# Simpliziale Flächen aus kongruenten Dreiecken : kombinatorische Grundlagen und geometrische Beispiele

• Simplicial surfaces with congruent faces : combinatorial foundations and geometric examples

Strzelczyk, Ansgar Werner; Plesken, Wilhelm (Thesis advisor); Niemeyer, Alice Catherine (Thesis advisor)

Aachen (2019)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2019

Abstract

The goal of this thesis is the studies of simplicial surfaces with congruent faces. There are two aspects to this: the theory of combinatorial simplicial surfaces and the theory of embeddings into the three-dimensional Euclidean space. This thesis starts with a review of the fundamental properties of simplicial surfaces. On the combinatorial side simplicial surfaces are incidence geometries. An approach with operators for construction and analysis of simplicial surfaces allows for a new handling of the surfaces. With these operators simplicial surfaces are not only comparable to each other but also constructable in small detail. For realizations of a simplicial surface the operators transport only a schem-taically construction. But in contrast to the operators there is procedure to naturally transfer the decomposition of a combinatorial simplicial surface to the decomposition of the corresponding realizations. The necessary property of simplicial surfaces are special closed edge paths of the lengths 2 and 3. They are called 2-waist or 3-waist respectively. On these waists a decomposition of a simplicial surface into smaller blocks is possible. Every realization in Euclidean three-space can uniquely be deduced by realizations of those blocks. Therefore, simplicial surfaces without 2-waists and 3-waists form a special class of simplicial surfaces. For spherical simplicial surfaces of this type a construction will be given. Up to this point, arbitrary simplicial surfaces were studied. From the non simplicial surfaces with congruent faces are investigated. Accordingly, the congruence of Euclidean triangles will be defined for abstract triangles. It is achieved by an edge colouring of the simplicial surface. The colouring imitates the criteria of Euclidean triangles to be congruent, if and only if they share three edge lengths. The colourings and therefore the combinatorial congruence is then understood in a group theoretical sense as involutions in the symmetric group of the triangles. This approach gives rise to algorithms to compute all admissable edge colouring. Additionally, algorithms to compute all simplicial surfaces with or without an edge colouring are presented. Afterwards, the theory of embeddings of simplicial surfaces into Euclidean three-space is studied more closely. The problem of finding realizations of a given simplicial surface is expressed in ideals. There are two presented possibilities doing so. The first one is to express an ideal in the polynomial ring over the coordinates. The second one is to express an ideal in the entries of the associated gram matrix. To find all realization turns out to be a computationally hard problem. To tackle the problem more efficiently additional generators are added. These generators specialize the solution set, for example to find only realizations that are injective on the vertices. Another specialization of the problem is given by adding symmetry equations. Lastly, an extensive data base of spherical simplicial surfaces without 2- and 3-waists is given. They data base contains several realizations.