Polytopes, the convex hulls of finitely many vertices, are a subject of mathematical study since antiquity. The Platonic solids were the culmination point of antique Greek mathematics: They are polytopes admitting a particularly high type of symmetry. While the cube, the tetrahedron and the octahedron can be realized with full symmetry using integer coordinates, this is impossible for the icosahedron and the dodecahedron. Realizing them with full symmetry requires the use of a sqrt(5) in the coordinate field. Here two combinatorial requirements, namely the type of a polytope (being a dodecahedron) and the requirement to realize it symmetrically, create structural constraints on the geometric realization of the polytope (the coordinates cannot be rational). Starting in dimension four there are combinatorial types of polytopes that (even without symmetry requirements) cannot be realized with integers as coordinates. Another characteristic property shared by all the Platonic solids is that they can be represented with all their vertices on a sphere. Not every polytope has such a realization, and it is still a challenging question to decide which polytopes do.
It is a central topic in research project A03 to study the various types of interplay of combinatorial properties of polytopes (face lattice, symmetry, etc) and their geometric properties (coordinates, shapes, etc). Questions like:
- "Which integer realizable polytope with n vertices requires the largest integer coordinates?",
- "What is the most compact way to describe a specific realization of a polytope?",
- "Does the 'roundness' of a polytope have influence on its combinatorial type?",
- "Which combinatorial types of polytopes can be inscribed in a sphere?"
are of great interest and still widely open. It is even necessary to define the right concepts of 'complexity' and 'roundness' to speak about these problems in proper mathematical terms. Attacking these fundamental problems is at the core of A03.