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

application fan

This application deals with polyhedral fans. You can define a fan, e.g. via its RAYS and MAXIMAL_CONES and compute several properties like HASSE_DIAGRAM and F_VECTOR.

imports from:

uses:

Consistency check

These clients provide consistency checks, e.g. whether a given list of rays and cones defines a polyhedral fan.


check_fan(Matrix rays, IncidenceMatrix cones)

Checks whether a given set of rays together with a list cones defines a polyhedral fan. If this is the case, the ouput is the PolyhedralFan defined by rays as INPUT_RAYS, cones as INPUT_CONES, lineality_space as LINEALITY_SPACE if this option is given.

Parameters:

Matrix rays

Options:

Matrix lineality_space: Common lineality space for the cones.

Bool verbose: prints information about the check.

Returns:

check_fan_objects<Coord>(Array<Cone> cones)

Checks whether the Cone objects form a polyhedral fan. If this is the case, returns that PolyhedralFan.

Type Parameters:

Coord

Parameters:

Array<Cone> cones

Options:

Bool verbose: prints information about the check.

Returns:

Finite metric spaces

All around Tight spans of finite metric spaces and their conections to polyhedral geometry


max_metric(Int n)

Compute a metric such that the f-vector of its tight span is maximal among all metrics with n points.

See Herrmann and Joswig: Bounds on the f-vectors of tight spans, Contrib. Discrete Math., Vol.2, (2007)
Parameters:

Int n: the number of points

Returns:
Example:

To compute the max-metric of five points and display the f-vector of its tight span, do this:

 > $M = max_metric(5);
 > $PC = metric_tight_span($M,extended=>1);
 > print $PC->POLYTOPAL_SUBDIVISION->TIGHT_SPAN->F_VECTOR;
 16 20 5


metric_extended_tight_span(Matrix<Rational> M)

Computes a extended tight span which is a PolyhedralComplex with induced from a mertic.

Parameters:

Matrix<Rational> M: a metric

Returns:
Example:

To compute the thrackle-metric of five points and display the f-vector of its tight span, do this:

 > $M = thrackle_metric(5);
 > $PC = metric_extended_tight_span($M);
 > print $PC->F_VECTOR;
 16 20 5


metric_tight_span(Matrix<Rational> M)

Computes a SubdivisionOfPoints with a weight function which is induced from a mertic.

Parameters:

Matrix<Rational> M: a metric

Options:

Bool extended: If true, the extended tight span is computed.

Returns:
Example:

To compute the thrackle-metric of five points and display the f-vector of its tight span, do this:

 > $M = thrackle_metric(5);
 > $PC = metric_tight_span($M,extended=>1);
 > print $PC->POLYTOPAL_SUBDIVISION->TIGHT_SPAN->F_VECTOR;
 16 20 5


min_metric(Int n)

Compute a metric such that the f-vector of its tight span is minimal among all metrics with n points.

See Herrmann and Joswig: Bounds on the f-vectors of tight spans, Contrib. Discrete Math., Vol.2, (2007)
Parameters:

Int n: the number of points

Returns:
Example:

To compute the min-metric of five points and display the f-vector of its tight span, do this:

 > $M = min_metric(5);
 > $PC = metric_tight_span($M,extended=>1);
 > print $PC->POLYTOPAL_SUBDIVISION->TIGHT_SPAN->F_VECTOR;
 16 20 5


thrackle_metric(Int n)

Compute a thrackle metric on n points. This metric can be interpreted as a lifting function for the thrackle triangulation.

See De Loera, Sturmfels and Thomas: Gröbner bases and triangulations of the second hypersimplex, Combinatorica 15 (1995)
Parameters:

Int n: the number of points

Returns:
Example:

To compute the thrackle-metric of five points and display the f-vector of its tight span, do this:

 > $M = thrackle_metric(5);
 > $PC = metric_extended_tight_span($M);
 > print $PC->F_VECTOR;
 16 20 5


tight_span_max_metric(Int n)

Compute a SubdivisionOfPoints with a tight span of a metric such that the f-vector is maximal among all metrics with n points.

