Monthly, November 2004
Brunnian Clothes on the Runway: Not for the Bashful
by Colin Adams, Thomas Fleming, and Christopher Koegel
Colin.Adams@williams.edu, email@example.com, Chis.Koegel@Bain.com
Do you have the confidence to wear clothes where the slightest friction might cause them to disintegrate? Brunnian clothes are exactly that, made up of links such that if one component of the link is broken, what remains is the trivial link. In this article, Ralf Laurent explains his latest Brunnian garments and how the mathematics impacts fashion.
Hypergraphs, Entropy, and Inequalities
by Ehud Friedgut
What do hypergraphs, entropy, and inequalities have to do with each other? Many common inequalities such as the Cauchy-Schwarz inequality have a surprising information theoretic interpretation. In this paper this theme is identified in various other well-known, lesser known, and (perhaps?) new examples, and encoded neatly via hypergraphs. As an example of an aesthetic inequality dealt with by this method consider the following. Let A, B, and C be matrices with real entries such that the product ABC is well defined, and let tM denote the transpose of M. Then [Trace(ABC)]2 is less or equal to Trace(AtA)Trace(BtB)Trace(CtC).
The Characteristic Polynomial and Determinant Are Not Ad Hoc Constructions
by Skip Garibaldi
How do you define the determinant of a matrix? As an alternating sum of products of entries in the matrix, like Jacobi did? Where does that magical formula come from? What about the characteristic polynomial? The definitions one sees in linear algebra don't apply to other algebraic structures like the quaternions, but the quaternions do have determinants and characteristic polynomials. In fact, the determinant and characteristic polynomial can be defined for any finite-dimensional algebra over a field (e.g., n-by-n matrices, the quaternions, a finite-degree field extension). In the case of matrices, one gets Jacobi’s "magical formula" as a consequence.
How Conics Govern Möbius Transformations
by Marc Frantz
Students of complex analysis learn that Möbius (linear fractional) transformations map circles to circles, provided that straight lines are considered as special cases of circles. But how do the points map from one circle to another? When the image and preimage are true circles we can visualize this, thanks to a surprising fact: for each Möbius transformation of the unit circle onto itself, there exists a conic that is the envelope of the family of lines that connect points on the circle with their respective images. In a sense the conic "guides" the points along tangent lines to their destinations. We give an elementary proof and illustrations, including "graph paper" for a Möbius transformation from one circle to another. We also discuss applications to pedal curves, Blaschke products, and the famous theorem from geometry known as Poncelet’s porism.
How Cauchy Missed Ramanujan’s 1-psi-1 summation
by Warren P. Johnson
In 1843 Cauchy tried to generalize Jacobi’s triple product identity of 1829. The attempt failed, but only through inadvertence—it is not difficult to derive a correct generalization along the same lines, using two other results in Cauchy’s paper. This theorem is, in a sense, a missing link between Jacobi’s triple product and one of Ramanujan’s best identities, the 1-psi-1 summation formula. I try to argue that if Cauchy had thought more carefully about what he was doing, he might have found the 1-psi-1 formula some seventy years before Ramanujan probably did.
Problems and Solutions
A Note on the Effros Theorem
by Jan van Mill
An Elementary Proof of Jacobi’s Six Squares Theorem
by Song Heng Chan
Computing Cavalieri’s Quadrature Formula By a Symmetry of the n-Cube
by Nils R. Barth
A Property of Normal Tilings
by Deniz Kazanci and Andrew Vince
On the Appearance of Moving Bodies
by Andrzej Nowojewski, Jakub Kallas, and Andrzej Dragan
Knots: Mathematics with a Twist
by Alexei Sossinsky
Reviewed by Louis H. Kauffman