Thilo Rörig (geb. Schröder)

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

Room:	MA 879
Phone:	+49-30-314-29486
Fax:	+49-30-314-79282
Email:	roerig"AT"math.tu-berlin.de

Member of the Research Group Polyhedral Surfaces

Junior Faculty of the Berlin Mathematical School

Co-author and developer of the software package polymake

Lehre
publications
preprints
Schülervorträge
more

Lehre

Research Interests

discrete geometry, polyhedral surfaces, applications to architecture

Publications

Discretization of asymptotic line parametrizations using hyperboloid surface patches
with Emanuel Huhnen-Venedey
accepted in Geometriae Dedicata available at arXiv: math.DG/1112.3508.

Two-dimensional affine A-nets in 3-space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The characterizing property of A-nets is planarity of vertex stars, so for generic A-nets the elementary quadrilaterals are skew. We classify the simply connected affine A-nets that can be extended to continuously differentiable surfaces by gluing hyperboloid surface patches into the skew quadrilaterals. The resulting surfaces are called "hyperbolic nets" and are a novel piecewise smooth discretization of surfaces parametrized along asymptotic lines. It turns out that a simply connected affine A-net has to satisfy one combinatorial and one geometric condition to be extendable - all vertices have to be of even degree and all quadrilateral strips have to be "equi-twisted". Furthermore, if an A-net can be extended to a hyperbolic net, then there exists a 1-parameter family of such C¹-surfaces. It is briefly explained how the generation of hyperbolic nets can be implemented on a computer. The article uses the projective model of Plücker geometry to describe A-nets and hyperboloids.
Topology Optimisation of Regular and Irregular Elastic Gridshells by Means of a Non-linear Variational Method
with Elisa Lafuente Hernandez, Stefan Sechelmann, Christoph Gengnagel
in Proceedings of Advances in Architectural Geometry, 2012, pp. 147-160

Gridshells composed of elastically-bent profiles offer significant cost and time advantages during the production, transport and construction processes. Nevertheless, the shaping of the initially flat grid also generates important bending stresses on the structures, reducing therewith their bearing capacity against external loads. An optimisation of the grid topology in order to minimise the profiles curvature and, with it, the initial stresses is therefore crucial. In this paper a non-linear variational method for optimising topologies of elastic gridshells with regular and irregular meshes is presented. Different case studies of double-curved gridshells show the advantages and capacity of this method.
Quasiisothermic Mesh Layout
with Stefan Sechelmann, Alexander Bobenko
in Proceedings of Advances in Architectural Geometry, 2012, pp. 243-258

The quality of a quad-mesh depends on the shape of the individual quadrilaterals. The ideal shape from an architectural point of view is the planar square or rectangles with fixed aspect ratio. A parameterization that divides a surface into such shapes is called isothermic, i.e., angle-preserving and curvaturealigned. Such a parameterization exists only for the special class of isothermic surfaces. We extend this notion and introduce quasiisothermic parameterizations for arbitrary triangulated surfaces. We describe an algorithm that creates quasiisothermic meshes. Interestingly many surfaces appearing in architecture are close to isothermic surfaces, namely those coming from form finding methods and physical simulation. For those surfaces our method works particularly well and gives a high quality and robust mesh layout. We show how to optimize such meshes further to obtain disk packing representations. The quadrilaterals of these meshes are planar and possess touching incircles.
Polytope mit vielen Splits und ihre Sekundärfächer
with Michael Joswig
in Mathematische Semesterberichte 2012, Volume 59, Issue 2, pp. 145-152, springer link
Publisher's Erratum: October 2012, Volume 59, Issue 2, p. 153, springer link

Konvexe Polytope, also konvexe Hüllen endlich vieler Punkte im euklidischen Raum, stehen an der Schnittstelle zwischen Geometrie und Kombinatorik. Ziel dieses Textes ist es, ein Ergebnis vorzustellen, zu motivieren und zu visualisieren, das darauf abzielt, Polytope zu klassifizieren, die sich auf besonders einfache Weise in Teilpolytope zerlegen lassen (in Herrmann und Joswig, Discrete Comput. Geom. 44:149-166, 2010). Um diese Klassifikation formulieren und illustrieren zu können, beginnt der Text mit einer Einführung in die Theorie der Sekundärfächer.
Non-projectability of polytope skeleta
with Raman Sanyal,
in Advances in Mathematics Volume 229 (1), 2012, pp. 79-101, available at link.

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.
On the Materiality and Structural Behaviour of highly-elastic Gridshell Structures
with Elisa Lafuente Hernandez, Christoph Gengnagel, Stefan Sechelmann,
in Computational Design Modeling: Proceedings of the Design Modeling Symposium Berlin 2011

Gridshell structures made of highly elastic materials provide significant advantages thanks to their cost-effective and rapid erection process, whereby the initially in-plane grid members are progressively bent elastically until the desired structural geometry is achieved. Despite the strong growing interest that architects and engineers have in such structures, the complexity of generating grid configurations that are developable into free-form surfaces and the limitation of suitable materials restrict the execution of elastically bent gridshells. Over the past ten years, several research studies have focused on methodologies to generate developable grid configurations and to calculate their resulting geometry after the erection process. However, the same curved shell surface can be reproduced by various developable grid configurations which, in combination with their material properties, exhibit different structural behaviours not only during the shaping process but also on the gridshell load-bearing capacity. In this paper, the structural consequences of the choice of the grid configuration for an anticlastic surface have been analysed by means of FEM-Modelling combined with an geometrical optimisation of the initial bending stresses. In addition, the potential of using natural fibre-reinforced composites as a lightweight and environmentally friendly alternative has been investigated.
Polyhedral Surfaces in Wedge Products
with Günter Ziegler,
in Geometriae Dedicata, Volume 151, Number 1, 2011, pp. 155-173, available at springer link

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.
Drawing polytopal graphs with polymake
with Ewgenij Gawrilow, Michael Joswig, and Nikolaus Witte,
in Computing and Visualization in Science, Volume 13, Number 2 / Februrary 2010; available at springer link
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

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