> $p = new Polytope(POINTS=>[[1,-1,-1],[1,1,-1],[1,-1,1],[1,1,1],[1,0,0]]);
The ''%%POINTS%%'' can be any set of coordinates, they are not required to be irredundant nor vertices of their convex hull. To compute the actual vertices of our polytope, we do this:
> print $p->VERTICES;
1 -1 -1
1 1 -1
1 -1 1
1 1 1
You can also add a lineality space via the input property ''%%INPUT_LINEALITY%%''.
> $p2 = new Polytope(POINTS=>[[1,-1,-1],[1,1,-1],[1,-1,1],[1,1,1],[1,0,0]],INPUT_LINEALITY=>[[0,1,0]]);
To take a look at what that thing looks like, you can use the ''%%VISUAL%%'' method:
> $p2->VISUAL;
> $p3 = new Polytope(VERTICES=>[[1,-1,-1],[1,1,-1],[1,-1,1],[1,1,1]], LINEALITY_SPACE=>[]);
==== H-Description ====
It is also possible to define a polytope as an intersection of finitely many halfspaces, i.e., a matrix of inequalities.
An inequality a0 + a1 x1 + ... + ad xd >= 0 is encoded as a row vector (a0,a1,...,ad), see also [[coordinates|Coordinates for Polyhedra]]. Here is an example:
> $p4 = new Polytope(INEQUALITIES=>[[1,1,0],[1,0,1],[1,-1,0],[1,0,-1],[17,1,1]]);
To display the inequalities in a nice way, use the ''%%print_constraints%%'' method.
> print_constraints($p4->INEQUALITIES);
0: x1 >= -1
1: x2 >= -1
2: -x1 >= -1
3: -x2 >= -1
4: x1 + x2 >= -17
5: 0 >= -1
The last inequality means 17+x1+x2 >= 0, hence it does not represent a facet of the polytope. If you want to take a look at the acutal facets, do this:
> print $p4->FACETS;
1 1 0
1 0 1
1 -1 0
1 0 -1
If your polytope lies in an affine subspace then you can specify its equations via the input property ''%%EQUATIONS%%''.
> $p5 = new Polytope(INEQUALITIES=>[[1,1,0,0],[1,0,1,0],[1,-1,0,0],[1,0,-1,0]],EQUATIONS=>[[0,0,0,1],[0,0,0,2]]);
Again, if you are sure that all your inequalities are facets, you can use the properties ''%%FACETS%%'' and ''%%AFFINE_HULL%%'' instead. Note that this pair of properties is dual to the pair ''%%VERTICES%%'' / ''%%LINEALITY_SPACE%%'' described above.
===== Convex Hulls =====
Of course, ''%%polymake%%'' can convert the V-description of a polytope to its H-description and vice versa. Depending on the individual configuration polymake chooses one of the several convex hull computing algorithms that have a ''%%polymake%%'' interface. Available algorithms are double description ([[http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html|cdd]] of [[http://bugseng.com/products/ppl|ppl]]), reverse search ([[http://cgm.cs.mcgill.ca/~avis/C/lrs.html|lrs]]), and beneath beyond (internal). It is also possible to specify explicitly which method to use by using the ''%%prefer%%'' command:
> prefer "lrs"; # use lrs until revoked by another 'prefer' or 'reset_preference "lrs"'
> $p = new Polytope(POINTS=>[[1,1],[1,0]]);
> print $p->FACETS;
1 -1
0 1
===== A Neighborly Cubical Polytope =====
''%%polymake%%'' provides a variety of standard polytope constructions and transformations. This example construction introduces some of them. Check out the [[documentation:latest:polytope|documentation]] for a comprehensive list.
The goal is to construct a 4-dimensional cubical polytope which has the same graph as the 5-dimensional cube. It is an example of a //neighborly cubical// polytope as constructed in
* Joswig & Ziegler: Neighborly cubical polytopes. Discrete Comput. Geom. 24 (2000), no. 2-3, 325--344, [[http://www.springerlink.com/content/m73pqv6kr80rw4b1/|DOI 10.1007/s004540010039]]
This is the entire construction in a few lines of ''%%polymake%%'' code:
> $c1 = cube(2);
> $c2 = cube(2,2);
> $p1x2 = product($c1,$c2);
> $p2x1 = product($c2,$c1);
> $nc = conv($p1x2,$p2x1);
Let us examine more closely what this is about. First we constructed a square ''%%$c1%%'' via calling the function ''%%cube%%''. The only parameter ''%%2%%'' is the dimension of the cube to be constructed. It is not obvious how the coordinates are chosen; so let us check.
> print $c1->VERTICES;
1 -1 -1
1 1 -1
1 -1 1
1 1 1
The four vertices are listed line by line in homogeneous coordinates, where the homogenizing coordinate is the leading one. As shown the vertices correspond to the four choices of ''%%+/-1%%'' in two positions. So the area of this square equals four, which is verified as follows:
> print $c1->VOLUME;
4
Here the volume is the Euclidean volume of the ambient space. Hence the volume of a polytope which is not full-dimensional is always zero.
The second polytope ''%%$c2%%'' constructed is also a square. However, the optional second parameter says that ''%%+/-2%%''-coordinates are to be used rather than ''%%+/-1%%'' as in the default case. The optional parameter is also allowed to be ''%%0%%''. In this case a cube with ''%%0/1%%''-coordinates is returned. You can access the documentation of functions by typing their name in the ''%%polymake%%'' shell and then hitting F1.
The third command constructs the polytope ''%%$p1x2%%'' as the cartesian product of the two squares. Clearly, this is a four-dimensional polytope which is combinatorially (even affinely) equivalent to a cube, but not congruent. This is easy to verify:
> print isomorphic($p1x2,cube(4));
true
> print congruent($p1x2,cube(4));
0
Both return values are boolean, represented by the numbers ''%%1%%'' and ''%%0%%'', respectively. This questions are decided via a reduction to a graph isomorphism problem which in turn is solved via ''%%polymake%%'''s interface to ''%%nauty%%''.
The polytope ''%%$p2x1%%'' does not differ that much from the previous. In fact, the construction is twice the same, except for the ordering of the factors in the call of the function ''%%product%%''. Let us compare the first vertices of the two products. One can see how the coordinates are induced by the ordering of the factors.
> print $p1x2->VERTICES->[0];
1 1 1 2 2
> print $p2x1->VERTICES->[0];
1 2 2 1 1
In fact, one of these two products is obtained from the other by exchanging coordinate directions. Thats is to say, they are congruent but distinct as subsets of Euclidean 4-space. This is why taking their joint convex hull yields something interesting. Let us explore what kind of polytope we got.
> print $nc->SIMPLE, " ", $nc->SIMPLICIAL;
false false
This says the polytope is neither simple nor simplicial. A good idea then is to look at the f-vector. Beware, however, this usually requires to build the entire face lattice of the polytope, which is extremely costly. Therefore this is computationally infeasible for most high-dimensional polytopes.
> print $nc->F_VECTOR;
32 80 72 24
This is a first hint that our initial claim is indeed valid. The polytope constructed has 32 vertices and 80 = 32*5/2 edges, as many as the 5-dimensional cube:
> print cube(5)->F_VECTOR;
32 80 80 40 10
What is left is to check whether the vertex-edge graphs of the two polytopes actually are the same, and if all proper faces are combinatorially equivalent to cubes.
> print isomorphic($nc->GRAPH->ADJACENCY,cube(5)->GRAPH->ADJACENCY);
true
> print $nc->CUBICAL;
true
See the [[apps_graph|tutorial on graphs]] for more on that subject.