See Herrmann and Joswig: Bounds on the f-vectors of tight spans, Contrib. Discrete Math., Vol.2, (2007)
Parameters:

Int n: the number of points

Returns:
Example:

To compute the f-vector of the tight span with maximal f-vector, do this:

 > print tight_span_max_metric(5)->POLYTOPAL_SUBDIVISION->TIGHT_SPAN->F_VECTOR;
 11 15 5


tight_span_min_metric(Int n)

Compute a SubdivisionOfPoints with a tight span of a metric such that the f-vector is minimal among all metrics with n points.

See Herrmann and Joswig: Bounds on the f-vectors of tight spans, Contrib. Discrete Math., Vol.2, (2007)
Parameters:

Int n: the number of points

Returns:
Example:

To compute the f-vector of the tight span with minimal f-vector, do this:

 > print tight_span_min_metric(5)->POLYTOPAL_SUBDIVISION->TIGHT_SPAN->F_VECTOR;
 11 15 5


tight_span_thrackle_metric(Int n)

Compute SubdivisionOfPoints with a tight span of th thrackle metric on n points. This metric can be interpreted as a lifting function which induces the thrackle triangulation of the second hypersimplex.

See De Loera, Sturmfels and Thomas: Gröbner bases and triangulations of the second hypersimplex, Combinatorica 15 (1995)
Parameters:

Int n: the number of points

Returns:
Example:

To compute the $f$-vector, do this:

 > print tight_span_min_metric(5)->POLYTOPAL_SUBDIVISION->TIGHT_SPAN->F_VECTOR;
 11 15 5


Geometry

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


cell_decomposition_brute_force

This function computes the CELL_DECOMPOSITION of a given hyperplane arrangement in a brute force way, by just considering every possible signature. Since not every signature gives a valid cell, it is much cheaper to traverse the cells of CELL_DECOMPOSITION by flipping the walls. This method is here for verifying results of our other algorithms.


generating_polyhedron_facets(PolyhedralFan P)

The facets of a polyhedron that has the fan P as its normal fan, or the empty matrix if no such polyhedron exists.

Parameters:
Returns:
Matrix<Scalar>

induced_subdivision<Scalar>(VectorConfiguration<Scalar> pc, Matrix<Scalar> R, Set I)

Calculate the subdivision induced on a point configuration by a height function h. The height function is specified as the sum of a set of rows of a matrix. Using the RAYS of the secondary_fan of the configuration works well.

Type Parameters:

Scalar: the underlying number type

Parameters:

VectorConfiguration<Scalar> pc: (or polytope/cone) the input configuration

Matrix<Scalar> R: a matrix such that R→cols() == pc→N_VECTORS

Set I: (or ARRAY) a set of indices that select rows from R

Options:

Bool verbose: print the final height function used=? Default 0

Returns:
Set<Set>
induced_subdivision

Calculate the subdivision induced on a polytope by a height function h.


Producing a fan

These clients provide standard constructions for PolyhedralFan objects, e.g. from polytopes (face_fan or normal_fan) or from other fans (via projection, refinement or product).


common_refinement(PolyhedralFan f1, PolyhedralFan f2)

Computes the common refinement of two fans.

Parameters:
Returns:

face_fan<Coord>(Polytope p, Vector v)

Computes the face fan of p.

Type Parameters:

Coord

Parameters:

Vector v: a relative interior point of the polytope

Returns:
face_fan<Coord>(Polytope p)

Computes the face fan of p. the polytope has to be CENTERED

Type Parameters:

Coord

Parameters:
Returns:

gfan_secondary_fan(Matrix M)

Call gfan to compute the secondary fan of a point configuration.

Parameters:

Matrix M: a matrix whose rows are the vectors in the configuration

Returns:
gfan_secondary_fan(PointConfiguration P)

Call gfan to compute the secondary fan of a point configuration.

Parameters:
Returns:

graph_associahedron_fan(Graph G)

Produce the dual fan of a graph associahedron.

