURE

The polyhedral fan is pure if all maximal cones are of the same dimension.

Type:

Bool

Example:

Generating a fan from two distinct 2-dimensional cones gives a pure fan:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0],[0,1],[-1,0],[0,-1]],INPUT_CONES=>[[0,1],[2,3]]);
 > print $f->PURE;
 true

Example:

Gnerating a fan from a 2-dimensional and a 1-dimensional cone results in the fan not being pure if the latter is a ray not contained in the former:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,1],[-1,1],[0,-1]],INPUT_CONES=>[[0,1],[2]]);
 > print $f->PURE;
 false

---

SIMPLICIAL

The polyhedral fan is simplicial if all maximal cones are simplicial.

Type:

Bool

Example:

The normal fan of a d-cube is simplicial; here we see this for the 3-cube:

 > print normal_fan(cube(3))->SIMPLICIAL;
 true

---

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

---

FACET_NORMALS

The possible facet normals of all maximal cones.

Type:

Matrix<Scalar,NonSymmetric>

Example:

The two facet normals of the complete fan dividing the plane into the four quadrants (which can be obtained by computing the normal fan of the 2-cube) point in x- and in y-direction, respectively:

 > print normal_fan(cube(2))->FACET_NORMALS;
 1 0
 0 1

---

FAN_AMBIENT_DIM

Dimension of the space which contains the polyhedral fan. Note: To avoid confusion in context of (in-)homogenuous coordinates it is generally advised to use the method AMBIENT_DIM.

Type:

Int

Example:

The fan living in the plane containing only the cone which consists of the ray pointing in positive x-direction is by definition embedded in the plane:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0]],INPUT_CONES=>[[0]]);
 > print $f->FAN_AMBIENT_DIM;
 2

---

FAN_DIM

Dimension of the polyhedral fan. # Note: To avoid confusion in context of (in-)homogenuous coordinates it is generally advised to use the method DIM.

Type:

Int

Example:

The fan living in the plane containing only the cone which consists of the ray pointing in positive x-direction is 1-dimensional:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0]],INPUT_CONES=>[[0]]);
 > print $f->FAN_DIM;
 1

---

FULL_DIM

A fan is full-dimensional if its dimension and ambient dimension coincide.

Type:

Bool

Example:

The normal fan of a polytope is always full-dimensional, which we see here for the 5-dimensional cross polytope:

 > $nf = normal_fan(cross(5));
 > print $nf->FULL_DIM;
 true

---

LINEALITY_DIM

Dimension of the lineality space.

Type:

Int

Example:

A pointed fan has no lineality, thus the dimension of its lineality space is 0. The most simple example is a fan only consisting of one ray, here in the plane:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0]],INPUT_CONES=>[[0]]);
 > print $f->LINEALITY_DIM;
 0

---

LINEALITY_SPACE

Since we do not require our cones to be pointed: a basis of the lineality space of the fan. Co-exists with RAYS.

Type:

Matrix<Scalar,NonSymmetric>

Example:

We can create a fan in 3-space from the two 2-dimensional cones which are the (x,y)- and (x,z)-halfspaces where y and z are non-negative, respectively. Both cones and thus the fan contain lineality in x-direction:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0,0],[0,1,0],[0,0,1],[-1,0,0]],INPUT_CONES=>[[0,1,3],[0,2,3]]);
 > print $f->LINEALITY_SPACE;
 1 0 0

---

LINEAR_SPAN_NORMALS

The possible linear span normals of all maximal cones. Empty if PURE and FULL_DIM, i.e. each maximal cone has the same dimension as the ambient space.

Type:

Matrix<Scalar,NonSymmetric>

Example:

We construct a fan in 3-space with two 2-dimensional maximal cones and a 1-dimensional maximal cone. Note that there are two possible normals for the latter, but one of these also is a normal for one of the 2-dimensional cones; thus we receive three normals:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0,0],[0,1,0],[-1,0,1],[-1,-1,0]],INPUT_CONES=>[[0,1],[1,2],[3]]);
 > print $f->LINEAR_SPAN_NORMALS;
 0 0 1
 1 0 1
 -1 1 0

---

MAXIMAL_CONES_FACETS

Tells for each maximal cone what are its facet normals, thus implying the facets. Each row corresponds to a maximal cone and each column to the row with the same index of FACET_NORMALS. A negative number means that the corresponding row of FACET_NORMALS has to be negated.

Type:

SparseMatrix<Int,NonSymmetric>

Example:

The two facet normals of the complete fan dividing the plane into the four quadrants (which can be obtained by computing the normal fan of the 2-cube) point in x- and in y-direction, respectively. Each quadrant has both of these as normals only differing by the combination of how they point in- or outside:

 > $f = normal_fan(cube(2));
 > print $f->MAXIMAL_CONES;
 {0 2}
 {1 2}
 {0 3}
 {1 3}

 > print rows_numbered($f->FACET_NORMALS);
 0:1 0
 1:0 1

 > print $f->MAXIMAL_CONES_FACETS;
 1 1
 -1 1
 1 -1
 -1 -1

---

MAXIMAL_CONES_LINEAR_SPAN_NORMALS

Tells for each maximal cone what is its linear span by naming its normals. Indices refer to LINEAR_SPAN_NORMALS. Rows correspond to MAXIMAL_CONES. An empty row corresponds to a full-dimensional cone.

Type:

IncidenceMatrix<NonSymmetric>

Example:

We construct a fan in 3-space with two 2-dimensional maximal cones and a 1-dimensional maximal cone. This way we receive two cones with one normal and one cone with two normals:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0,0],[0,1,0],[-1,0,1],[-1,-1,0]],INPUT_CONES=>[[0,1],[1,2],[3]]);
 > print $f->MAXIMAL_CONES;
 {0 1}
 {1 2}
 {3}

 > print rows_numbered($f->LINEAR_SPAN_NORMALS);
 0:0 0 1
 1:1 0 1
 2:-1 1 0

 > print $f->MAXIMAL_CONES_LINEAR_SPAN_NORMALS;
 {0}
 {1}
 {0 2}

---

N_FACET_NORMALS

The number of possible facet normals of all maximal cones.

Type:

Int

Example:

The facets of the normal fan of the 3-cube are each spanned by a ray pointing in positive or negative direction of one axis and another ray pointing in positive or negative direction of another axis; thus, the third axis is normal to this facet. This leaves us with three possible facet normals:

 > print normal_fan(cube(3))->N_FACET_NORMALS;
 3

---

N_INPUT_RAYS

Number of INPUT_RAYS.

Type:

Int

Example:

To determine the combined amount of redundant and unused rays given by INPUT_RAYS, we compare this number to N_RAYS:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0],[0,1],[-1,0],[0,-1],[2,0],[1,1]],INPUT_CONES=>[[0,1],[1,2],[2,3],[3,4]]);
 > print ($f->N_INPUT_RAYS-$f->N_RAYS);
 2

---

N_RAYS

Number of RAYS.

Type:

Int

Example:

The number of facets of a polytope is the number of rays of the corresponding normal fan. Here we see this for the 3-cube:

 > print normal_fan(cube(3))->N_RAYS;
 6

---

ORTH_LINEALITY_SPACE

A basis of the orthogonal complement to LINEALITY_SPACE.

Type:

Matrix<Scalar,NonSymmetric>

Example:

In 3-space, we build a fan containing only the ray pointing in positive x-direction, together with lineality in y-direction. The definition of the cones yields no additional lineality, thus the orthogonal complement to the lineality space is spanned by (1,0,0) and (0,0,1):

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0,0]],INPUT_CONES=>[[0]],INPUT_LINEALITY=>[[0,1,0]]);
 > print $f->ORTH_LINEALITY_SPACE;
 1 0 0
 0 0 1

---

POINTED

A fan is pointed if the lineality space is trivial.

Type:

Bool

Example:

The normal fan of a polytope is pointed if and only if the polytope is full-dimensional. Here we confirm this for the normal fans of the 2-cube living in 2- and in 3-space, respectively:

 > $nf_full = normal_fan(cube(2));
 > print $nf_full->POINTED;
 true

 > $nf_not_full = normal_fan(product(cube(2),new Polytope(POINTS=>[[1,0]])));
 > print $nf_not_full->POINTED;
 false

---

PSEUDO_REGULAR

True if the fan is a subfan of a REGULAR fan

Type:

Bool

Example:

Here we build a fan in the plane with the non-negative quadrant and the ray pointing in negative y-direction as its maximal cones. This is a subfan of the fan with all four quadrants as its maximal cones which is the normal fan of the 2-cube; thus, our fan is pseudo regular:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0],[0,1],[-1,0]],INPUT_CONES=>[[0,1],[2]]);
 > print $f->PSEUDO_REGULAR;
 true

---

RAYS

Rays of the PolyhedralFan. Non-redundant. Co-exists with LINEALITY_SPACE.

Type:

Matrix<Scalar,NonSymmetric>

Example:

The rays of the face fan of a 3-dimensioanl cube. This fan has one ray in the direcction of each vertex of the cube.

