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- Math Club Event, 4/23/13

BSU Mathematics Department Events, BSU Math Club, and more

The purpose of the Math Club is to promote the learning, understanding, and enjoyment of mathematics on the Bridgewater State University campus. The Club currently hosts a Game Night annually in the fall and Math Chats annually in the spring.

*Students, faculty, and staff from all disciplines are all welcome to join the club!*

If you would like to participate, organize additional activities, or help obtain funds please e-mail **Sarah Milligan** at SMILLIGAN@student.bridgew.edu .

Sarah Milligan, President

Megan Lalumiere, Vice President

Alanna Nucci, Secretary

Savanah Seay, Treasurer

Tameka, Tiago, Yaqin, Jamie, Terry, & Olivia

**Previous Advisors:**

Dr. Shelley Stahl, 2017-2019

Dr. Laura Gross 2011 - 2015

Dr. Vignon Oussa, 2012 - 2014

Dr. Jacqueline Anderson, 2013 - 2014

Professor Thomas Moore

Dr. Rebecca Metcalf

Dr. Uma Shama

**Tuesday, 23 April, 2013**

**3:30 - 5:00 p.m.**

**Conant Science and Math Building, 461**

(Event Flyer)

**Faculty Mentors
Dr. Vignon Oussa
Dr. Matthew Salonmone**

by

**Ashley Erwin**

by

**James Marcotte**

**Description:**

M.C. Escher's 1958 woodcut Sphere Surface is an image of black and white fish that swim on a spiral path out from a central point on the surface of a sphere. The goal of this project is to investigate whether or not the spirals that run through the fish in Sphere Surface are logarithmic, which would uncover the geometry of the entire sphere. In order to do this the original image of Sphere Surface was first adjusted for orthographic view and then stereographically projected onto the Riemann sphere. The Riemann sphere was then rotated by a specific Möbius transformation such that the limit point or pole in the image was centered at the origin. The principal complex logarithm was then applied to the image to determine whether or not the spirals from the original image would become straight lines in the exponential covering space. The spirals did in fact become straight lines in the exponential covering space which is proof that the spirals in Sphere Surface are logarithmic spirals. This shows that the fish in Sphere Surface follow loxodromic spirals on the surface of the Riemann sphere, viewed in orthographic projection. The results of this project will help show whether any regular tiling that starts with straight lines in the design will lead to an Escher like image once the complex exponential function is applied to it and then projected back onto the Riemann sphere.

by

**Joshua Bernard**

**Abstract:**

This honors thesis examines the consequences of abandoning specific underlying assumptions of economic models used to describe the distribution of goods among individual agents or parties and the information about each one's preferences. In microeconomic theory, the Edgeworth Box, Pareto-optimal trade, and convex (especially Cobb-Douglas) preference structures are used to model the process in which consumers and producers make trade-off choices that allocate limited resources among competing agents. This thesis investigates the common underlying assumptions of these economic models by drawing upon mathematical theory to develop both an analytical framework and the tools that help us establish boundaries for these economic problems. The means of investigation involves extensive use of mathematical reasoning and computer simulation. The main focus of this investigation is to determine the consequences of relaxing the theoretical assumption stating that agents participating in Pareto-efficient exchange always operate with complete and correct information. The objective is to first determine the changes in Pareto optimization and price-setting that occur as a result of differences in perception regarding marginal rates of exchange and then to determine which trades are and are not Pareto-efficient.

*Loxodromes: A Rhumb Way to Go*by James Alexander.*Mathematics Magazine*, Vol. 77, No. 5 (Dec., 2004), pp. 349-356.*M.C. Escher : visions of symmetry*by Doris Schattschneider. New York : H.N. Abrams, c2004.*The magic of M.C. Escher*by M.C. Escher. New York : Harry N. Abrams, 2000.*The Miquel Point*from Wolfram MathWorld*The Tessellating World of M.C. Escher*by Doris Schattscheider.*Odyssey*, Vol. 8 Issue 8, pp. 18-20.*Three-Circle Problems in Modern Geometry*by C. D. Smith.*National Mathematics Magazine*, Vol. 14, No. 6 (Mar., 1940), pp. 299-307.*Visions of symmetry : notebooks, periodic drawings, and related work of M.C. Escher*text and commentary by Doris Schattschneider. New York : W.H. Freeman, c1990.*The world of M. C. Escher*Edited by J. L. Locher. New York, H. N. Abrams [c1971].