Parameters:

Graph G: the input graph

Returns:

groebner_fan(Ideal I)

Call gfan to compute the greobner fan of an ideal.

Parameters:

Ideal I: input ideal

Returns:

hyperplane_arrangement(Matrix H)

Compute the fan given by a bunch of hyperplanes H.

Parameters:

Matrix H

Returns:

intersection(PolyhedralFan F, Matrix H)

Construct a new fan as the intersection of given fan with a subspace.

Parameters:

Matrix H: equations of subspace

Returns:

k_skeleton<Coord>(PolyhedralFan F, Int k)

Computes the k-skeleton of the polyhedral fan F, i.e. the subfan of F consisting of all cones of dimension ⇐k.

Type Parameters:

Coord

Parameters:

Int k: the desired top dimension

Returns:

normal_fan<Coord>(Polytope p)

Computes the normal fan of p.

Type Parameters:

Coord

Parameters:
Returns:

planar_net(Polytope p)

Computes a planar net of the 3-polytope p. Note that it is an open problem if such a planar net always exists.

  • PROGRAM MIGHT TERMINATE WITH AN EXCEPTION *

If it does, please, notify the polymake team! Seriously.

Parameters:
Returns:

product(PolyhedralFan F1, PolyhedralFan F2)

Construct a new polyhedral fan as the product of two given polyhedral fans F1 and F2.

Parameters:
Options:

Bool no_coordinates: only combinatorial information is handled

Returns:

project_full(PolyhedralFan P)

Orthogonally project a fan to a coordinate subspace such that redundant columns are omitted, i.e., the affine hull of the support of the projection is full-dimensional, without changing the combinatorial type.

Parameters:
Options:

Bool no_labels: Do not copy VERTEX_LABELS to the projection. default: 0

Returns:
Example:

x and y axis in 3-space

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


Producing a polyhedral complex

These clients provide constructions for PolyhedralComplex objects.


mixed_subdivision(Polytope P_0, Polytope P_1, Array<Set> VIF, Scalar t_0, Scalar t_1)

Create a weighted mixed subdivision of the scaled Minkowski sum of two polytopes, using the Cayley trick. The polytopes must have the same dimension, at least one of them must be pointed. The vertices of the first polytope P_0 are weighted with t_0, and the vertices of the second polytope P_1 with t_1. Default values are t_0=t_1=1.

Parameters:

Polytope P_0: the first polytope

Polytope P_1: the second polytope

Array<Set> VIF: the indices of the vertices of the mixed cells

Scalar t_0: the weight for the vertices of P_0; default 1

Scalar t_1: the weight for the vertices of P_1; default 1

Options:

Bool no_labels: Do not copy VERTEX_LABELS from the original polytopes. default: 0

Returns:
mixed_subdivision(Int m, Polytope C, Array<Set> a)

Create a weighted mixed subdivision of a Cayley embedding of a sequence of polytopes. Each vertex v of the i-th polytope is weighted with t_i, the i-th entry of the optional array t.

Parameters:

Int m: the number of polytopes giving rise to the Cayley embedding

Polytope C: the Cayley embedding of the input polytopes

Array<Set> a: triangulation of C

Options:

Vector<Scalar> t: scaling for the Cayley embedding; defaults to the all-1 vector

Bool no_labels: Do not copy VERTEX_LABELS from the original polytopes. default: 0

Returns:
mixed_subdivision(Array<Polytope> A, Array<Set> VIF)

Create a weighted mixed subdivision of a sequence (P1,…,Pm) of polytopes, using the Cayley trick. All polytopes must have the same dimension, at least one of them must be pointed. Each vertex v of the i-th polytope is weighted with t_i, the i-th entry of the optional array t.

Parameters:

Array<Polytope> A: the input polytopes

Array<Set> VIF: the indices of the vertices of the mixed cells

Options:

Vector<Scalar> t: scaling for the Cayley embedding; defaults to the all-1 vector

Bool no_labels: Do not copy VERTEX_LABELS from the original polytopes. default: 0

Returns:

tiling_quotient<Coord>(Polytope P, Polytope Q)

Calculates the quotient of P by Q+L, where Q+L is a lattice tiling. The result is a polytopal complex inside Q.