 > print face_fan(cube(3))->RAYS;
 -1 -1 -1
 1 -1 -1
 -1 1 -1
 1 1 -1
 -1 -1 1
 1 -1 1
 -1 1 1
 1 1 1

---

REGULAR

True if the fan is the normal fan of a bounded polytope.

Type:

Bool

Example:

The face fan of the centered d-dimensional cross polytope w.r.t. the origin is the normal fan of the d-cube; here we confirm the regularity of the face fan for d=7:

 > print face_fan(cross(7))->REGULAR;
 true

Example:

A normal fan always is complete; thus a fan which is not complete can not be regular:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0],[0,1]],INPUT_CONES=>[[0,1]]);
 > print $f->REGULAR;
 false

---

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

---

CONES_ORBIT_SIZES

Number of CONES_REPS in each orbit.

Type:

Array<Array<Int>>

---

CONES_REPS

List of all cones of all dimensions of the fan, one from each orbit. Indices refer to RAYS.

Type:

Array<Array<Set<Int>>>

---

GROUP

Type:

Group

Methods of GROUP:

REPRESENTATIVE_INPUT_CONES

explicit representatives of equivalence classes of INPUT_CONES under a group action

REPRESENTATIVE_INPUT_RAYS

explicit representatives of equivalence classes of INPUT_RAYS under a group action

Properties of GROUP:

MATRIX_ACTION

Type:

MatrixActionOnVectors<Scalar>

MAXIMAL_CONES_ACTION

derived from:

PermutationAction

Type:

PermutationAction<Int,Rational>

Properties of MAXIMAL_CONES_ACTION:

REPRESENTATIVE_COMBINATORIAL_DIMS

dimensions of representatives of maximal cones

Type:

Array<Int>

REPRESENTATIVE_F_VECTOR

counts how many representatives of maximal cones there are in each dimension

Type:

Array<Int>

REPRESENTATIVE_CONES

Type:

IncidenceMatrix<NonSymmetric>

REPRESENTATIVE_MAXIMAL_CONES

Type:

IncidenceMatrix<NonSymmetric>

REPRESENTATIVE_RAYS

Type:

Matrix<Scalar,NonSymmetric>

---

MAXIMAL_CONES_IN_ORBITS

Tells which maximal cone is in the orbit of which representative, indices refers to rows of MAXIMAL_CONES.

Type:

Array<Set<Int>>

---

MAXIMAL_CONES_ORBIT_SIZES

Number of MAXIMAL_CONES_REPS in each orbit.

Type:

Array<Int>

---

MAXIMAL_CONES_REPS

Non redundant list of maximal cones, one from each orbit. Indices refer to RAYS.

Type:

Array<Set<Int>>

---

MAXIMAL_CONES_REPS_DIMS

The dimensions of MAXIMAL_CONES_REPS.

Type:

Array<Int>

---

MAXIMAL_CONES_REPS_FACETS

Tells for each maximal cone representative what are its facets. A negative number means that the corresponding row of REPS_FACET_NORMALS has to be negated.

Type:

SparseMatrix<Int,NonSymmetric>

---

MAXIMAL_CONES_REPS_LINEAR_SPAN_NORMALS

Tells for each maximal cone representative what is its linear span. Indices refer to REPS_LINEAR_SPAN_NORMALS. Rows correspond to MAXIMAL_CONES_REPS_FACETS

Type:

IncidenceMatrix<NonSymmetric>

---

N_MAXIMAL_CONE_ORBITS

Number of orbits of MAXIMAL_CONES.

Type:

Int

---

N_RAY_ORBITS

Number of orbits of RAYS.

Type:

Int

---

N_SYMMETRIES

Number of elements of the symmetry group.

Type:

Int

---

ORBITS_F_VECTOR

f_k is the number of k-dimensional cones up to symmetry.

Type:

Vector<Integer>

---

RAYS_IMAGES

Each row contains the image of all RAYS under one element of the symmetry group.

Type:

Array<Array<Int>>

---

RAYS_IN_ORBITS

Tells which ray is in the orbit of which representative, indices refers to rows of RAYS.

Type:

Array<Set<Int>>

---

RAYS_ORBIT_SIZES

Number of elements of each orbit of RAYS.

Type:

Array<Int>

---

RAYS_REPS

One ray from each orbit. Non-redundant.

Type:

Matrix<Scalar,NonSymmetric>

---

RAYS_REPS_LABELS

Unique names assigned to the RAYS_REPS.

Type:

Array<String>

---

REPS_FACET_NORMALS

The possible facet normals of all maximal cone representatives.

