Polyhedral complexes are contained in the application
fan. There are two ways to define polyhedral complexes in
polymake. First, a polyhedral complex can be defined by an array of input points and, firstly, by a list of polytopes or, secondly, by a list of cells. In both cases the indices refer to the given points. The distinction between them is done due to the definition of regular subdivisions, where a point projected back to the subdivision may be an inner point of a cell and not a vertex of it. In both cases the indices refer to the given points.
fan > $pc1 = new PolyhedralComplex(POINTS=>[[1,0,0],[1,0,1],[1,1,0],[1,1,1]],INPUT_POLYTOPES=>[[0,1,2],[2,3],]); fan > $pc2 = new PolyhedralComplex(POINTS=>[[1,0],[1,1],[1,2]], MAXIMAL_CELLS=>[[0,2],[0,1,2]]);
Since some of the input polytopes may be redundant, you should ask for the
fan > print $pc->MAXIMAL_POLYTOPES;
Triangulations of polytopes form an important special class of polytopal complexes. In polymake they are objects of type
SimplicialComplex (and thus belong to the application
topaz). However, it is easy to convert them as follows:
fan > $c=cube(3); fan > $triangulation=new PolyhedralComplex(VERTICES=>$c->VERTICES,MAXIMAL_POLYTOPES=>$c->TRIANGULATION->FACETS);
In the future, the polytope object
VoronoiDiagram and regular subdivisions will be handled as polyhedral subdivisions.