Type Parameters:

Coord

Parameters:

Polytope P: a polytope

Polytope Q: a polytope that tiles space

Returns:

Symmetry

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


combinatorial_symmetries(PolyhedralFan f)

Compute the combinatorial symmetries (i.e., automorphisms of the face lattice) of a given fan f. They are stored in terms of a GROUP.RAYS_ACTION and a GROUP.MAXIMAL_CONES_ACTION property in f, and the GROUP.MAXIMAL_CONES_ACTION is also returned.

Parameters:
Returns:
Example:

To get the ray symmetry group of the square and print its generators, type the following:

 > print combinatorial_symmetries(normal_fan(polytope::cube(2)))->GENERATORS;
 2 3 0 1
 1 0 3 2
 0 2 1 3
 > $f = normal_fan(polytope::cube(2)); combinatorial_symmetries($f);
 > print $f->GROUP->RAYS_ACTION->GENERATORS;
 0 1 3 2
 1 0 2 3
 2 3 0 1
 > print $f->GROUP->MAXIMAL_CONES_ACTION->GENERATORS;
 2 3 0 1
 1 0 3 2
 0 2 1 3


cones_action(PolyhedralFan f, Int k)

Returns the permutation action induced by the symmetry group of the fan f on the set of k-dimensional cones. This action is not stored as a property of f, because polymake doesn't support dynamic names of properties. Be aware that the set of k-dimensional cones itself is $f->CONES->[$k-1] .

Parameters:

PolyhedralFan f: the input fan

Int k: the dimension of the cones to induce the action on

Returns:
Example:

Consider a 3-cube c. To calculate the induced action of Aut(c) on the set of 2-dimensional cones of the normal fan, type

 > $f = fan::normal_fan(polytope::cube(3, group=>1));
 > print fan::cones_action($f,2)->properties();
 name: CONES_ACTION(2)
 type: PermutationAction<Int, Rational>
 description: action induced on 2-dimensional cones
 
 GENERATORS
 0 3 4 1 2 5 7 6 8 10 9 11
 1 0 2 5 6 3 4 7 9 8 11 10
 0 2 1 4 3 8 9 10 5 6 7 11
 > print $f->CONES->[1];
 {2 4}
 {0 4}
 {0 2}
 {1 4}
 {1 2}
 {3 4}
 {0 3}
 {1 3}
 {2 5}
 {0 5}
 {1 5}
 {3 5}


orbit_complex(PolyhedralComplex input_complex, Array<Array<Int>> gens)

Constructs the orbit complex of a given polyhedral complex input_complex with respect to a given set of generators gens.

Parameters:

PolyhedralComplex input_complex: the generating complex of the orbit complex

Array<Array<Int>> gens: the generators of a permutation group that acts on the coordinates of the ambient space

Returns:
Example:

To calculate an orbit complex with respect to a group of coordinate permutations, follow these steps: First specify a seed complex:

 > $f=new PolyhedralComplex(VERTICES=>[[1,1,1],[1,1,0],[1,-1,-1]], MAXIMAL_POLYTOPES=>[[0,1],[1,2]]);

Then define the orbit complex by specifying a permutation action on coordinates:

 > $oc = orbit_complex($f, [[1,0]]);

The only properties of $oc defined so far reside in GROUP:

 > print $oc->GROUP->properties();
 type: Group as PolyhedralComplex<Rational>::GROUP
 
 COORDINATE_ACTION
 type: PermutationAction<Int, Rational> as PolyhedralComplex<Rational>::GROUP::COORDINATE_ACTION
 
 MAXIMAL_POLYTOPES_ACTION
 type: PermutationAction<Int, Rational> as PolyhedralComplex<Rational>::GROUP::MAXIMAL_POLYTOPES_ACTION

Now you can calculate the VERTICES and MAXIMAL_POLYTOPES of the orbit fan. IMPORTANT: You must ask for VERTICES before MAXIMAL_POLYTOPES.

