Getting started with the application fan

Besides the name giving polyhedral fans this application covers a few other big objects and related functions. An overview can be found in the documentation or the interactive help.

However, this tutorial focuses on PolyhedralFan objects. Subdivisions have their own tutorial here and some notes on polyhedral complexes can be found here.

Most of the following code snippets will only work in your polymake shell after switching to the application fan with the command

> application 'fan';

A primal description containing rays and rays-cones incidence relations can be passed to the constructor like this:

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

Former are assigned to INPUT_RAYS as an array of row vectors (which is a matrix). All input rays must not be zero but redundancies are allowed. The latter are assigned to INPUT_CONES and encoded as an array of index sets. Each index set refers to a subset of INPUT_RAYS that forms a cone in the fan, indexing starts with zero. Input rays that do not belong to any of the input cones are ignored. Input cones do not need to be inclusion-wise maximal. Subcones of input cones are, however, implicitly included. Indeed, for our fan $f we obtain:

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

You can specify a fan with lineality by additionally passing INPUT_LINEALITY. Nevertheless, a fan given by input rays and input cones can have lineality as well. Please remind yourself, that all cones in a fan share the same lineality space.

The properties RAYS, MAXIMAL_CONES and LINEALITY_SPACE are giving a non-redundant primal description:

> print rows_labeled($f->RAYS),"\n";
> print $f->MAXIMAL_CONES,"\n";
> print "lineality dimensions: ", $f->LINEALITY_SPACE->rows() ."x". $f->LINEALITY_SPACE->cols();
0:1 0
1:0 1
2:-1 0
3:0 -1
 
{0 1}
{1 2}
{2 3}
{0 3}
 
lineality dimensions: 0x2

Note that, even though LINEALITY_SPACE is an empty matrix, its number of columns is equal to the ambient dimension of $f.

Instead of the input properties, you may right away use RAYS, MAXIMAL_CONES and LINEALITY_SPACE for construction purposes but keep in mind:

Unlike input rays and input cones, only providing rays and maximal cones may not describe a fan with lineality. In this case polymake assumes an empty lineality space. All given rays must be non-redundant and in case of non-pointed fans LINEALITY_SPACE stores a basis of the lineality space.

The following properties give rise to a dual description:

> print rows_labeled($f->FACET_NORMALS),"\n";
> print rows_labeled($f->MAXIMAL_CONES_FACETS);
0:1 0
1:0 1
 
0:1 1
1:-1 1
2:-1 -1
3:1 -1

Where FACET_NORMALS is an array of row vectors, the facet normals of all maximal cones. Incidence relations between them are stored in the sparse matrix MAXIMAL_CONES_FACETS. Each row corresponds to a maximal cone and each column to a facet normal. Its entries are 0, 1 or -1 encoding either no incidence, an inner or and outer facet normal of the cone, respectively. For example, the second row of MAXIMAL_CONES_FACETS shows that the first one is an outer and the second one is an inner facet normal of the second maximal cone.

The dual description requires additional information on the linear span of each maximal cone. This is stored in LINEAR_SPAN_NORMALS and MAXIMAL_CONES_LINEAR_SPAN_NORMALS. An empty index set in the latter corresponds to a full dimensional maximal cone. Check out the documentation for more information. All maximal cones in $f are full dimensional, hence LINEAR_SPAN_NORMALS is empty:

> print $f->LINEAR_SPAN_NORMALS->rows."\n\n";
> print $f->MAXIMAL_CONES_LINEAR_SPAN_NORMALS;
0
 
{}
{}
{}
{}

As an example one can extract the second and fourth maximal cone of $f:

> $c1 = $f->cone(1);
> $c3 = $f->cone(3);

and pass them to the user method check_fan_objects, which returns the corresponding PolyhedralFan object if and only if the set of provided cones defines a valid polyhedral fan, id est satisfies the intersection property.

> $checkedfan = check_fan_objects($c1,$c3);
> print $checkedfan->MAXIMAL_CONES;
{0 1}
{2 3}

Polymake provides several clients doing this job.

The inner normal fan of a polytope can be produced with this client. For example the normal fan of the 3-dimensional +/-1 cube:

> $nf = normal_fan(cube(3));

Normal fans of bounded feasible polytopes always satisfy the following properties:

> foreach my $prop (qw(regular pure complete full_dim)) {
>     print ucfirst($prop),": ", $nf->give(uc($prop)),"\n";
> }
Regular: true
Pure: true
Complete: true
Full_dim: true

If the given polytope is not full-dimensional, its normal fan will have lineality.

Face fans of polytopes are always constructed with respect to a certain point in the polytopes relative interior. Providing it is optional if the polytope is centered. Zero will be used as default. If the polytope is not centered you have to pass such a point as a second argument (in homogeneous coordinates). For example:

> $v = new Vector([1,0,0,1/2]);
> $ff = face_fan(cross(3), $v);

This client can be used to obtain a subfan consisting of all cones up to a certain dimension. As an example we construct the skeleton of $nf with $k=2$:

> $nf2skel = k_skeleton($nf,2);

By taking a look at the f-vectors one can see that the latter has no cones of dimension 3.

> print "normal fan: ",$nf->F_VECTOR,"\n";
> print "skeleton:   ",$nf2skel->F_VECTOR;
normal fan: 6 12 8
skeleton:   6 12

This can also be seen in the Hasse diagram of the skeleton.

Note that the Hasse diagram of a polyhedral fan will always contain an artifical node at the top which is marked in black and does not correspond to any cone.

> svg($nf2skel->HASSE_DIAGRAM->VISUAL);