Source: https://www.discretization.de/en/projects/A03/
Timestamp: 2019-04-19 20:17:43+00:00

Document:
This project is based on the observation that combinatorial and geometric features of polytopes are interlocked in many different, conceptually independent, ways. This interaction in both directions divides our project into two main strands, “Geometry → Combinatorics” and “Combinatorics → Geometry,” where the arrows may be read as “constrains,” “impacts,” or “restricts.” Both directions are pursued in parallel, and the focus of understanding the interactions was considerably furthered during the first funding period. We now joined by our Einstein Visiting Fellow Francisco Santos.
This project studies the interaction between geometric properties of polytopes (such as “roundness” measured in various ways) and combinatorial data (such as given by face/flag vectors or adjacency information).
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.
"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.
Authors: Sjöberg, Hannah and Ziegler, Günter M.
Authors: Brinkmann, Philip and Ziegler, Günter M.
Authors: Blagojević, Pavle V. M. and Rote, Günter and Steinmeyer, Johanna and Ziegler, Günter M.
Authors: Blagojević, Pavle V. M. and Palić, Nevena and Ziegler, Günter M.
Authors: Labbé, Jean-Philippe and Rote, Günter and Ziegler, Günter M.
Authors: Blagojević, Pavle V. M. and Ziegler, Günter M.
Authors: Blagojević, Pavle V. M. and Blagojević, Aleksandra S. Dimitrijević and Ziegler, Günter M.
Note: Israel J. Math., to appear.
Authors: Loiskekoski, Lauri and Ziegler, Günter M.
Authors: Blagojević, Pavle V. M. and Haase, Albert and Ziegler, Günter M.
Authors: Blagojević, Pavle V. M. and Frick, Florian and Haase, Albert and Ziegler, Günter M.
Authors: Bárány, Imre and Blagojević, Pavle V. M. and Ziegler, Günter M.
Authors: Blagojevic, Pavle V. M. and Frick, Florian and Haase, Albert and Ziegler, Günter M.
Authors: Blagojevic, Pavle V. M. and Frick, Florian and Matschke, Benjamin and Ziegler, Günter M.
Authors: Blagojevic, Pavle V. M. and Frick, Florian and Ziegler, Günter M.
Many neighborly polytopes and oriented matroids.
Authors: Blagojevic, Pavle V. M. and Lück, Wolfgang and Ziegler, Günter M.
Authors: Gonska, Bernd and Ziegler, Günter M.

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