Available versions of this document: latest release, release 3.5, nightly master
Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2
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 nontrivial 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 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
Properties
Input property
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 ofINPUT_RAYS
it contains. 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 forMAXIMAL_CONES
if you want to know the maximal cones (indexed byRAYS
). Type:

INPUT_CONES_REPS
Maybe redundant list of not necessarily maximal cones, one from each orbit. All vectors in the input must be nonzero. Indices refer to
INPUT_RAYS
. Type:

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>

INPUT_RAYS
Rays from which the cones are formed. May be redundant. All vectors in the input must be nonzero. You also need to provide
INPUT_CONES
to define a fan completely. Input section only. Ask forRAYS
if you want a list of nonredundant rays. Type:
Matrix<Scalar,NonSymmetric>

INPUT_RAYS_REPS
One Ray from each orbit. May be redundant.
 Type:
Matrix<Scalar,NonSymmetric>
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.

COMBINATORIAL_DIM
Combinatorial dimension.
 Type:

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:

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

CONES_COMBINATORIAL_DIMS
The combinatorial dimensions of the cones.
 Type:

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

F2_VECTOR
f_{ik} is the number of incident pairs of idimensional cones and kdimensional cones; the main diagonal contains the
F_VECTOR
. Type:

F_VECTOR
f_{k} is the number of kdimensional cones starting from dimension k=1.
 Type:

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 twodimensional cone.
 Type:

HASSE_DIAGRAM

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:

MAXIMAL_CONES_COMBINATORIAL_DIMS
The combinatorial dimensions of the maximal cones.
 Type:

MAXIMAL_CONES_INCIDENCES
Array of incidence matrices of all maximal cones.
 Type:

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

N_CONES
Number of
CONES
. Type:

N_MAXIMAL_CONES
Number of
MAXIMAL_CONES
. Type:

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

SIMPLICIAL
The polyhedral fan is simplicial if all maximal cones are simplicial.
 Type:
Geometry
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>

FAN_AMBIENT_DIM
Dimension of the space which contains the polyhedral fan.
 Type:

FAN_DIM
Dimension of the polyhedral fan.
 Type:

FULL_DIM
True if
FAN_DIM
andFAN_AMBIENT_DIM
coincide. Type:

LINEALITY_DIM
Dimension of
LINEALITY_SPACE
. Type:

LINEALITY_SPACE
Since we do not require our cones to be pointed: a basis of the lineality space of the fan. Coexists with
RAYS
. Type:
Matrix<Scalar,NonSymmetric>

LINEAR_SPAN_NORMALS
The possible linear span normals of all maximal cones.
 Type:
Matrix<Scalar,NonSymmetric>

MAXIMAL_CONES_FACETS
Tells for each maximal cone what are its facets. A negative number means that the corresponding row of
FACET_NORMALS
has to be negated. Type:

MAXIMAL_CONES_LINEAR_SPAN_NORMALS
Tells for each maximal cone what is its linear span. Indices refer to
LINEAR_SPAN_NORMALS
. Rows correspond toMAXIMAL_CONES_FACETS
. Type:

N_FACET_NORMALS
The number of possible facet normals of all maximal cones.
 Type:

N_INPUT_RAYS
Number of
INPUT_RAYS
. Type:

N_RAYS
Number of
RAYS
. Type:

ORTH_LINEALITY_SPACE
A basis of the orthogonal complement to
LINEALITY_SPACE
. Type:
Matrix<Scalar,NonSymmetric>

POINTED
True if
LINEALITY_SPACE
is trivial. Type:

PSEUDO_REGULAR
True if the fan is a subfan of a
REGULAR
fan Type:

RAYS
Rays from which the cones are formed. Nonredundant. Coexists with
LINEALITY_SPACE
. Type:
Matrix<Scalar,NonSymmetric>

REGULAR
True if the fan is the normal fan of a bounded polytope.
 Type:
Symmetry
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:

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

GROUP
 Type:
 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:
 Type:
 Properties of MAXIMAL_CONES_ACTION:

REPRESENTATIVE_COMBINATORIAL_DIMS
dimensions of representatives of maximal cones
 Type:

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


REPRESENTATIVE_CONES
 Type:

REPRESENTATIVE_MAXIMAL_CONES
 Type:

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:

MAXIMAL_CONES_ORBIT_SIZES
Number of
MAXIMAL_CONES_REPS
in each orbit. Type:

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

MAXIMAL_CONES_REPS_DIMS
The dimensions of
MAXIMAL_CONES_REPS
. Type:

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:

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 toMAXIMAL_CONES_REPS_FACETS
 Type:

N_MAXIMAL_CONE_ORBITS
Number of orbits of
MAXIMAL_CONES
. Type:

N_RAY_ORBITS
Number of orbits of
RAYS
. Type:

N_SYMMETRIES
Number of elements of the symmetry group.
 Type:

ORBITS_F_VECTOR
f_{k} is the number of kdimensional cones up to symmetry.
 Type:

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

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

RAYS_ORBIT_SIZES
Number of elements of each orbit of
RAYS
. Type:

RAYS_REPS
One ray from each orbit. Nonredundant.
 Type:
Matrix<Scalar,NonSymmetric>

RAYS_REPS_LABELS
Unique names assigned to the
RAYS_REPS
. Type:

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 ith entry is the image of the ith coordinate.
 Type:

SYMMETRY_GROUP
Each element of the array is an element of the symmetry group.
 Type:
Topology
The following properties are topological invariants.

HOMOLOGY
The homology of the intersection of the fan with the unit sphere.
 Type:

INTERSECTION_COMPLEX
If the fan is
SIMPLICIAL
the simplicial complex obtained by intersection the fan with the unit sphere. If the fan is notSIMPLICIAL
the crosscut complex of the intersection. Type:
Toric Varieties
Properties coming from associated toric varieties

GORENSTEIN
A fan is Gorenstein if it is
Q_GORENSTEIN
withQ_GORENSTEIN_INDEX
equal to one. Type:

Q_GORENSTEIN
A fan is QGORENSTEIN if each maximal cone is Q_Gorenstein.
 Type:

Q_GORENSTEIN_INDEX
If a fan is
Q_GORENSTEIN
, then its QGorenstein index is the least common multiple of the QGorenstein indices of its maximal cones. Otherwise Q_GORENSTEIN_INDEX is undefined. Type:

SMOOTH_FAN
A fan is smooth if all cones of the fan are smooth.
 Type:
Visualization
These properties are for visualization.

INPUT_RAY_LABELS
Unique names assigned to the
INPUT_RAYS
. Similar toRAY_LABELS
forRAYS
. Type:

MAXIMAL_CONE_LABELS
Unique names assigned to the
MAXIMAL_CONES
. Similar toRAY_LABELS
forRAYS
. Type:

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:
Methods
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.

CONES_DIMS()
The dimensions of the cones.
 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.
 Returns:

DIM()
Returns the dimension of the linear space spanned by the fan.
 Returns:
Symmetry
These methods capture information of the object that is concerned with the action of permutation groups.
Visualization
These methods are for visualization.

VISUAL()
Visualizes the fan, intersected with the unit ball.
 Options:
 option list
geometric_options_linear
 Returns: