Dreifach periodische simpliziale Flächen

Wisotzky, Matthias; Plesken, Wilhelm (Thesis advisor); Niemeyer, Alice Catherine (Thesis advisor)

Aachen (2020)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2020

Abstract

Simplicial surfaces are an important part of many areas in math and simplicial approximations are often times used to compute properties of surfaces. Further they are the base of mathematical origami. This thesis offers an easy and fast possibility to create a multitude of simplicial surfaces and also to visualize them. For the construction crystallographic space groups are used, so the results have many symmetries and are at least three-periodic. The construction starts with a polyedric fundamental domain of a space group - such can be computed via the Voronoi-Dirichlet-construction. A simplicial surface clamped into such a fundamental domain, whose trace in the walls the of the fundamental domain is either a line segment connecting two edge middlepoints or empty, is called a cut of the fundamental domain. These cuts are the building blocks of the constructed surfaces. A three-periodic surface can be constructed from two space groups (with compatible fundamental domains) as follows: One combines the fundamental domains of the first groups to get a fundamental domain for the second one. After selecting cuts for the fundamental domains of the first group one applies the second group to get the desired surface as the orbit. This only works if some constraints are met which are presented in definitions of compatible fundamental domains respectively compatible cuts. This always succeeds if one selects a reflection group as the first space group and a subgroup of it as the second. Now a suitable transversal of the subgroup in the group provides a connected fundamental domain and every locally fitting cut induces a surface. Next to the automatic construction a classification of such surfaces is started. The main result is an analogon to the Euler-charateristic, the so called Euler-Torus-characteristic, which uses the periodicity of the surface. The computation of this invariant is easily done because a version of the Gauss-Bonnet-theorem is given. In special surfaces created from cuts in a tetrahedron and a space group which is generated by reflections are examined. In this case the surface can be described with ease and from the edge graph of the fundamental domain of the bigger group one obtains a one-dimensional skeleton of a simplicial complex. This complex contains all information to decide whether the surface is connected and to easily compute the Euler-Torus-characteristic. In line with this thesis a computer program was written which allows the user to create surfaces in the data structure and compute their invariants. This program is documented in one chapter and was used for the creation and visualization of the examples.