March 2011 Contents
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Subtle Secrets of Singular Vectors
Joan Remski, Mike Lachance & David James
Social scientists use adjacency tables to discover influence networks within and among groups. Building on work by Moler and Morrison, we use ordered pairs from the components of the first and second singular vectors of adjacency matrices as tools to distinguish these groups and to identify particularly strong or weak individuals in them.
Mohammad Torabi Dashti
Like Pascal's triangle, Faulhaber's triangle is easy to draw: all you need is a little recursion. The rows are the coefficients of polynomials representing sums of integer powers. Such polynomials are often called Faulhaber formulae, after Johann Faulhaber (1580--1635); hence we dub the triangle Faulhaber's triangle.
How Spherical are the Archimedean Solids and their Duals?
P. K. Aravind
The Isoperimetric Quotient, or IQ, introduced by G. Polya, characterizes the degree of sphericity of a convex solid. This paper obtains closed form expressions for the surface area and volume of any Archimedean polyhedron in terms of the integers specifying the type and number of regular polygons around each vertex. Similar results are obtained for the duals, the Catalan solids. These results are used to compute the IQ of each Archimedean and Catalan solid and it is found that nine of them have greater sphericity than the truncated icosahedron, the solid which serves as the geometric framework for a molecule of C-60, or “Buckyball.”
The Symmedian Point Constructed and Applied
A post WWII assignment from the U. S. Navy analyzing data from a test of a harbor mine-detecting algorithm leads to a new construction of the symmedian point.
Dynamics of Simple Folds in the Plane
Nikolai A. Krylov, Edwin L. Rogers
Take a strip of paper and fold a crease intersecting the long edges, creating two angles. Choose one edge and consider the angle with the crease. Fold the opposite edge along the crease, creating a new crease that bisects the angle. Fold again, this time using the newly created crease and the initial edge, creating a new angle along the chosen edge. It is well known that the angles thus constructed approach a limiting value independent of the initial angle. In this article, we generalize this process relaxing the requirement that the long edges be straight and that the folds be angle bisectors.
Given a segment joining two lattice points in R3, when is it possible to form a lattice cube with this segment as one of its twelve edges? A necessary and sufficient condition is that the length of the segment be an integer. This paper presents an algorithm for finding such a cube when the prime factors of this length are known, and shows that the cube is essentially unique if the given segment does not contain another lattice point.
The Symmetry Group of the Permutahedron
Although it can be visualized fairly easily and its symmetry group is easy to calculate, the permutahedron is a somewhat neglected combinatorial object. We propose it as a useful case study in abstract algebra. It supplies concrete examples of group actions, the difference between right and left actions, and how geometry and algebra can work together.
A Midsummer Knot’s Dream
Oliver Pechenik, J. Townsend, A. Henrich, N. MacNaughton, R. Silversmith
We introduce playing games on the shadows of knots and demonstrate two novel games, namely, "To Knot or Not to Knot" and "Much Ado about Knotting." We discuss winning strategies for these games on certain families of knot shadows and go on to suggest variations of these games for further study.
STUDENT RESEARCH PROJECT
Golden Matrix Families
Anne Fontaine and Sue Hurley
This student research project explores the properties of a family of matrices of zeros and ones that arises from the study of the diagonal lengths in a regular polygon. There is one family for each n > 2. A series of exercises guides the student to discover the eigenvalues and eigenvectors of the matrices, which leads in turn to formulas for products and inverses. For more advanced students there is a section on the connection to cyclotomic fields. A number of open-ended questions are given as avenues for further investigation.
An Application of Sylvester’s Rank Inequality
Using two well-known criteria for the diagonalizability of a square matrix and an extended form of Sylvester's Rank Inequality, the author presents a new condition for the diagonalization of a real matrix. From this result one can obtain the eigenvectors themselves by simply multiplying some associated matrices, i.e., without solving a system of equations.
PROBLEMS AND SOLUTIONS
Logical Labryrinths by Raymond Smullyan
Reviewed by Ken Shilling
Roads to Infinity: The Mathematics of Truth and Proof by John Stillwell
Reviewed by Stan Wagon