## polymake

### Summary

polymake is an object-oriented system for experimental discrete mathematics. The typical working cycle of a polymake user starts with the construction of an object of interest, auch as a convex polytope, a finite simplicial complex, a graph, etc. It is then possible to ask the system for some of the object’s properties or for some form of visualization. Further steps might include more elaborate constructions based on previously defined objects.

Each class of polymake objects comes with a set of rules which describe how a new property of an object can be derived from previously known ones. It is a key feature that the user can extend or modify the set of rules, add further properties or even new classes of objects (with entirely new rule bases).

The functions provided include: several convex hull algorithms, face lattices of convex polytopes, Voronoi diagrams and Delaunay decompositions (in arbitrary dimensions), simplicial homology (with integer coefficients), simplicial cup and cap products, intersection forms of triangulated 4-manifolds. Several forms of (interactive) visualization via interfaces to Geomview, JavaView and other programs.

### Authors

Ewgenij Gawrilow and Michael Joswig with contributions by Thilo Schröder and Nikolaus Witte

### Status

officially approved by the authors### Aims and scope

### Mathematical Classification

### Keywords

- 4-dimensional visualization
- boundary complexes of simplicial polytopes
- building spheres
- combinatorics
- convex hulls
- convex polyhedra
- coxeter groups
- curve fitting
- Delaunay triangulations
- experimental mathematics
- face lattices
- finite simplicial complexes
- flag f-vectors
- flag h-vectors
- graphs
- halfspace intersections
- homology
- homology groups
- Isomorphism checking
- isomorphism testing
- Klein bottle
- Knot recognition
- linear algebra
- linear programming
- manifold recognition
- Newton polytopes
- object-oriented system
- optimization
- polyhedral surfaces
- polytopes
- projective plane
- simplicial complex
- solution spaces
- surfaces
- tight spans of finite metric spaces
- topaz
- topology application zoo
- triangulations
- visualization
- Voronoi diagrams