January 15, 2008
Widely regarded as one of the finest mathematical expositors around, Harvey Mudd College Professor Art Benjamin brought his entertaining and energetic presentation technique to the MAA’s Carriage House on Saturday, January 12, 2008 to talk about Combinatorial Trigonometry and a method known as D.I.E. (Audio)
Benjamin’s talk centered on combinations and permutations dealing with the number of ways a set of squares (1 X 1 tiles) and dominos (1 x 2 tiles) could be arranged and how they could be used to cover a “board” of length 1 x n. He began the talk by using simple combinatorics to prove the Pythagorean Theorem and then progressed into more elaborate mathematics involving trigonometry and Fibonacci Numbers. As is custom with any Benjamin lectures, the crowd, ranging from high school students to professional mathematicians, was engaged in his easy to understand presentation and delighted by his well-timed jokes.
Benjamin took several known identities involving Fibonacci numbers and proved their truth using a method called D.I.E., which stands for description, involution, and exception. Benjamin mentioned the D.I.E. method can offer new insights to identities involving not only Fibonacci numbers, but also binomial coefficients, derangements, and as he would later demonstrate, Chebyshev polynomials. (To hear Art Benjamin describe the D.I.E. method, click here)
Benjamin’s talk concluded with a nifty proof involving a trigonometric identity and the aforementioned Chebyshev polynomials. Benjamin took what looked to be a rather complicated identity and gave a beautiful, straightforward proof of it using the D.I.E. method.
Benjamin is co-author of Proofs That Really Count: The Art of Combinatorial Proof, available at the MAA Bookstore.—Ryan Miller & Robert Vallin