The stately chords and melody of Pachelbel's "Canon in D" are familiar to millions of listeners. But Alissa Crans, associate professor of mathematics at Loyola Marymount University, hears something more that makes a connection between mathematics and music. For her work in that area, Crans was among several LMU professors who won distinctions at the Mathematical Association of America meeting in Lexington, Ky.
Crans and two co-contributors won a share of the Merten M. Hasse Prize for the paper "Musical Actions of Dihedral Groups." She also was awarded one of three Henry L. Alder Awards for Distinguished Teaching by a Beginning College or University Mathematics Faculty Member. "It is truly an honor to be recognized with a Henry L. Alder award by the Mathematical Association of America," Crans said.
Also honored at the summer meeting were mathematics professors Curtis D. Bennett and Patrick D. Shanahan, and Blake Mellor, associate professor of mathematics, for their paper "Drawing a Triangle on the Thurston Model of Hyperbolic Space," which appeared in Mathematics Magazine.
Crans was recognized with the distinguished teaching award for increasing her students' appreciation and enthusiasm for mathematics through the Pacific Coast Undergraduate Mathematics Conference, which she helped organize, and for mentoring young women at mathematics workshops.
The paper Crans wrote with Thomas M. Fiore, of the University of Michigan-Dearborn, and Ramon Satyendra, of University of Michigan, describes a connection between operations suggested by music theory and chord progressions in Beethoven's Symphony No. 9. That connection leads to a proof that the set of operations has a mathematical structure similar to the set of symmetries of an n-sided polygon. Another musical example of this is "Canon in D." The paper "gives a very pretty explanation of what we commonly hear in tonal music" in terms of abstract algebra, the award citation reads.
The non-Euclidian geometry paper by Bennett, Shanahan and Mellor was described in the award citation as a "skillfully written article [that] carefully compares the classical Poincaré disk model of the hyperbolic plane with a physical paper model due to Thurston," a model approximation of the hyperbolic plane.
Responding to the award, the three authors wrote, "This paper started as a conversation about a surprisingly difficult homework problem in a class for future elementary teachers and then took on a life of its own; we hope the results can be brought back to inspire future classroom discussions."