2024-07-12 –, If (1.1)
Polygonal complexes are combinatoric objects that arise from polyhedra and facilitate various combinatoric and geometric studies. In this talk, we present the Julia package GeoCombSurfX which enables the study of polygonal complexes and the construction of their corresponding embedding. Moreover, we demonstrate the usage of our package by targeting the problem of deciding whether a given asssembly of 3D-blocks gives rise to a topological interlocking.
By denoting the incidence structure of the vertices, edges and facets of a given polyhedron, a polygonal complex is obtained. In our talk we give an overview of the basic definitions and the underlying theory of convex and non convex polyhedron and their corresponding polygonal complex. A given polygonal complex can be analysed from a combinatoric point of view by examining the corresponding isomorphism group, different vertex- and edge-colorings and manipulations of the incidence structure that give rise to new polygonal complexes, whereas geometric properties that can be highlighted are convexity properties of the given complex and the problem of embedding the complex into 3-space as a polyhedron. Julia's diverse package ecosystem and great performance makes it a suitable candidate to develop a software that contains implementations to analyse all aspects that arise in the study of polygonal complexes.
As an example for the usage of our package we formulate the interlocking problem of an assembly of 3D-blocks. This problem can be summarised as follows: Suppose you are given a collection of blocks, which are in contact with one another and a subset of the blocks (frame) that is fixed in space. Are there non-zero simultaneous motions of the remaining blocks such that applying these motions to the blocks of the assembly does not result in penetrations of any blocks? If there are no such motions, the assembly is called topologically interlocking. Furthermore, we describe a sufficient criterion for a given assembly to be topologically interlocked and outline how Julia and its ecosystem, in particular the packages Polyhedron.jl, JuMP.jl and HiGHS.jl, can be used to tackle this problem.
Finally, we describe future projects and features that we aim to include into GeoCombSurfX-package. We want to make use of the OSCAR project to answer algebraic questions about polygonal complexes. In particular, OSCAR offers an interface to the computer-algebra system GAP that is being developed at RWTH Aachen University and for which a package has already been implemented that deals with combinatorial aspects of polygonal complexes. Further, we want to parameterise all embeddings of polygonal complexes into 3-space with triangular faces and given edge lengths algebraically, which requires the exact solutions of multivariate polynomial systems over the reals.
I am a PHD student at RWTH Aachen University under the supervision of Prof. Dr. Daniel Robertz. My main research interests lie in geometric aspects of simplicial surfaces. In particular I study how to embed a combinatorial simplicial surface into 3-space under edge lenghts constraints and investigate different rigidity properties.
I am a PhD student in the field of simplicial surfaces under the supervision of Prof. Dr. Alice C. Niemeyer and Prof. Dr. Wilhelm Plesken.