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: common, graph, polytope
uses: group, ideal, topaz

User Functions

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Consistency check

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

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check_fan (rays, cones) → PolyhedralFan

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 IncidenceMatrix cones
Options
 Matrix lineality_space Common lineality space for the cones. Bool verbose prints information about the check.
Returns
 PolyhedralFan
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check_fan_objects <Coord> (cones) → PolyhedralFan

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

Type Parameters
 Coord
Parameters
 Array cones
Options
 Bool verbose prints information about the check.
Returns
 PolyhedralFan
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Finite metric spaces

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

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max_metric (n) → Matrix

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
 Matrix

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`
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metric_extended_tight_span (M) → PolyhedralComplex

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

Parameters
 Matrix M a metric
Returns
 PolyhedralComplex

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`
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metric_tight_span (M) → SubdivisionOfPoints

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

Parameters
 Matrix M a metric
Options
 Bool extended If true, the extended tight span is computed.
Returns
 SubdivisionOfPoints

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`
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min_metric (n) → Matrix

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
 Matrix

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`
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thrackle_metric (n) → Matrix

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
 Matrix

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`
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tight_span_max_metric (n) → SubdivisionOfPoints

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
 SubdivisionOfPoints

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`
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tight_span_min_metric (n) → SubdivisionOfPoints

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
 SubdivisionOfPoints

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`
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tight_span_thrackle_metric (n) → SubdivisionOfPoints

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
 SubdivisionOfPoints

Example:
• To compute the \$f\$-vector, do this:`> print tight_span_min_metric(5)->POLYTOPAL_SUBDIVISION->TIGHT_SPAN->F_VECTOR;`` 11 15 5`
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Geometry

These functions 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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generating_polyhedron_facets (P) → Matrix<Scalar>

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

Parameters
 PolyhedralFan P
Returns
 Matrix
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Other

Special purpose functions.

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building_set (generators, n) → PowerSet

Produce a building set from a family of sets.

Parameters
 Array generators the generators of the building set Int n the size of the ground set
Returns
 PowerSet the induced building set
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cone_of_tubing (G, T) → Cone

Output the cone of a tubing

Parameters
 Graph G the input graph Graph T the input tubing
Returns
 Cone
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flip_tube (G, T, t) → Graph

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
 Graph
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is_building_set (check_me, n) → Bool

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
 Bool is check_me really a building set?
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is_B_nested (check_me, B) → Bool

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

Parameters
 Set check_me the would-be nested sets PowerSet B the building set
Returns
 Bool is the family of sets really nested wrt B?
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tubes_of_graph (G) → Set<Set>

Output the set of all tubes of a graph

Parameters
 Graph G the input graph
Returns
 Set
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tubes_of_tubing (G, T) → Set<Set>

Output the tubes of a tubing

Parameters
 Graph G the input graph Graph T the input tubing
Returns
 Set
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tubing_of_graph (G) → Set<Set>

Output one tubing of a graph

Parameters
 Graph G the input graph
Returns
 Set
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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).

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common_refinement (f1, f2) → PolyhedralFan

Computes the common refinement of two fans.

Parameters
 PolyhedralFan f1 PolyhedralFan f2
Returns
 PolyhedralFan
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face_fan <Coord> (p, v) → PolyhedralFan

Computes the face fan of p.

Type Parameters
 Coord
Parameters
 Polytope p Vector v a relative interior point of the polytope
Returns
 PolyhedralFan
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face_fan <Coord> (p) → PolyhedralFan

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

Type Parameters
 Coord
Parameters
 Polytope p
Returns
 PolyhedralFan
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graph_associahedron_fan (G) → PolyhedralFan

Produce the dual fan of a graph associahedron.

Parameters
 Graph G the input graph
Returns
 PolyhedralFan
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groebner_fan (I) → PolyhedralFan

Call wiki:external_software#gfan to compute the greobner fan of an ideal.

Parameters
 ideal::Ideal I input ideal
Returns
 PolyhedralFan
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hyperplane_arrangement (H) → PolyhedralFan

Compute the fan given by a bunch of hyperplanes H.

Parameters
 Matrix H
Returns
 PolyhedralFan
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k_skeleton <Coord> (F, k) → PolyhedralFan

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
 PolyhedralFan F Int k the desired top dimension
Returns
 PolyhedralFan
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normal_fan <Coord> (p) → PolyhedralFan

Computes the normal fan of p.

Type Parameters
 Coord
Parameters
 Polytope p
Returns
 PolyhedralFan
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planar_net (p) → PlanarNet

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
 Polytope p
Returns
 PlanarNet
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product (F1, F2) → PolyhedralFan

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

Parameters
 PolyhedralFan F1 PolyhedralFan F2
Options
 Bool no_coordinates only combinatorial information is handled
Returns
 PolyhedralFan
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projection (P, indices) → PolyhedralFan

Orthogonally project a pointed fan to a coordinate subspace.

The subspace the fan P is projected on is given by indices in the set indices. The option revert inverts the coordinate list.

Parameters
 PolyhedralFan P Array indices
Options
 Bool revert inverts the coordinate list
Returns
 PolyhedralFan
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project_full (P) → PolyhedralFan

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

Parameters
 PolyhedralFan P
Options
 Bool no_labels VERTEX_LABELS]] to the projection. default: 0
Returns
 PolyhedralFan
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secondary_fan (M) → PolyhedralFan

Call wiki:external_software#gfan to compute the secondary fan of a point configuration.

