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@PHDTHESIS{Grtzen:992149,
author = {Görtzen, Tom Frederik},
othercontributors = {Niemeyer, Alice Catherine and Robertz, Daniel},
title = {{C}onstructing simplicial surfaces with given geometric
constraints},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
publisher = {RWTH Aachen University},
reportid = {RWTH-2024-08038},
pages = {1 Online-Ressource : Illustrationen},
year = {2024},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University; Dissertation, RWTH Aachen University, 2024},
abstract = {Simplicial surfaces encode the incidence relationships
between vertices, edges, and faces of triangulated surfaces,
providing a combinatorial description of these structures.
By assigning a three-dimensional point to each vertex, we
again obtain a triangulation, which we view as an embedded
simplicial surface in the context of this thesis.The primary
aim of this thesis is the construction of simplicial
surfaces under specified geometric constraints, with a
particular emphasis on symmetry. Symmetry, in mathematical
terms, can be expressed using the language of group theory.
Starting with a geometric object, we can determine its
automorphism group by identifying all transformations that
leave the object invariant. Conversely, from a group
theoretic perspective, we can study groups independently and
explore whether a geometric object exists such that its
automorphism group matches a given group.Our first main
result demonstrates that for any given finite group, we can
construct a simplicial surface whose automorphism group is
isomorphic to that group. In specific instances, these
vertices can be embedded to produce embedded simplicial
surfaces with given symmetry.Additionally, embedded
simplicial surfaces can characterise other symmetric
properties. For example, we explore systems of interlocked
three-dimensional bodies, known as topological interlocking
assemblies, which rely solely on their geometric properties.
We demonstrate that the theory of planar crystallographic
groups can be applied to construct a wide variety of
interlocking assemblies. Moreover, we develop the
mathematical foundations of interlocking assemblies
containing a definition, a method for verifying the
interlocking property and many examples. Furthermore,
extending the action of planar crystallographic groups
allows the creation of surfaces with doubly periodic
symmetry.Embedded simplicial surfaces can be useful in
various applications. In the final chapter, we illustrate
how the theory of simplicial surfaces can be applied in the
context of 3D printing: even if an initial model exhibits
degenerations, we can modify it to produce a 3D printable
file. Through these explorations, we highlight the practical
and theoretical significance of simplicial surfaces in both
mathematical research and technological applications,
underscoring their versatility and potential for future
developments.},
cin = {115320 / 114410 / 110000},
ddc = {510},
cid = {$I:(DE-82)115320_20140620$ / $I:(DE-82)114410_20140620$ /
$I:(DE-82)110000_20140620$},
pnm = {DFG project 444414437 - Algebra, Kinematik und
Kompatibilität triangulierter Geometrien (A04) (444414437)
/ DFG project 417002380 - TRR 280: Konstruktionsstrategien
für materialminimierte Carbonbetonstrukturen – Grundlagen
für eine neue Art zu bauen (417002380)},
pid = {G:(GEPRIS)444414437 / G:(GEPRIS)417002380},
typ = {PUB:(DE-HGF)11},
doi = {10.18154/RWTH-2024-08038},
url = {https://publications.rwth-aachen.de/record/992149},
}
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