> $M = new Matrix([[1,0,0],[1,2,0],[1,1,1],[1,0,2],[1,2,2],[1,1,3]]);
> $w = new Vector([0,0,1,0,0,3]);
> $S = new fan::SubdivisionOfPoints(POINTS=>$M,WEIGHTS=>$w);
> print $S->MAXIMAL_CELLS;
{0 1 3 4}
{3 4 5}
The six points (given by the six rows of the matrix ''%%M%%'') define a pentagon with interior point ''%%(1,1)%%''. The weights ''%%w%%'' must be given in the same order as the vertices. So in our case the interior point has weight ''%%1%%''. Switching to the ''%%fan%%'' application we may create the regular subdivision via the polymake object ''%%SubdivisionOfPoints%%'', whose input parameters are the points ''%%M%%'' and weights ''%%w%%''. A maximal cell of the subdivision is given by a set of indices, representing the points that cell contains. In our example we got two maximal cells and the first one is a quadrilateral with vertices 0, 1, 3 and 4.
We may visualize the regular subdivision if it is at most three-dimensional.
> $S->VISUAL;
> $w = new Vector([0,0,0,0,0,3]);
> $S = new fan::SubdivisionOfPoints(POINTS=>$M,WEIGHTS=>$w);
> print $S->MAXIMAL_CELLS;
{0 1 2 3 4}
{3 4 5}
> $S->VISUAL;
> $w = new Vector([0,0,-1,0,0,3]);
> $S = new fan::SubdivisionOfPoints(POINTS=>$M,WEIGHTS=>$w);
> print $S->MAXIMAL_CELLS;
{0 2 3}
{0 1 2}
{3 4 5}
{1 2 4}
{2 3 4}
> $S->VISUAL;
> $SC = new topaz::GeometricSimplicialComplex(COORDINATES=>$M, INPUT_FACES=>regular_subdivision($M,$w));
Note that ''%%regular_subdivision($M,$w)%%'' produces a list of the maximal cells in the regular subdivision of the point configuration ''%%M%%'' respecting the weights ''%%w%%''.\\
Now we can ask for many other properties, e.g.
> print $SC->F_VECTOR;
6 10 5
> print $SC->H_VECTOR;
1 3 1 0
Check the properties of ''%%SimplicialComplex%%'' for more available properties.
==== Non-generic Weights ====
If your weights are not generic or you are unsure, then you should create a ''%%PolyhedralComplex%%''. Since every ''%%SubdivisionOfPoints%%'' Object has its underlying ''%%PolyhedralComplex%%'' as a property this can be done simply by:
> $M = new Matrix([[1,0,0,0],[1,0,1,0],[1,1,0,0],[1,1,1,0],[1,0,0,1],[1,1,0,1],[1,0,1,1],[1,1,1,1],[1,0,0,2]]);
> $w = new Vector([1,0,0,1,0,1,1,0,1]);
> $S = new fan::SubdivisionOfPoints(POINTS=>$M,WEIGHTS=>$w);
> $PC = $S->POLYHEDRAL_COMPLEX;
> print $PC->MAXIMAL_POLYTOPES;
{1 2 4 7}
{1 2 3 7}
{1 4 6 7 8}
{2 4 5 7 8}
{0 1 2 4}
As with an simplicial complex you can do some computation with it. For example:
> print $PC->F_VECTOR;
9 20 17 5
> print $PC->SIMPLICIAL;
false
> $PC->VISUAL;
> $p=new Vector([1,0,0,0,0,0]);
> $msd=regular_subdivision(hypersimplex(2,4)->VERTICES,$p);
Each cell of this subdivision is a //matroid polytope//, that is, the convex hull of characteristic vectors of the bases of some matroid on //n// elements of rank //d//. The vertices of the hypersimplices happen to be listed in lexicographical ordering. With this extra knowledge we can cook up suitable labels for pretty-printing the maximal cells.
> print rows_labeled($msd,undef,["12","13","14","23","24","34"]);
0:12 13 14 23 24
1:13 14 23 24 34
In this case the //(2,4)//-hypersimplex (a.k.a. the regular octahedron) is split into two egyptian pyramids.
===== Secondary cone and regularity of subdivisions =====
The set of all the weight vectors which induce the same regular subdivision form a convex cone. In this section we will learn how to compute that cone with polymake and even how we can check whether a given subdivision is regular or not.
