Thesis--Kaiser-Wilhelms-Universität.

Inaug.-diss.--Marburg. / Lebenslauf.

Rational tetrahedra are tetrahedra with rational edges. Heron tetrahedra are tetrahedra with integer edges, integer faces areas and integer volume --- the three-dimensional analogue of Heron triangles. Of course, if a rational tetrahedron has rational face areas and volume then it is easy to scale it up to get a Heron tetrahedron. So we also use `Heron tetrahedra' when we mean tetrahedra with rational edges, areas and volume. The work in this thesis is motivated by Buchholz's paper {\it Perfect Pyramids} [4]. Buchholz examined certain configurations of rational tetrahedra, looking first for tetrahedra with rational volume, and then for Heron tetrahedra. Buchholz left some of the cases he examined unsolved and Chapter 2 is largely devoted to the resolution of these. In Chapters 3 and 4 we expand upon some of Buchholz's results to find infinite families of Heron tetrahedra corresponding to rational points on certain elliptic curves. In Chapters 5 and 6 we blend the ideas of Buchholz in [4] and of Buchholz and MacDougall in [7], and consider rational tetrahedra with edges in arithmetic (AP) or geometric (GP) progression. It turns out that there are no Heron AP or GP tetrahedra, but AP tetrahedra can have rational volume. They can also have one rational face area, although only one AP tetrahedron has been found with a rational face area and rational volume. For GP tetrahedra there are still unsolved cases, but we show that if GP tetrahedra with rational volume exist, then there are only finitely many. The faces of a rational GP tetrahedron are never rational. Much of the work in these two chapters also appeared in the author's Honours thesis, but has been refined and extended here, and is included to give a more complete picture of the work on Heron tetrahedra which has been done to date. In the final chapter we use a different approach and concentrate on the face areas first, instead of the volume. To make it easier (hopefully) to find tetrahedra with all faces having rational area, we place restrictions on the types of faces and number of different faces the tetrahedra have. / Masters Thesis

Thesis (doctoral)--Universität Leipzig, 1914. / Vita.

No description available.

Thesis (M.S.)--Rochester Institute of Technology, 1992. / Typescript. Includes bibliographical references.

If one has three sticks (lengths), when can you make a triangle with the sticks? As long as any two of the lengths sum to a value strictly larger than the third length one can make a triangle. Perhaps surprisingly, if one is given 6 sticks (lengths) there is no simple way of telling if one can build a tetrahedron with the sticks. In fact, even though one can make a triangle with any triple of three lengths selected from the six, one still may not be able to build a tetrahedron. At the other extreme, if one can make a tetrahedron with the six lengths, there may be as many 30 different (incongruent) tetrahedra with the six lengths. Although tetrahedra have been studied in many cultures (Greece, India, China, etc.) Over thousands of years, there are surprisingly many simple questions about them that still have not been answered. This thesis answers some new questions about tetrahedra, as well raising many more new questions for researchers, teachers, and students. It also shows in an appendix how tetrahedra can be used to illustrate ideas about arithmetic, algebra, number theory, geometry, and combinatorics that appear in the Common Cores State Standards for Mathematics (CCSS -M). In particular it addresses representing three-dimensional polyhedra in the plane. Specific topics addressed are a new classification system for tetrahedra based on partitions of an integer n, existence of tetrahedra with different edge lengths, unfolding tetrahedra by cutting edges of tetrahedra, and other combinatorial aspects of tetrahedra.

Thesis (Ph.D.)--Memorial University of Newfoundland, 2003. / Restricted until May 2004. Bibliography: leaves 162-176.

The minimum energy structures of H₆Si₂O₇, H₇SiAlO₇, and H₈Al₂O₇ have been calculated using quantum mechanical molecular orbital techniques. The calculated bond lengths and angles of H₆Si₂O₇ and H₇SiAlO₇ agree with those found in silicate and aluminosilicate minerals, but no such comparison is possible for H₈Al₂O₇ since we know of no aluminates having such an underbonded bridging oxygen (Pauling bond strength sum of 1.5). The total energies of the three molecules were used to model the stability of the SiOAI unit relative to the SiOSi and AIOAI units in framework aluminosilicates such as the feldspars and the zeolites. The calculated electronic energy for the reaction H₆Si₂O₇ + H₈Al₂O₇ = 2H₇SiAlO₇ is positive (ca. + 20 kJ mol⁻¹). This result docs not adequately model the "aluminum avoidance rule," but the value is closer than previous calculations performed on energy optimized molecules (which gave ΔE = - 484 kJ moI⁻¹) to experimental enthalpies of mixing for the reaction 2M_1/nⁿ⁺AlO₂ + 2SiO₂ = 2M_1/nⁿ⁺AlSiO₄. For this reaction ΔH_mix = -100.4 kJ mol⁻¹ for M = Na, and ΔH_mix = -75.6 kJ mol⁻¹ for M = Ca. The calculated relative order of stability for reactions of water and hydronium ion with the hydroxyacid molecules used in this study was found to be H₃O⁺ + SiOSi > H₃O⁺ + SiOAl > H₂O + SiOAl > H₃O⁺ + AlOAl ≃ H₂O + AlOAl > H₂O + SiOSi The results model the hydrophilic nature of aluminosilicate zeolites and the hydrophobic nature of the silicate zeolite silicalite. / M.S.

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.