Lafayette Campus News (www.lafayette.edu), January 29, 2008 — Mathematics major Daniel D’Argenio ’10 (Yardley, Pa.) uncovered three parametric families of parallelpipeds that are nearly perfect while working as an EXCEL scholar last summer under Cliff Reiter, professor of mathematics. Reiter and D’Argenio recently published an article about their findings titled, “Families of Nearly Perfect Parallelpipeds” in the JP Journal of Algebra, Number Theory and Applications.
“It is unknown whether there is a perfect parallelepiped,” explains Reiter. “This paper studies organized families of examples that contain much of the structure needed by a perfect parallelepiped in the hope that they would lead to an example of a perfect parallelepiped, or an understanding of why a perfect parallelepiped does not exist.”
Reiter and D’Argenio’s research demonstrates how to extend integer-length integer vectors, required for stronger versions of perfect parallelpipeds, in any dimension up to one dimension higher. With this construction, they presented three parametric families of parallelpipeds that are nearly perfect with the exception of two conditions.
Several computer searches that Reiter and D’Argenio have done show various examples where either, but not both, of these two conditions may be satisfied.
D’Argenio’s research is an extension of related EXCEL research conducted in 2005 and 2006 by fellow mathematics major Jordan Tirrell ’08 (West Grove, Pa.) on the perfect cuboid, also under Reiter’s guidance.
“[The perfect parallelpiped problem] is related to the perfect cuboid problem, which is more famous and studied,” Reiter explains. “If there is not a perfect parallelepiped, then there is not a perfect cuboid either.”
Using Tirrell’s research as a reference, D’Argenio and Reiter discussed known examples and, from these, developed some theorems and examples of their own to test.
Reiter describes D’Argenio as an exceptionally bright student to work with.
“I always looked forward to his questions,” says Reiter. “He ran his own searches, had his own favorite strategies and his questions always forced me to clarify my thinking and my own questions so that progress occurred.”
