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:
HyperplaneArrangement
:
A hyperplane arrangement. The hyperplane arrangement is given by a matrix HYPERPLANES
whose rows are the linear equations of the hyperplanes and an optional support cone. The support cone defaults to being the whole space. Duplicate hyperplanes are ignored, as well as hyperplanes that intersect the support cone trivially. The support cone is subdivided by the hyperplanes resulting in a fan CELL_DECOMPOSITION
.
PlanarNet
:
A special big object class devoted to planar unfoldings of 3-polytopes. Its main functionality is the visualization.
PolyhedralComplex
:
A polyhedral complex. The derivation from PolyhedralFan
works like the derivation of Polytope
from Cone
.
PolyhedralFan
:
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
.
SubdivisionOfPoints
:
The inhomogeneous variant of SubdivisionOfVectors
, similar to the derivation of PointConfiguration
from VectorConfiguration
.
SubdivisionOfVectors
:
A subdivision of vectors, in contrast to PolyhedralFan
this allows cells with interior points. Similar to the distinction between Cone
and VectorConfiguration
.
Visual::PlanarNet
:
Visualization of a 3-polytope as a planar net.
Visual::PolyhedralFan
:
Visualization of a polyhedral fan as a graph
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.
Matrix
rays
IncidenceMatrix
cones
Matrix
lineality_space
: Common lineality space for the cones.
Bool
verbose
: prints information about the check.
check_fan_objects<Coord>(Array<Cone> cones)
Checks whether the Cone
objects form a polyhedral fan. If this is the case, returns that PolyhedralFan
.
Coord
Bool
verbose
: prints information about the check.
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)
Int
n
: the number of points
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.
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.
Bool
extended
: If true, the extended tight span is computed.
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)
Int
n
: the number of points
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)
Int
n
: the number of points
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)
Int
n
: the number of points
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)
Int
n
: the number of points
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)
Int
n
: the number of points
To compute the $f$-vector, do this:
> print tight_span_min_metric(5)->POLYTOPAL_SUBDIVISION->TIGHT_SPAN->F_VECTOR; 11 15 5
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.
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.
Scalar
: the underlying number type
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
Bool
verbose
: print the final height function used=? Default 0
induced_subdivision
Calculate the subdivision induced on a polytope by a height function h.
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.
face_fan<Coord>(Polytope p, Vector v)
Computes the face fan of p.
face_fan<Coord>(Polytope p)
Computes the face fan of p. the polytope has to be CENTERED
Coord
Polytope
p
gfan_secondary_fan(Matrix M)
Call gfan to compute the secondary fan of a point configuration.
Matrix
M
: a matrix whose rows are the vectors in the configuration
gfan_secondary_fan(PointConfiguration P)
Call gfan to compute the secondary fan of a point configuration.
graph_associahedron_fan(Graph G)
Produce the dual fan of a graph associahedron.
Graph
G
: the input graph
groebner_fan(Ideal I)
Call gfan to compute the greobner fan of an ideal.
Ideal
I
: input ideal
hyperplane_arrangement(Matrix H)
Compute the fan given by a bunch of hyperplanes H.
Matrix
H
intersection(PolyhedralFan F, Matrix H)
Construct a new fan as the intersection of given fan with a subspace.
Matrix
H
: equations of subspace
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.
Coord
Int
k
: the desired top dimension
normal_fan<Coord>(Polytope p)
Computes the normal fan of p.
Coord
Polytope
p
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.
If it does, please, notify the polymake team! Seriously.
Polytope
p
product(PolyhedralFan F1, PolyhedralFan F2)
Construct a new polyhedral fan as the product of two given polyhedral fans F1 and F2.
Bool
no_coordinates
: only combinatorial information is handled
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.
Bool
no_labels
: Do not copy VERTEX_LABELS
to the projection. default: 0
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}
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.
Polytope
P_0
: the first polytope
Polytope
P_1
: the second polytope
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
Bool
no_labels
: Do not copy VERTEX_LABELS
from the original polytopes. default: 0
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.
Int
m
: the number of polytopes giving rise to the Cayley embedding
Polytope
C
: the Cayley embedding of the input polytopes
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
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.
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
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.
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.
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]
.
PolyhedralFan
f
: the input fan
Int
k
: the dimension of the cones to induce the action on
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.
PolyhedralComplex
input_complex
: the generating complex of the orbit complex
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.
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
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.
PolyhedralFan
input_fan
: the generating fan of the orbit fan
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.
Scalar
: underlying number type
PolyhedralFan
input_fan
: the generating fan of the orbit fan
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
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.
VectorConfiguration
V
: (or polytope) the input configuration
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
PolyhedralFan<Scalar>
secondary_fan
These functions are for visualization.
splitstree(Visual::Object vis_obj …)
Call SplitsTree with the given visual objects.
Visual::Object
vis_obj …
: objects to display
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.
String
name
: Name of the file
Special purpose functions.
building_set(Array<Set> generators, Int n)
Produce a building set from a family of sets.
Int
n
: the size of the ground set
is_B_nested(Set<Set> check_me, PowerSet B)
Check if a family of sets is nested wrt a given building set.
PowerSet
B
: the building set
tubes_of_graph(Graph G)
Output the set of all tubes of a graph
Graph
G
: the input graph