Type:

Matrix<Scalar,NonSymmetric>

---

REPS_LINEAR_SPAN_NORMALS

The possible linear span normals of all maximal cone representatives.

Type:

Matrix<Scalar,NonSymmetric>

---

SYMMETRY_GENERATORS

Each element of the array is a generator of the subgroup of the symmetric group acting on the coordinates. Each generator is represented by an Array whose i-th entry is the image of the i-th coordinate.

Type:

Array<Array<Int>>

---

SYMMETRY_GROUP

Each element of the array is an element of the symmetry group.

Type:

Array<Array<Int>>

---

The following properties are topological invariants.

---

HOMOLOGY

The homology of the intersection of the fan with the unit sphere. # @example When intersecting, the normal fan of the 3-cube gives us a simplicial complex which is a topological 2-sphere: > print normal_fan(cube(3))→HOMOLOGY; | ({} 0) | ({} 0) | ({} 1)

Type:

Array<HomologyGroup<Integer>>

---

INTERSECTION_COMPLEX

If the fan is SIMPLICIAL the simplicial complex obtained by intersecting the fan with the unit sphere. If the fan is not SIMPLICIAL the crosscut complex of the intersection.

Type:

SimplicialComplex

Example:

When intersecting, the normal fan of the 3-cube gives us a simplicial complex which is a topological 2-sphere:

 > $ic = normal_fan(cube(3))->INTERSECTION_COMPLEX;

print $ic→SPHERE;

---

Properties coming from associated toric varieties

---

GORENSTEIN

A fan is Gorenstein if it is Q_GORENSTEIN with Q_GORENSTEIN_INDEX equal to one.

Type:

Bool

---

Q_GORENSTEIN

A fan is Q-GORENSTEIN if each maximal cone is Q_Gorenstein.

Type:

Bool

---

Q_GORENSTEIN_INDEX

If a fan is Q_GORENSTEIN, then its Q-Gorenstein index is the least common multiple of the Q-Gorenstein indices of its maximal cones. Otherwise Q_GORENSTEIN_INDEX is undefined.

Type:

Int

---

SMOOTH_FAN

A fan is smooth if all cones of the fan are smooth.

Type:

Bool

---

These properties are for visualization.

---

INPUT_RAY_LABELS

Unique names assigned to the INPUT_RAYS. Similar to RAY_LABELS for RAYS.

Type:

Array<String>

---

MAXIMAL_CONE_LABELS

Unique names assigned to the MAXIMAL_CONES. Similar to RAY_LABELS for RAYS.

Type:

Array<String>

---

RAY_LABELS

Unique names assigned to the RAYS. If specified, they are shown by visualization tools instead of vertex indices. For a polyhedral fan built from scratch, you should create this property by yourself, either manually in a text editor, or with a client program.

Type:

Array<String>

Example:

The normal fan of the 2-cube has 4 rays; to assign the labels 'a', 'b' and 'any label' to the first three entries of RAYS, do this (note that the unspecified label of the last ray is blank when visualizing, and that, depending on your visualization settings, not the whole string of the third label might be displayed):

 > $f = normal_fan(cube(2));
 > $f->RAY_LABELS=['a','b','any label'];

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

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CONES_DIMS()

The dimensions of the cones.

Returns:

Array<Int>

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cone(Int i)

Returns the i-th maximal cone.

Parameters:

Int i

Returns:

Cone

Example:

To obtain the cone generated by the x-, y- and z-axis, we can take the first cone of the normal fan of the 3-cube:

 > $c = normal_fan(cube(3))->cone(0);
 > print $c->RAYS;
 1 0 0
 0 1 0
 0 0 1

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

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AMBIENT_DIM()

Returns the dimension of the ambient space.

Returns:

Int

Example:

The fan living in the plane containing only the cone which consists of the ray pointing in positive x-direction is by definition embedded in the plane:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0]],INPUT_CONES=>[[0]]);
 > print $f->AMBIENT_DIM;
 2

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DIM()

Returns the dimension of the linear space spanned by the fan.

Returns:

Int

Example:

The fan living in the plane containing only the cone which consists of the ray pointing in positive x-direction is 1-dimensional:

 > $f = new PolyhedralFan(INPUT_RAYS=>[[1,0]],INPUT_CONES=>[[0]]);
 > print $f->DIM;
 1

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

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cone_representative(Int i)

Returns the i-th maximal cone representative.

Parameters:

Int i

Returns:

Cone

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These methods are for visualization.

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VISUAL()

Visualizes the fan, intersected with the unit ball.

Options:

option list geometric_options_linear

Returns:

Visual::PolyhedralFan

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