==== Secondary Cone of a Regular Subdivision ====
Let us look at the following regular subdivision (using homogeneous coordinates):
> $points = new Matrix([[1,2,0],[1,0,2],[1,-2,0],[1,0,-2],[1,1,0],[1,0,1],[1,-1,0],[1,0,-1]]);
> $cells = new Array>([[0,1,4,5],[0,3,4,7],[2,3,6,7],[1,2,5,6],[4,5,6,7]]);
> $S = new fan::SubdivisionOfPoints(POINTS=>$points, MAXIMAL_CELLS=>$cells);
we can visualize the subdivision with the following command:
> $S->VISUAL;
> $sc = $S->secondary_cone();
> print $sc->DIM;
4
> print $sc->RAYS;
1 1 1 1 0 0 0 0
> print $sc->LINEALITY_SPACE;
3/2 3/2 -1/2 -1/2 1 1 0 0
-13/14 15/14 23/14 -5/14 -2/7 5/7 1 0
5/9 -1/3 5/9 13/9 5/9 1/9 5/9 1
We see that this cone is 4-dimensional with a 3-dimensional lineality space. Every vector of the **interior** of that cone induces weights that produces the same regular subdivision. The vectors on the boundary give rise to a coarsening. As you see by looking at the coordinates, we do not need to lift the inner square to 0, there are also weight vectors which lift them in a different fashion but still produce the same subdivision.
With the options ''%%lift_to_zero%%'' or ''%%lift_face_to_zero%%'' one can specify points or a face which must be lifted to zero. Let us see what happens if we force polymake to only allow weights that lift the inner square to zero. As we see from our input, the face which describes the inner square is the fifth one (but since polymake starts counting at zero, it has number 4).
> $sc_fixed = $S->secondary_cone(lift_face_to_zero=>4);
> print $sc_fixed->DIM;
1
> print $sc_fixed->RAYS;
1 1 1 1 0 0 0 0
> print $sc_fixed->LINEALITY_SPACE;
Now you see that the lineality space of that cone is gone. The only way to lift our points now is by raising all points on the outer square to the same height which should be greater than 0.
==== Regularity of Subdivisions ====
You might run into the situation where you want to check whether a subdivision is regular or not. Don't worry, polymake comes to rescue along with it's side-kick the ''%%is_regular%%'' function. The function takes the same input as the ''%%secondary_cone%%'' function. The output is a pair of a boolean and a vector. The boolean tells you whether or not the subdivision is regular and the vector is a weight vector which induces your subdivision. Let us check if our subdivision of the example above is regular.
> print is_regular($points, $cells);
1 <0 0 1 1 0 0 1/2 1/2>
> $lambda = new Vector("0 0 1 1 0 0 1/2 1/2");
> print regular_subdivision($points, $lambda);
{1 2 5 6}
{0 1 4 5}
{0 3 4 7}
{2 3 6 7}
{4 5 6 7}
As you can see polymake tells us that our subdivision is indeed regular. And the vector ''%%0 0 1 1 0 0 1/2 1/2%%'' produces the same subdivision we started with. Now let us see what happens if we use a subdivision which is not regular. We use the same points as above but different cells.
> $nreg_cells = new Array>([[0,1,5],[0,4,5],[0,3,4],[3,4,7],[2,3,7],[2,6,7],[1,2,6],[1,5,6],[4,5,6,7]]);
> $nreg_pc = new fan::PolyhedralComplex(POINTS=>$points, INPUT_POLYTOPES=>$nreg_cells);
> $nreg_pc->VISUAL;
> print is_regular($points, $nreg_cells);
0 <>
Just as expected polymake tells us that the subdivision is not regular. But although there is no weight function which induces this particular subdivision we can still create a ''%%SubdivisionOfPoints%%'' object with the given cells. What happens if we ask for the secondary cone of that subdivision?
> $nreg_sub = new fan::SubdivisionOfPoints(POINTS=>$points, MAXIMAL_CELLS=>$nreg_cells);
> $nreg_sc = $nreg_sub->secondary_cone();
> print $nreg_sc->DIM;
4
This might be weird at first. Although our subdivision is not regular (meaning there does not exist a secondary cone for that subdivision) the secondary cone seems to be 4 dimensional. Let us investigate.
> print $nreg_sc->RAYS;
1 1 1 1 0 0 0 0
> print $nreg_sc->LINEALITY_SPACE;
3/2 3/2 -1/2 -1/2 1 1 0 0
-13/14 15/14 23/14 -5/14 -2/7 5/7 1 0
5/9 -1/3 5/9 13/9 5/9 1/9 5/9 1
This looks familiar, doesn't it? We got the same secondary cone as we did in our first example. The reason for this is that the first example is the finest regular coarsening of our non regular subdivision. And this is exactly what we see here. In other words, the subdivision of our first example is the closest regular thing similar to our non regular subdivision and this is exactly what the ''%%secondary_cone%%'' function produces. If a subdivision is not regular it gives you the secondary cone of the next best regular subdivision.