Discrete Geometry Group

Thilo Rörig (geb. Schröder)

New address

Technische Universität Berlin
Fakultät II: Institut für Mathematik, MA 8-3,
Straße des 17. Juni 136
10623 Berlin
Germany

Room:	MA 874
Phone:	+49-30-314-24778
Fax:	+49-30-314-79282
Email:	roerig"AT"math.tu-berlin.de

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Member of the Research Group Polyhedral Surfaces

Member of the Berlin Mathematical School

Co-author and developer of the software package polymake

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• Lehre
• publications
• preprints
• Schülervorträge
• more

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Lehre

• Sommersemester 2009: Lineare Algebra II für Mathematiker
• Wintersemester 2008/09: Lineare Algebra I für Mathematiker

Research Interests

• discrete geometry, polyhedral surfaces, applications to architecture

Publications

• Drawing polytopal graphs with polymake
  with Ewgenij Gawrilow, Michael Joswig, and Nikolaus Witte,
  to appear in Computing and Visualization in Science; available at arXiv: 0711.2397.
  

  This note wants to explain how to obtain meaningful pictures of (possibly high-dimensional) convex polytopes, triangulated manifolds, and other objects from the realm of geometric combinatorics such as tight spans of finite metric spaces and tropical polytopes. In all our cases we arrive at specific, geometrically motivated, graph drawing problems. The methods displayed are implemented in the software system polymake.

• Polyhedral Surfaces, Polytopes, and Projections (PhD-Thesis)
  published online at Technische Universität Berlin, available as pdf

• Zonotopes With Large 2D Cuts with Nikolaus Witte and Günter Ziegler
  in Discrete and Computational geometry, 2008, available at springer link,
  preprint at arXiv: 0710.3116.
  

  There are d-dimensional zonotopes with n zones for which a 2-dimensional central section has \Omega(n^{d-1}) vertices. For d=3 this was known, with examples provided by the "Ukrainian easter eggs" by Eppstein et al. Our result is asymptotically optimal for all fixed d>=2.

• Polyhedral Surfaces in Wedge Products with Raman Sanyal and Günter Ziegler
  in Oberwolfach Report, Vol. 4, No. 1, 2007, pp. 244 pdf
  

  We introduce the wedge product of two polytopes which is dual to the wreath product of Joswig and Lutz. The wedge product of a p-gon and a (q-1)-simplex contains many p-gon faces of which we select a subcomplex corresponding to a surface. This surface is regular of type {p,2q}, that is, all faces are p-gons, all vertices have degree 2q, and the combinatorial automorphism group acts transitively on the flags of the surface. We show that for certain choices of parameters $p$ and $q$ there exists a realization of the wedge product such that the surface survives the projection to R^4. For a different choice of parameters such a realization does not exist.

• Neighborly Cubical Polytopes and Spheres with Michael Joswig
  in Israel Journal of Mathematics , Vol. 159, 2007, pp. 221, available at arXiv: math.CO/0503213.
  

  We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first author arise as a special case of neighborly cubical spheres constructed by Babson, Billera, and Chan. By relating the two constructions we obtain an explicit description of a non-polytopal neighborly cubical sphere and, further, a new proof of the fact that the cubical equivelar surfaces of McMullen, Schulz, and Wills can be embedded into R^3.

• EG-Models
  
  • Zonotopes with Large 2D Cuts, with Nikolaus Witte and Günter M. Ziegler
    in EG-Models number 2008.10.002.
    

    This model complements the above article on Zonotopes with large 2D-cuts.

Preprints

• Polyhedral Surfaces in Wedge Products
  with Günter Ziegler,
  available at arXiv: 0908.3159.
  

  We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated "subdirect product" as introduced by McMullen (1976); it is dual to the "wreath product" construction of Joswig and Lutz (2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially regular polyhedral surfaces as subcomplexes. These generalize known classes of surfaces "of unusually large genus" that first appeared in works by Coxeter (1937), Ringel (1956), and McMullen, Schulz, and Wills (1983). Via "projections of deformed wedge products" we obtain realizations of some of the surfaces in the boundary complexes of 4-polytopes, and thus in R³. As additional benefits our construction also yields polyhedral subdivisions for the interior and the exterior, as well as a great number of local deformations ("moduli") for the surfaces in R3. In order to prove that there are many moduli, we introduce the concept of "affine support sets" in simple polytopes. Finally, we explain how duality theory for 4-dimensional polytopes can be exploited in order to also realize combinatorially dual surfaces in R³ via dual 4-polytopes.

• Non-projectability of polytope skeleta
  with Raman Sanyal,
  available at arXiv: 0908.0845.
  

  We investigate necessary conditions for the existence of projections of polytopes that preserve the full k-skeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the given combinatorial type such that a linear projection to e-space strictly preserves the k-skeleton. Building on the work of Sanyal (2009), we develop a general framework to calculate obstructions to the existence of such realizations using topological combinatorics. Our obstructions take the form of graph colorings and linear integer programs. We focus on polytopes of product type and calculate the obstructions for products of polygons, products of simplices, and wedge products of polytopes. Our results show the limitations of constructions for the deformed products of polygons of Sanyal & Ziegler (2009) and the wedge product surfaces of Rörig & Ziegler (2009) and complement their results.

Master Thesis (Diplomarbeit)

• Neighborly cubical spheres and polytopes (.pdf, .ps, .ps.gz)

Schülervorträge

• Der Vortrag über die Buddy-Bären im Rahmen der Mathredaktion.

Urania Vortrag

• Einige zusätzliche Informationen zu meiner Geometrischen Reise in höhere Dimensionen in der Urania.

Mathekalender

• Der Mathekalender des Matheon enthält auch dieses Jahr viele Aufgaben, die zum Knobeln einladen. Also einfach anmelden und mitmachen.

More...

• An applet to compute equivelar surfaces of type M_{4,5} constructed by McMullen, Schulz, and Wills. An equivelar surface of type {p,q} is a polyhedral surface consisting of p-gons, q meeting at each vertex.
• Decomposing a truncated octahedron was a problem I found in the Open Problem Garden. I found a solution via zonotopes and their tilings.
• The finest Illustration and Graphic Design at sonja-roerig.de