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Mathematics major Jordan Tirrell ’08 (West Grove, Pa.) has found a proof for an algebraic structure that generalizes Pythagorean triples in multiple dimensions.
Tirrell and Cliff Reiter, professor of mathematics, are hoping that the results from their EXCEL research will be published in a scholarly journal.
“We are looking for methods to put a (matrix algebra) structure on integer solutions to equations that generalize the Pythagorean triples,” says Reiter.
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. It is derived from the Pythagorean theorem. A well-known example of a triple is (3, 4, 5).
Tirrell has done computer searches to get ideas for what matrix generators might work for their theorem, and, after testing numerous proofs, has found one in which all the solutions in dimensions up to eight can be generated by using a small number of matrices.
“Jordan is smart, enthusiastic and enjoys doing math, so it is pleasant and thought provoking to work with him,” says Reiter.
The two collaborated on another EXCEL project last summer and this is an outgrowth of that research. The results from last year’s work, “Pursuing the Perfect Parallelepiped,” have been published in JP Journal of Algebra, Number Theory and Applications and presented at the AMS-MAA-SIAM Joint Mathematics Meetings Jan. 12-15 in San Antonio.
“I think I’ve been able to do a lot more on my own this summer because of my research experience last summer,” says Tirrell. “Professor Reiter is an excellent research advisor. I can’t remember a day when he didn’t have an interesting question to pose.”
Tirrell plans to pursue a doctorate in mathematics after he graduates from Lafayette and would like to teach at a college or university. He believes that the time he has spent as an EXCEL scholar has helped solidify his future career path.
“EXCEL has certainly contributed significantly to my Lafayette experience,” he says. “Being able to do research gave me an entirely new perspective on the mathematical community.”
Reiter also thinks the EXCEL experience is rewarding. He has mentored over 30 students in various academic projects, including work with Prince Chidyagwai ’05 on the mathematical generation of snowflake patterns. The project was featured as the lead story on Nature magazine’s web site and also appeared in Chaos, Solutions and Fractals.
“I find the interaction with EXCEL students to be one of the most engaging parts of working as a teacher/scholar at Lafayette,” he says.
Tirrell recently achieved national distinction as recipient of the Goldwater Scholarship. Awarded for academic merit, the Goldwater is the premier undergraduate award of its type in the fields of mathematics, science, and engineering.
He is a member of the tennis club, table tennis club, math club, and works as a tutor for the math department. He plans to spend the next spring semester in Budapest, Hungary studying math.
Lafayette’s distinctive EXCEL Scholars program, in which students conduct research with faculty while earning a stipend, has helped make Lafayette a national leader in undergraduate research. Many of the more than 160 students who participate each year share their work through articles in academic journals and/or conference presentations.