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D0%DMLMDMDLDD%M : February 22, 2008
To the RHIT Mathematics Journal Editors:
Im writing to request the honor of having my research paper, Fundamental Reflection Domains for Hyperbolic Tessellations, be published in your journal. I became interested in non-Euclidean geometry after receiving a brief introduction to hyperbolic geometry in a geometry class at the University of Utah. My research in hyperbolic tessellations started at the University of Utah in the summer (starting in June) of 2006 with Domingo Toledo (a full professor in the Mathematics Department) as my advisor. I started consolidating my research into a paper in the spring of 2007.
Biographical Sketch:
After graduating high school in 2000, I studied Engineering Science at Broome Community College (Binghamton, NY) for two years. I was on the tennis team both years I was there. After finishing my associate in science degree, I served a two year, Spanish-speaking mission for The Church of Jesus Christ of Latter-Day Saints in Salt Lake City, UT. After my mission, I chose to continue my studies at the University of Utah, double-majoring in both Mathematics and Physics. Im currently in my final semester at the University of Utah, enjoying the great outdoor activities that Utah has to offer, such as skiing and mountain biking.
This summer I will be doing research for the US Air Force in NJ with the Test andEvaluation Squadron. Within the next year or two, I plan pursuing a doctorates degree in Mathematics. I hope to become a professional researcher; however I havent been able to narrow down one specific area of research to focus on.
Sponsor:
Domingo Toledo
Additional Instructions Concerning My Paper:
My Tex compiler gives me several warnings which pertain to the fact that some of the tables are wider than the compiler likes, and some of the pages only contain tables and no text. These warnings can be ignored. The Tex document should be compiled as a PDF file since all my pictures (there are about 30 of them) are PDFs.
Abstract:
This paper summarizes Vinberg's algorithm for finding the subgroup generated by reflections of the group of integral matrices that preserve particular quadratic forms of signature (n,1). Also, many fundamental reflection domains of different hyperboloids, found by the author using Vinberg's algorithm, are listed in this paper. Plus, Matlab code, written by the author, is included, which serves to help one discover potential perpendicular vectors to the hyperplanes (mirrors) that enclose the fundamental domain.
Introduction:
Finding a fundamental reflection domain that can tessellate hyperbolic space can be both a lengthy and a complex problem. In this paper, Vinberg's algorithm for finding the subgroup generated by reflections of the group of integral matrices that preserve particular quadratic forms of signature (n,1) is described in detail. However, this process is simply stated and not proven. Then it is shown how to write the equations of the hyperplanes that correspond to this subgroup, and how to characterize their symmetries in a Coxeter Diagram. This process is centered around preserving integrality through reflections and the many conditions that arise from attempting to do so. The fundamental reflection domains of different hyperboloids that were found by the author using Vinberg's algorithm will also be shown. Code, written by the author, to a Matlab program has also been included in order to aide in the tedious process of finding the perpendicular vectors to the hyperplanes (mirrors) that enclose the fundamental domain.
Thanks for your consideration,
Jacob R. Grosek
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