 > print $oc->VERTICES;
 1 1 1
 1 1 0
 1 -1 -1
 1 0 1
 > print $oc->N_MAXIMAL_POLYTOPES;
 4
orbit_complex(PolyhedralFan input_fan, PermutationAction a)

Constructs the orbit fan of a given fan input_fan with respect to a given group action a.

Parameters:

PolyhedralFan input_fan: the generating fan of the orbit fan

PermutationAction a: the action of a permutation group on the coordinates of the ambient space

Returns:
Example:

To calculate an orbit complex with respect to a group of coordinate permutations, follow these steps: First specify a seed complex:

 > $f=new PolyhedralComplex(VERTICES=>[[1,1,1],[1,1,0],[1,1/2,1/4]], MAXIMAL_POLYTOPES=>[[0,2],[1,2]]);

Then define the orbit complex by specifying a matrix group action on the coordinates:

 > $oc = orbit_complex($f, polytope::cube(2,group=>1)->GROUP->MATRIX_ACTION);

The only properties of $oc defined so far reside in GROUP:

Now you can calculate the VERTICES and MAXIMAL_POLYTOPES of the orbit fan. IMPORTANT: You must ask for VERTICES before MAXIMAL_POLYTOPES.

 > print $oc->VERTICES;
 1 1 1
 1 1 0
 1 1/2 1/4
 1 -1 -1
 1 -1 1
 1 1 -1
 1 -1 0
 1 0 -1
 1 0 1
 1 -1/2 -1/4
 1 -1/2 1/4
 1 -1/4 -1/2
 1 -1/4 1/2
 1 1/4 -1/2
 1 1/4 1/2
 1 1/2 -1/4
 > print $oc->N_MAXIMAL_POLYTOPES;
 16


orbit_fan(PolyhedralFan input_fan, Array<Array<Int>> gens)

Constructs the orbit fan of a given fan input_fan with respect to a given set of generators gens.

Parameters:

PolyhedralFan input_fan: the generating fan of the orbit fan

Array<Array<Int>> gens: the generators of a permutation group that acts on the coordinates of the ambient space

Returns:
Example:

To calculate an orbit fan, follow these steps: First specify a seed fan:

 > $f=new PolyhedralFan(RAYS=>[[1,1],[1,0],[-1,-1]], MAXIMAL_CONES=>[[0,1],[1,2]]);

Then define the orbit fan by specifying coordinate permutations:

 > $of = orbit_fan($f,[[1,0]]);

The only properties of $of defined so far reside in GROUP:

 > print $of->GROUP->properties();
 name: unnamed#0
 type: Group as PolyhedralFan<Rational>::GROUP
 
 HOMOGENEOUS_COORDINATE_ACTION
 type: PermutationAction<Int, Rational>
 
 MAXIMAL_CONES_ACTION
 type: PermutationAction<Int, Rational> as PolyhedralFan<Rational>::GROUP::MAXIMAL_CONES_ACTION

Now you can calculate the RAYS and MAXIMAL_CONES of the orbit fan. IMPORTANT: You must ask for RAYS before MAXIMAL_CONES.

 > print $of->RAYS;
 1 1
 1 0
 -1 -1
 0 1
 > print $of->N_MAXIMAL_CONES;
 4
orbit_fan<Scalar>(PolyhedralFan input_fan, Array<Matrix<Scalar>> gens)

Constructs the orbit fan of a given fan input_fan with respect to a given set of matrix group generators gens.

