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. .

Author Bio

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. I'm 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 haven’t been able to narrow down one specific area of research to focus on.