The computer programs SnapPea by Weeks and Geo by Casson have proven to be powerful tools in the study of hyperbolic 3-manifolds. Manifolds are special examples of spaces called orbifolds, which are modelled locally on R^n modulo finite groups of symmetries. SnapPea can also be used to study orbifolds but it is restricted to those whose singular set is a link.One goal of this thesis is to lay down the theory for a computer program that can work on a much larger class of 3-orbifolds. The work of Casson is generalized and implemented in a computer program Orb which should provide new insight into hyperbolic 3-orbifolds.The other main focus of this work is the study of 2-handle additions. Given a compact 3-manifold M and an essential simple closed curve α on ∂M, then we define M[α] to be the manifold obtained by gluing a 2-handle to ∂M along α. If α lies on a torus boundary component, we cap off the spherical boundary component created and the result is just Dehn filling.The case when α lies on a boundary surface of genus ≥ 2 is examined and conditions on α guaranteeing that M[α] is hyperbolic are found. This uses a lemma of Scharlemann and Wu, an argument of Lackenby, and a theorem of Marshall and Martin on the density of strip packings. A method for performing 2-handle additions is then described and employed to study two examples in detail.This thesis concludes by illustrating applications of Orb in studying orbifoldsand in the classification of knotted graphs. Hyperbolic invariants are used to distinguish the graphs in Litherland’s table of 90 prime θ-curves and provide access to new topological information including symmetry groups. Then by prescribing cone angles along the edges of knotted graphs, tables of low volume orbifolds are produced.

The fascinating field of molecular capsules has recently begun to see the creation of structures that, medicated by the encapsulation of guest molecules within their central cavity, are able to change the properties or reactivity of the substrate. The current capsule designs are however, prone to exchange of either part or whole ligands. This exchange or the capsule's subsequent disassembly can lead to loss of the cavity or modification of their external properties, and is a barrier to their more widespread application, a problem this work seeks to address by creating more a robust capsule structure. This thesis presents the design, synthesis and properties displayed by three novel capsules. All the species presented share a similar supramolecular tetrahedral structure, but their properties deviate significantly, showing either switchable behaviour, spin-crossover or a novel synthetic route to a kinetically inert structure. Improvements in the design have led to a final capsule that is water-soluble, robust, non-toxic and has been shown to encapsulate a range of guests. Chapter 1 includes an overview of the types of capsule constructed in literature and their possible application. The fundamental properties of these capsules are identified, with emphasis given to a discussion of mechanisms underlying their encapsulation phenomena. Chapter 2 describes efforts to construct a tetrahedral capsule based on iron(II) and an oxime ligand. While the use of an oxime motif achieved the aim of preventing exchange of the external groups, the capsule also displayed the surprising property of possessing a solvent responsive assembly-disassembly process. This potentially provides a basis for 'on-demand' encapsulation by being able to choose when to have hydrophobic cavity available for guests. Chapter 3 details the synthesis of a tetrahedral capsule containing iron (II) coordinated by a pyridyl-triazole bonding motif. the spin-crossover properties of the complex were initially demonstrated in the solid state, however, when in solution the capsule displayed the unusual ability of spin-crossover mediated structural rearrangement. Chapter 4 demonstrates the synthesis of a robust capsule. The synthetic route shown alleviates the problems surrounding the construction of inert species in a self-assembly process. Based around a cobalt (III) cation, the stability of the capsule to carious conditions is examined and its host-guest chemistry is explored, revealing some insights into the encapsulation behaviour of this structure.

