Polyhedral complexes in polymake

Polyhedral complexes are contained in the application fan, so you hanve to switch application to access the full functionality. 

> application "fan";

To define polyhedral complexes in polymake, you need to provide an array of input points and a list of polytopes represented as an array of arrays of point indices. 

> $pc1 = new PolyhedralComplex(POINTS=>[[1,0,0],[1,0,1],[1,1,0],[1,1,1]],INPUT_POLYTOPES=>[[0,1,2],[2,3],[1]]);

Since some of the input polytopes may be redundant, you should ask for the MAXIMAL_POLYTOPES. 

> print $pc1->MAXIMAL_POLYTOPES;
{0 1 2}
{2 3}
> $pc1->VISUAL;
pcom:pc1
Explode
Automatic explosion
Exploding speed
Transparency
depthWrite
Rotation
x-axis
y-axis
z-axis
Rotation speed
Display
Objects
Camera
SVG
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Triangulations

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: 

> $c=cube(3);
> $triangulation=new PolyhedralComplex(VERTICES=>$c->VERTICES,MAXIMAL_POLYTOPES=>$c->TRIANGULATION->FACETS);

Voronoi Diagrams and regular subdivisions

There are separate tutorials for Voronoi diagrams and regluar subdivisions of point sets.