Realisierungen endlich gestörter hexagonaler Dreieckspflasterungen durch Faltungsmuster

Niehenke, Björn; Plesken, Wilhelm (Thesis advisor); Niemeyer, Alice Catherine (Thesis advisor)

Aachen (2018, 2019)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2018

Abstract

The discipline of mathematical origami is quite young. A large part of its research has been devoted to the numerical realization of crease pattern, to flat foldable pattern, and the description and analysis of crease patterns of small finite surfaces with but a few folds - such as the single vertex folds. This thesis aims to broaden the perspective: With algebraical methods we strive to analyse the global structure of infite folded surfaces as forced upon it by its interior geometry. To take a first step into that direction we simplify the problem as follows: 1. The given surface permits a triangulation consisting of equilateral triangles. 2. In all but finitely many corners there are exactly six triangles meeting. 3. The creases coincide with the sides of the triangles. 4. The number of creases is finite. The easiest example of such a surface does not have any creases: The Euclidean plane triangulated with respect to the Eisenstein lattice by equilateral triangles such that in each corner exactly six triangles meet. If there are more or less than six triangles meeting at a given point we call this a defect which corresponds to a conic singularity of the surface. An octahedron, for example, has the defect 2 in each of its corners. Those defects are local disturbances with consequences for global structure. The main goal of this thesis is the examination of the surfaces triangulated by equilateral triangles with defects adding up to 0 with respect to the existance of realisations induced by folding patterns that are in certain aspects similar in their global structure to the Euclidean plane. To that end the large two dimensional pieces of the embedded surface containing a two dimensional affine cone are sought to be parallel to each other. We will call this quality quasi-plane. This thesis takes a first step toward the classification of the quasi-plane realisations of a given surface. The surface is modelled as a simplicial surface and in an intermediate step the realisation is modelled as an abstract crease pattern by attaching weights to the edges of the underlying graph. Spherical mechanisms then enable us to set up local conditions for quasi-plane realisations and those allow us to proof an exisitance theorem for quasi-plane realisations of surfaces with exactly two defects adding up to 0. As those algebraical methods have a very limited reach we restrict ourselves to realisations on the surfaces of a regular tilings of the three dimensional Euclidean space by regular octahedrons and tetrahedrons. In this scenario the creases follow a simple grammar allowing us to construct an algorithm that for any given finitely disturbed simplicial surface will construct a generating set of quasi-plane realisations. The main work here is done on the abstract folding patterns. The main idea is that the regular case is quite simple and that any given simplicial surface can be deconstructed into a finite part containing all defects and infinite parts that behave very similar to the regular case. The finite part can be handled combinatorically where as the infinite parts can be handled due to the simple grammar for crease patterns in the regular case. The grammar also allows us to bridge the passage from the finite to the infinite parts. With some simple criteria for parallel planar parts we are then able to in this setting classify all crease patterns that permit quasi-plane realisations.