Spherical, Euclidean, and hyperbolic simplices can be characterized by the dihedral angles on their codimension-two faces. These characterizations analyze the Gram matrix, a matrix with entries given by cosines of dihedral angles. Hyperideal hyperbolic simplices are non-compact generalizations of hyperbolic simplices wherein the vertices lie outside hyperbolic space. We extend recent characterization results to include fully general hyperideal simplices. Our analysis utilizes the Gram matrix, however we use inversive distances instead of dihedral angles to accommodate fully general hyperideal simplices.For two-dimensional triangulations, an angle structure is an assignment of three face angles to each triangle. An angle structure permits a globally consistent scaling provided the faces can be simultaneously scaled so that any two contiguous faces assign the same length to their common edge. We show that a class of symmetric Euclidean angle structures permits globally consistent scalings. We develop a notion of virtual scaling to accommodate spherical and hyperbolic triangles of differing curvatures and show that a class of symmetric spherical and hyperbolic angle structures permit globally consistent virtual scalings.The double tetrahedron is a triangulation of the three-sphere obtained by gluing two congruent tetrahedra along their boundaries. The pentachoron is a triangulation of the three-sphere obtained from the boundary of the 4-simplex. As piecewise flat manifolds, the geometries of the double tetrahedron and pentachoron are determined by edge lengths that gives rise to a notion of a metric. We study notions of Einstein metrics on the double tetrahedron and pentachoron. Our analysis utilizes Regge's Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds.A notion of conformal structure on a two dimensional piecewise flat manifold is given by a set of edge constants wherein edge lengths are calculated from the edge constants and vertex based parameters. A conformal variation is a smooth one parameter family of the vertex parameters. The analysis of conformal variations often involves the study of degenerating triangles, where a face angle approaches zero. We show for a conformal variation that remains weighted Delaunay, if the conformal parameters are bounded then no triangle degenerations can occur.

This diploma thesis Tetrahedra and their properties summarizes the basic properties of tetrahedron. The main goal is to introduce the topic to a reader of this thesis. Author would like to provide basic information about the particular types of tetrahedra and their properties. I try to examine there a spatial analogy of selected terms of a triangle. Conditions for the existence of the orthocenter of a tetrahedron are derived. Then for a non-orthocetric tetrahedron the Monge point as its generalization is introduced. By most properties their proofs are given. In the final part worksheets for pupils of primary and secondary schools are designed. Pictures in the thesis are created in a geometrical program called GeoGebra 3D. These pictures can help the reader to understand this problem.

Mesh generation is a fundamental precursor to finite element implementations for solution of partial differential equations in engineering and science. This dissertation advances the field in three distinct but coupled areas. A robust and fast three dimensional mesh generator for arbitrarily shaped geometries was developed. It deploys nodes throughout the domain based upon user-specified mesh density requirements. The system is integer and pixel based which eliminates round off errors, substantial memory requirements and cpu intensive calculations. Linked, but fully detachable, to the mesh generation system is a physical boundary recovery routine. Frequently, the original boundary topology is required for specific boundary condition applications or multiple material constraints. Historically, this boundary preservation was not available. An algorithm was developed, refined and optimized that recovers the original boundaries, internal and external, with fidelity. Finally, a node repositioning algorithm was developed that maximizes the minimum solid angle of tetrahedral meshes. The highly coveted 2D Delaunay property that maximizes the minimum interior angle of a triangle mesh does not extend to its 3D counterpart, to maximize the minimum solid angle of a tetrahedron mesh. As a consequence, 3D Delaunay created meshes have unacceptable sliver tetrahedral elements albeit composed of 4 high quality triangle sides. These compromised elements are virtually unavoidable and can foil an otherwise intact mesh. The numerical optimization routine developed takes any preexisting tetrahedral mesh and repositions the nodes without changing the mesh topology so that the minimum solid angle of the tetrahedrons is maximized. The overall quality enhancement of the volume mesh might be small, depending upon the initial mesh. However, highly distorted elements that create ill-conditioned global matrices and foil a finite element solver are enhanced significantly.

Nous étudions, dans la première partie de notre thèse, les polysimplexes d’un espace euclidien de dimension quelconque, c’est-à-dire les objets consistant en une juxtaposition de simplexes réguliers (de tétraèdres si la dimension est 3) accolés le long de leurs faces. Nous étudions principalement le groupe des symétries de ces polysimplexes. Nous présentons une façon de représenter un polysimplexe à l’aide d’un diagramme. Ceci fournit une classification complète des polysimplexes à similitude près. De plus, le groupe des symétries se déduit du groupe des automorphismes du diagramme. Il découle en particulier de notre étude qu’en dimension supérieure à 2, une telle structure ne possède jamais deux faces parallèles et ne contient jamais de circuit fermé de simplexes. Dans la seconde partie de notre thèse, nous abordons un problème classique de combinatoire : l’énumération des pavages d’un rectangle mxn à l’aide de dominos. Klarner et Pollack ont montré qu’en fixant m la suite obtenue vérifie une relation de récurrence linéaire à coefficients constants. Nous établissons une nouvelle méthode nous permettant d’obtenir la fonction génératrice correspondante et la calculons pour m <= 16, alors qu’elle n’était connue que pour m <= 10.