Parameters
 Matrix M a matrix whose rows are the vectors in the configuration
Returns
 PolyhedralFan
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secondary_fan (P) → PolyhedralFan

Call wiki:external_software#gfan to compute the secondary fan of a point configuration.

Parameters
 polytope::PointConfiguration P
Returns
 PolyhedralFan
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Producing a polyhedral complex

These clients provide constructions for PolyhedralComplex objects.

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mixed_subdivision (P_0, P_1, VIF, t_0, t_1) → PolyhedralComplex

Create a weighted mixed subdivision of the 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 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
 PolyhedralComplex
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mixed_subdivision (m, C, a) → PolyhedralComplex

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 a triangulation of C
Options
 Vector 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
 PolyhedralComplex
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mixed_subdivision (A) → PolyhedralComplex

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 A the input polytopes
Options
 Vector 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
 PolyhedralComplex
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tiling_quotient <Coord> (P, Q) → PolyhedralComplex

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
 PolyhedralComplex
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Symmetry

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

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combinatorial_symmetries (f) → group::PermutationAction

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
 PolyhedralFan f
Returns
 group::PermutationAction the action of the combinatorial symmetry group on the rays

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 (f, k) → group::PermutationAction

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
 fan::PolyhedralFan f the input fan Int k the dimension of the cones to induce the action on
Returns
 group::PermutationAction a the action induced by Aut(f) on the set of k-dimensional cones

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}`
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orbit_complex (input_complex, gens) → fan::PolyhedralComplex

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

Parameters
 fan::PolyhedralComplex input_complex the generating complex of the orbit complex Array> gens the generators of a permutation group that acts on the coordinates of the ambient space
Returns
 fan::PolyhedralComplex the orbit complex of input_complex w.r.t. the coordinate action generated by gens

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 \$oc->VERTICES before \$oc->MAXIMAL_POLYTOPES. `> print \$oc->VERTICES;`` 1 1 1`` 1 1 0`` 1 -1 -1`` 1 0 1``> print \$oc->N_MAXIMAL_POLYTOPES;`` 4`
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orbit_complex (input_fan, a) → Polytope

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

Parameters
 fan::PolyhedralFan input_fan the generating fan of the orbit fan group::PermutationAction a the action of a permutation group on the coordinates of the ambient space
Returns
 Polytope the orbit fan of input_fan w.r.t. the action a

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:` type: Group as PolyhedralComplex<Rational>::GROUP`` `` MATRIX_ACTION_ON_COMPLEX`` type: MatrixActionOnVectors<Rational> as PolyhedralComplex<Rational>::GROUP::MATRIX_ACTION_ON_COMPLEX`` `` 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 \$oc->VERTICES before \$oc->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 (input_fan, gens) → fan::PolyhedralFan

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

Parameters
 fan::PolyhedralFan input_fan the generating fan of the orbit fan Array> gens the generators of a permutation group that acts on the coordinates of the ambient space
Returns
 fan::PolyhedralFan the orbit fan of input_fan w.r.t. the coordinate action generated by gens

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 \$of->RAYS before \$of->MAXIMAL_CONES. `> print \$of->RAYS;`` 1 1`` 1 0`` -1 -1`` 0 1``> print \$of->N_MAXIMAL_CONES;`` 4`
•
orbit_fan <Scalar> (input_fan, gens) → fan::PolyhedralFan

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
 fan::PolyhedralFan input_fan the generating fan of the orbit fan Array> gens the generators of a matrix group that acts on the ambient space
Returns
 fan::PolyhedralFan the orbit fan of input_fan w.r.t. the matrix action generated by gens

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 \$of->RAYS before \$of->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`
•

Visualization

These functions are for visualization.

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splitstree (vis_obj ...)

Call wiki:external_software#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 (M) → Visual::Object

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

Parameters
 Matrix M Matrix defining a metric
Options
 Array taxa Labels for the taxa String name Name of the file
Returns
 Visual::Object

Common Option Lists

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Geometry

These options 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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secondary_cone_options

Parameters for user method secondary_cone.

Options
 Matrix equations system of linear equation the cone is cut with Set lift_to_zero restrict lifting function to zero at points designated Bool lift_face_to_zero restrict lifting functions to zero at the entire face spanned by points designated Bool test_regularity throws an exception if subdivision is not regular
•

Visualization

These options are for visualization.

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flap_options

Parameters to control the shapes of the flaps. The precise values are more or less randomly chosen. Should work fine for most polytopes whose edge lengths do not vary too much.

Options
 Float WidthMinimum absolute minimum width Float WidthRelativeMinimum minimum relative width (w.r.t. length of first edge) Float Width absolute width, overrides the previous Float FlapCutOff proportion by which outer edge of flap is shorter than the actual edge (on both sides)
•
geometric_options

Options for visualizing fans.

Options
 Matrix BoundingBox
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Visual::Graph::TightSpanDecorations
UNDOCUMENTED
imports from: Visual::Graph::decorations, Visual::Wire::decorations, Visual::PointSet::decorations

Options
 Array Taxa Labels for the taxa of the metric.