====== 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 [[https://polymake.org/release_docs/latest/fan.html|documentation]] or the [[https://polymake.org/doku.php/user_guide/intro_tutorial#getting_help|interactive help]].

However, this tutorial focuses on [[https://polymake.org/release_docs/latest/fan.html#fan__PolyhedralFan__27|PolyhedralFan]] objects. Subdivisions have their own tutorial [[https://polymake.org/doku.php/user_guide/tutorials/regular_subdivisions|here]] and some notes on polyhedral complexes can be found [[https://polymake.org/doku.php/user_guide/tutorials/pcom|here]].

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

<code perl>
> application 'fan';
</code>
===== Polyhedral fans =====

==== Construction from scratch ====

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

<code perl>
> $f = new PolyhedralFan(INPUT_RAYS=>[[1,0],[0,1],[-1,0],[0,-1],[2,0]], INPUT_CONES=>[[0,1,4],[1,2],[2,3],[3,0],[0]]);
</code>
Former are assigned to [[https://polymake.org/release_docs/latest/fan.html#fan__INPUT_RAYS__161|INPUT_RAYS]] as an array of row vectors (which is a matrix). All input rays must not be zero but redundancies are allowed. The latter are assigned to [[https://polymake.org/release_docs/latest/fan.html#fan__INPUT_CONES__160|INPUT_CONES]] and encoded as an array of index sets. Each index set refers to a subset of ''INPUT_RAYS'' that forms a cone in the fan, indexing starts with zero. Input rays that do not belong to any of the input cones are ignored. Input cones do not need to be inclusion-wise maximal. Subcones of input cones are, however, implicitly included. Indeed, for our fan ''$f'' we obtain:

<code perl>
> print $f->CONES;
<{0}
{1}
{2}
{3}
>
<{0 1}
{1 2}
{2 3}
{0 3}
>
</code>
You can specify a fan with lineality by additionally passing [[https://polymake.org/release_docs/latest/fan.html#fan__INPUT_LINEALITY__162|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 [[https://polymake.org/release_docs/latest/fan.html#fan__RAYS__176|RAYS]], [[https://polymake.org/release_docs/latest/fan.html#fan__MAXIMAL_CONES__150|MAXIMAL_CONES]] and [[https://polymake.org/release_docs/latest/fan.html#fan__LINEALITY_SPACE__180|LINEALITY_SPACE]] are giving a **non-redundant** primal description:

<code perl>
> 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
</code>
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 dual description ===

The following properties give rise to a dual description:

<code perl>
> 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
</code>
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 [[https://polymake.org/release_docs/latest/fan.html#fan__MAXIMAL_CONES_LINEAR_SPAN_NORMALS__172|documentation]] for more informations. All maximal cones in ''$f'' are full dimensional, hence ''LINEAR_SPAN_NORMALS'' is empty:

<code perl>
> print $f->LINEAR_SPAN_NORMALS->rows."\n\n";
> print $f->MAXIMAL_CONES_LINEAR_SPAN_NORMALS;
0

{}
{}
{}
{}
</code>
==== Construction from a set of cones ====

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

<code perl>
> $c1 = $f->cone(1);
> $c3 = $f->cone(3);
</code>
and pass them to the user method [[https://polymake.org/release_docs/latest/fan.html#fan__check_fan_objects__54|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.

<code perl>
> $checkedfan = check_fan_objects($c1,$c3);
> print $checkedfan->MAXIMAL_CONES;
{0 1}
{2 3}
</code>
==== Construction from other objects ====

Polymake provides several clients doing this job.

=== normal_fan ===

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

<code perl>
> $nf = normal_fan(cube(3));
</code>
Normal fans of bounded feasible polytopes always satisfy the following properties:

<code perl>
> 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
</code>
If the given polytope is not full-dimensional, its normal fan will have lineality.

=== face_fan ===

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:

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

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 $k=2$:

<code perl>
> $nf2skel = k_skeleton($nf,2);
</code>
By taking a look at the f-vectors one can see that the latter has no cones of dimension 3.

<code perl>
> print "normal fan: ",$nf->F_VECTOR,"\n";
> print "skeleton:   ",$nf2skel->F_VECTOR;
normal fan: 6 12 8
skeleton:   6 12
</code>
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.

<code perl>
> svg($nf2skel->HASSE_DIAGRAM->VISUAL);
requires PDFLaTeX and a PDF viewer;
please specify the output File option or call reconfigure("common::pdfviewer.rules");
</code>