abstract: The Dual Marching Tetrahedra algorithm is a generalization of the Dual Marching Cubes algorithm, used to build a boundary surface around points which have been assigned a particular scalar density value, such as the data produced by and Magnetic Resonance Imaging or Computed Tomography scanner. This boundary acts as a skin between points which are determined to be "inside" and "outside" of an object. However, the DMT is vague in regards to exactly where each vertex of the boundary should be placed, which will not necessarily produce smooth results. Mesh smoothing algorithms which ignore the DMT data structures may distort the output mesh so that it could incorrectly include or exclude density points. Thus, an algorithm is presented here which is designed to smooth the output mesh, while obeying the underlying data structures of the DMT algorithm. / Dissertation/Thesis / M.S. Computer Science 2011

In this thesis, analysis and comparison of different attitude control systems of a satelliteusing different reaction wheel configurations were investigated. Three different reactionwheel configurations (e.g. tetrahedron configuration, pyramid configuration, standardorthogonal 3-wheel configuration) and three control algorithms (Linear Quadratic Regulator,Sliding Mode, Integrator Backstepping) were analyzed and compared in terms of settlingtimes, power consumptions and actuator failure robustness. / <p>Validerat; 20121205 (global_studentproject_submitter)</p>

A way to obtain materials that show novel phenomena is to explore the interplay between geometry and interactions. When it is not geometrically possible to satisfy all the interactions by a given configuration, then to find the ground state becomes very complicated. This interplay between geometry and interactions defines geometrical frustration. One of the most popular examples of geometrical frustration in magnetism is spin ice. In this system, nearest neighbour ferromagnetic interactions between Ising spins in a pyrochlore structure emulate water ice by showing the same degree of frustration. This is manifested by the same ground state residual entropy. Although the clearest example of spin ice among magnets is shown by Dy₂Ti₂O₇, the behaviour of this material is richer than that of pure spin ice. The large magnetic moments of the rare earth Dy form a spin ice that also interacts via dipolar interactions. These long range interactions give rise to monopolar excitations which dramatically affect the dynamics of the system with respect to the pure spin ice case. In this thesis magnetisation experiments and numerical methods are used to explore the properties of the magnetic insulator Dy₂Ti₂O₇. We study its excitations at low temperature and describe the out-of-equilibrium characteristics of the magnetisation processes, below a temperature where the system freezes out. For temperatures above the freezing temperature, we describe and measure a 3D Kasteleyn transition and the concomitant Dirac strings associated to it, for the field in the [100] crystallographic direction. For temperatures below the freezing temperature, we find new out-of-equilibrium phenomena. Magnetic jumps are measured and their sweep rate dependence analysed. A deflagration theory is proposed and supported by simultaneous magnetisation and sample temperature measurements obtained by a new design of a Faraday magnetometer.

Die Grundform der Skulptur “VOYAGER“ ist der Tetraeder, einer der fünf platonischen Körper. Diese geometrische Figur taucht als Bauprinzip, für das menschliche Auge unsichtbar, in der Natur immer wieder auf. Wir finden es in Elektronenorbitalen, in der Gitterstruktur von Quarzkristallen oder als Molekülstruktur des Methans CH4. Nervenzellen und größere Zellverbände wie Radiolarien, Foraminiferen und andere denkbare Organismen lassen sich mit dieser Form assoziieren. Auch Raumsonden, die unter anderem mit Hilfe der Bionik entwickelt wurden, folgen diesem Bauprinzip. Die Phantasie und die Möglichkeiten sind grenzenlos. - Auf zu neuen Ufern und unbekannten Welten, auch in uns.