Geometry and Topology Commons^™

Evolution As Spatial Projection, Charles H. Smith, Ngoc Nguyen

Faculty/Staff Personal Papers

A theory combining explanations for the nature of three-dimensional systems and evolutionary processes is advanced, set to a geometric simulation, and then discussed in the context of several empirical studies bearing on its validity. The concept “evolution” is first discussed, then related to Baruch de Spinoza’s ideas on natural philosophy, including his concept of the conatus. These ideas are then extended by posing the possible existence of a general form of natural systems subsystemization leading to the extended space condition. A geometrical/topological simulation study projecting spatial relations among subsystem structures interpretable through entropy maximization and multidimensional scaling methods is presented, …

Möbius Transformations And Hyperbolic Geometry In The Upper Half-Plane, Emma Bunch

Mahurin Honors College Capstone Experience/Thesis Projects

In this thesis, we discuss several properties of Möbius transformations and hyperbolic geometry, a type of non-Euclidean geometry, in the upper half-plane using tools of complex analysis. We begin with preliminaries for our work, comprising the stereographic projection, the representation of circles and lines in the complex plane, conformal maps, and a result on cross-products, which we include for further development. We proceed to Möbius transformations and discuss their properties, cross-ratios, and various mappings. We additionally provide useful calculations. Lastly, we conclude with the hyperbolic metric in the upper half-plane and explore hyperbolic distance, including its invariance under Möbius transformations. …

Knot Theory In Virtual Reality, Donald Lee Price

Masters Theses & Specialist Projects

Throughout the study of Knot Theory, there have been several programmatic solutions to common problems or questions. These solutions have included software to draw knots, software to identify knots, or online databases to look up pre-computed data about knots. We introduce a novel prototype of software used to study knots and links by using Virtual Reality. This software can allow researchers to draw links in 3D, run physics simulations on them, and identify them. This technique has not yet been rigorously explored and we believe it will be of great interest to Knot Theory researchers. The computer code is written …

Properties Of Functionally Alexandroff Topologies And Their Lattice, Jacob Scott Menix

Masters Theses & Specialist Projects

This thesis explores functionally Alexandroff topologies and the order theory asso- ciated when considering the collection of such topologies on some set X. We present several theorems about the properties of these topologies as well as their partially ordered set.

The first chapter introduces functionally Alexandroff topologies and motivates why this work is of interest to topologists. This chapter explains the historical context of this relatively new type of topology and how this work relates to previous work in topology. Chapter 2 presents several theorems describing properties of functionally Alexandroff topologies ad presents a characterization for the functionally Alexandroff topologies …

Loop Numbers Of Knots And Links, Van Anh Pham

Masters Theses & Specialist Projects

This thesis introduces a new quantity called loop number, and shows the conditions in which loop numbers become knot invariants. For a given knot diagram D, one can traverse the knot diagram and count the number of loops created by the traversal. The number of loops recorded depends on the starting point in the diagram D and on the traversal direction. Looking at the minimum or maximum number of loops over all starting points and directions, one can define two positive integers as loop numbers of the diagram D. In this thesis, the conditions under which these loop numbers become …

Minimizing Travel Time Through Multiple Media With Various Borders, Tonja Miick

Masters Theses & Specialist Projects

This thesis consists of two main chapters along with an introduction and
conclusion. In the introduction, we address the inspiration for the thesis, which
originates in a common calculus problem wherein travel time is minimized across two media separated by a single, straight boundary line. We then discuss the correlation of this problem with physics via Snells Law. The first core chapter takes this idea and develops it to include the concept of two media with a circular border. To make the problem easier to discuss, we talk about it in terms of running and swimming speeds. We first address …

Bowles, John (Sc 593), Manuscripts & Folklife Archives

Manuscript Collection Finding Aids

Finding aid and full-text (click on "Additional Files" below) for Manuscripts Small Collection 593. Ciphering book kept by John Bowles of Granville County, North Carolina. Bowles used "Pike's Arithmetic" in making his ciphering book.

King, John (Sc 594), Manuscripts & Folklife Archives

Manuscript Collection Finding Aids

Finding aid and full-text (click on "Additional Files" below) for Manuscripts Small Collection 594. Ciphering book of John King including mathematical exercises, numeration of money, simple and compound reduction, weights and measures, and word problems.

Curd, Spencer, 1788-1832 (Sc 966), Manuscripts & Folklife Archives

Manuscript Collection Finding Aids

Finding aid and full text (click on "Additional Files" below) for Manuscripts Small Collection 966. Ciphering book (94 p.), focusing chiefly on surveying and navigating rules for solving problems as well as giving specific problems, of Spencer Curd, Logan County, Kentucky, with accompanying genealogical data.

Continued Radicals, Jamie Johnson

Masters Theses & Specialist Projects

If a1, a2, . . . , an are nonnegative real numbers and fj(x) = paj + x, then f1o f2o· · · fn(0) is a nested radical with terms a1, . . . , an. If it exists, the limit as n ! 1 of such an expression is a continued radical. We consider the set of real numbers S(M) representable as an infinite nested radical whose terms a1, a2, . . . are all from a finite set M. We give conditions on the set M for S(M) to be (a) an interval, and (b) homeomorphic to the …