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Full-Text Articles in Physical Sciences and Mathematics
Geometries Gon Wild, Naat Ambrosino
Geometries Gon Wild, Naat Ambrosino
Undergraduate Theses
A circle is mathematically defined as the collection of points a given distance away from a set point. Thus, the appearance of a circle varies dramatically across different metrics—for example, the taxicab metric (as popularized by Krause and Reynolds) has a circle that is a Euclidean square. As such, metrics can be partially defined by the appearance of their unit circles. This paper focuses on creating and analyzing an infinite set of metrics defined by their circles being regular polygons. Additionally, it provides a method of exactly generating a regular n-gon given a center, included point, and specified orientation.
A Tropical Approach To The Brill-Noether Theory Over Hurwitz Spaces, Kaelin Cook-Powell
A Tropical Approach To The Brill-Noether Theory Over Hurwitz Spaces, Kaelin Cook-Powell
Theses and Dissertations--Mathematics
The geometry of a curve can be analyzed in many ways. One way of doing this is to study the set of all divisors on a curve of prescribed rank and degree, known as a Brill-Noether variety. A sequence of results, starting in the 1980s, answered several fundamental questions about these varieties for general curves. However, many of these questions are still unanswered if we restrict to special families of curves. This dissertation has three main goals. First, we examine Brill-Noether varieties for these special families and provide combinatorial descriptions of their irreducible components. Second, we provide a natural generalization …
Unifications Of Pythagorean Triple Schema, Emily Hammes
Unifications Of Pythagorean Triple Schema, Emily Hammes
Undergraduate Honors Theses
Euclid’s Method of finding Pythagorean triples is a commonly accepted and applied technique. This study focuses on a myriad of other methods behind finding such Pythagorean triples. Specifically, we discover whether or not other ways of finding triples are special cases of Euclid’s Method.
Integrating Non-Euclidean Geometry Into High School, John Buda
Integrating Non-Euclidean Geometry Into High School, John Buda
Honors Thesis
The purpose of this project is to provide the framework for integrating the study of non-Euclidean geometry into a high school math class in such a way that both aligns with the Common Core State Standards and makes use of research-based practices to enhance the learning of traditional geometry. Traditionally, Euclidean geometry has been the only strand of geometry taught in high schools, even though mathematicians have developed several other strands. The non-Euclidean geometry that I focus on in this project is what is known as taxicab geometry. With the Common Core Standards for Math Practice pushing students to “model …
A Kleinian Approach To Fundamental Regions, Joshua L. Hidalgo
A Kleinian Approach To Fundamental Regions, Joshua L. Hidalgo
Electronic Theses, Projects, and Dissertations
This thesis takes a Kleinian approach to hyperbolic geometry in order to illustrate the importance of discrete subgroups and their fundamental domains (fundamental regions). A brief history of Euclids Parallel Postulate and its relation to the discovery of hyperbolic geometry be given first. We will explore two models of hyperbolic $n$-space: $U^n$ and $B^n$. Points, lines, distances, and spheres of these two models will be defined and examples in $U^2$, $U^3$, and $B^2$ will be given. We will then discuss the isometries of $U^n$ and $B^n$. These isometries, known as M\"obius transformations, have special properties and turn out to be …
Constructible Numbers: Euclid And Beyond, Joshua Scott Marcy
Constructible Numbers: Euclid And Beyond, Joshua Scott Marcy
Theses Digitization Project
The purpose of this project is to demonstrate first why trisection for an arbitrary angle is impossible with compass and straightedge and second how trisection does become possible if a marked ruler is used instead.
Geometric Theorem Proving Using The Groebner Basis Algorithm, Karla Friné Rivas
Geometric Theorem Proving Using The Groebner Basis Algorithm, Karla Friné Rivas
Theses Digitization Project
The purpose fo this project is to study ideals in polynomial rings and affine varieties in order to establish a connection between these two different concepts. Doing so will lead to an in depth examination of Groebner bases. Once this has been defined, step will be outlined that will enable the application of the Groebner Basis Algorithm to geometric problems.
Mordell-Weil Theorem And The Rank Of Elliptical Curves, Hazem Khalfallah
Mordell-Weil Theorem And The Rank Of Elliptical Curves, Hazem Khalfallah
Theses Digitization Project
The purpose of this thesis is to give a detailed group theoretic proof of the rank formula in a more general setting. By using the proof of Mordell-Weil theorem, a formula for the rank of the elliptical curves in certain cases over algebraic number fields can be obtained and computable.
History Of Applied Geometry, Evelyn Jackson
History Of Applied Geometry, Evelyn Jackson
Electronic Thesis and Dissertation
Mathematics: Just what does the word mean to us? After a moment of thought many different meanings may present themselves to our minds. At first we are inclined to say that the word mathematics covers a vast field. We are justified in so thinking because mathematics embraces a wide scope of study. Were we to say that it is a science we should place it in its proper genius, for it is truly a science of numbers and space. However, could not the science be the art of calculation or the art of computation?