Type Parameters:

Scalar: underlying number type

Parameters:

PolyhedralFan input_fan: the generating fan of the orbit fan

Array<Matrix<Scalar>> gens: the generators of a matrix group that acts on the ambient space

Returns:
Example:

To calculate an orbit fan, follow these steps: First specify a seed fan:

 > $f=new PolyhedralFan(RAYS=>[[1,1,1],[1,1,0],[1,1/2,1/4]],MAXIMAL_CONES=>[[0,2],[1,2]]);

Then define the orbit fan by specifying a matrix group action:

 > $of = orbit_fan($f,polytope::cube(2,group=>1)->GROUP->MATRIX_ACTION);

The only properties of $of defined so far reside in GROUP:

 > print $of->GROUP->properties();
 name: unnamed#0
 type: Group as PolyhedralFan<Rational>::GROUP
 
 MATRIX_ACTION
 type: MatrixActionOnVectors<Rational>
 
 MAXIMAL_CONES_ACTION
 type: PermutationAction<Int, Rational> as PolyhedralFan<Rational>::GROUP::MAXIMAL_CONES_ACTION

Now you can calculate the RAYS and MAXIMAL_CONES of the orbit fan. IMPORTANT: You must ask for RAYS before MAXIMAL_CONES.

 > print $of->RAYS;
 1 1 1
 1 1 0
 1 1/2 1/4
 1 -1 -1
 1 -1 1
 1 1 -1
 1 -1 0
 1 0 -1
 1 0 1
 1 -1/2 -1/4
 1 -1/2 1/4
 1 -1/4 -1/2
 1 -1/4 1/2
 1 1/4 -1/2
 1 1/4 1/2
 1 1/2 -1/4
 > print $of->N_MAXIMAL_CONES;
 16


Triangulations, subdivisions and volume

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


secondary_fan(VectorConfiguration V)

Calculate the secondary fan of a point or vector configuration, or polytope.

Parameters:

VectorConfiguration V: (or polytope) the input configuration

Options:

Array<Set> initial_subdivision: a seed subdivision of V

Matrix restrict_to: the equations defining a subspace that the secondary fan should be restricted to

Int seed: controls the outcome of the random number generator for generating a randomized initial subdivision

Returns:
PolyhedralFan<Scalar>
secondary_fan

Visualization

These functions are for visualization.


splitstree(Visual::Object vis_obj …)

Call SplitsTree with the given visual objects.

Parameters:

Visual::Object vis_obj …: objects to display

Options:

String File: “filename” or “AUTO” Only create a NEXUS format file, don't start the GUI. The .nex suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for the open function, including “-” for terminal output, “&HANDLE” for an already opened file handle, or “| program” for a pipe.


visual_splitstree(Matrix<Rational> M)

Visualize the splits of a finite metric space (that is, a planar image of a tight span). Calls SplitsTree.

Parameters:

Matrix<Rational> M: Matrix defining a metric

Options:

Array<String> taxa: Labels for the taxa

String name: Name of the file

Returns:

Other

Special purpose functions.


building_set(Array<Set> generators, Int n)

Produce a building set from a family of sets.

Parameters:

Array<Set> generators: the generators of the building set

Int n: the size of the ground set

Returns:

cone_of_tubing(Graph G, Graph T)

Output the cone of a tubing

Parameters:

Graph G: the input graph

Graph T: the input tubing

Returns:

flip_tube(Graph G, Graph T, Int t)

Flip a tubing in a tube

Parameters:

Graph G: the input graph

Graph T: the input tubing

Int t: the tube to flip, identified by its root

Returns:

is_B_nested(Set<Set> check_me, PowerSet B)

Check if a family of sets is nested wrt a given building set.

Parameters:

Set<Set> check_me: the would-be nested sets

PowerSet B: the building set

Returns:

is_building_set(PowerSet check_me, Int n)

Check if a family of sets is a building set.

Parameters:

PowerSet check_me: the would-be building set

Int n: the size of the ground set

Returns:

tubes_of_graph(Graph G)

Output the set of all tubes of a graph

Parameters:

Graph G: the input graph

Returns:
Set<Set>

tubes_of_tubing(Graph G, Graph T)

Output the tubes of a tubing

Parameters:

Graph G: the input graph

Graph T: the input tubing

Returns:
Set<Set>

tubing_of_graph(Graph G)

Output one tubing of a graph

Parameters:

Graph G: the input graph

Returns:
Set<Set>

  • documentation/latest/fan.txt
  • Last modified: 2019/08/13 10:31
  • (external edit)