Release date: June 20, 2011.
polytope: The system can now work with arbitrary cones, possibly with a lineality space of positive dimension. Polytopes are derived from these cones; this is backwards compatible with polymakes description of a polytope via its homogenization.
fan to deal with polyhedral fans and polyhedral complexes. Have a look at our tutorial on polyhedral complexes!
We have a first preliminary interface to Gfan which is used for computing secondary fans.
polytope > $sec_fan_of_square=fan::secondary_fan(cube(2)->VERTICES); polymake: used package gfan Gfan is a software package for computing Groebner fans and tropical varieties. Copyright by Anders Jensen http://www.math.tu-berlin.de/~jensen/software/gfan/gfan.html
The crosscut complex of a polytopal complex is homotopy equivalent.
polytope > $crosscut=new topaz::SimplicialComplex(FACETS=>rows($sec_fan_of_square->MAXIMAL_CONES));
This can be exploited to compute the (reduced integral) homology:
polytope > print rows_labeled($crosscut->HOMOLOGY);
0:{} 1
The two connected components are the two vertices of the 0-sphere corresponding to the two ways to triangulate a square.
polymake learned how to deal with permutation groups via an interface to the C++-library PermLib written by Thomas Rehn.
It is now possible to relate one (or more) GroupOfCone to a cone or a polytope via the new property GROUP which encapsulates symmetry information about the object. A GroupOfCone is a permutation group given by some GENERATORS that can act on three different domains: RAYS (DOMAIN⇒1), FACETS (DOMAIN⇒2), or COORDINATES (DOMAIN⇒3), see the following example:
polytope > $p=cube(3);
polytope > $g=new group::GroupOfCone(DOMAIN=>3,GENERATORS=>[[1,2,0],[0,2,1]]);
polytope > $p->GROUP=$g;
polytope > print $p->GROUP->RAYS_IN_ORBITS;
polymake: used package permlib
A callable C++ library for permutation computations.
Written by Thomas Rehn.
http://www.math.uni-rostock.de/~rehn/software/permlib.html
{0}
{1 2 4}
{3 5 6}
{7}
polytope > print $p->GROUP->FACETS_IN_ORBITS;
{0 2 4}
{1 3 5}
polytope > print $p->GROUP->COORDINATES_IN_ORBITS;
{0 1 2}
TPLib implements the tropical version of the double description method.
tropical > $p=new TropicalPolytope(POINTS=>[[1,0,0],[0,1,0],[0,0,1]]);
tropical > print $p->HALF_SPACES;
polymake: used package tplib
TPLib: Tropical Polyhedra Library
Copyright by Xavier Allamigeon
https://gforge.inria.fr/projects/tplib/
(<0 0 0> {0 2})
(<0 0 1> {2})
(<1 0 0> {0})
(<0 0 0> {1 2})
(<0 1 0> {1})
(<0 0 0> {0 1})