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American Mathematical Monthly -January 2001

January 2001

Mathematics Departments in the 21st Century: Role, Relevance, and Responsibility
by W.E. Kirwan
What opportunities exist for mathematics departments to create their enrollments and their role in the larger agenda of our universities? This question deserves the serious attention of every department in the nation. For, like a sequence of comets striking the Earth, external and internal factors are combining to alter the environment in which mathematics departments operate. Under these new conditions, we can either meet the fate of the dinosaurs, or we can adopt new strategies and thrive in the coming decades. We need to address the challenges of the new academic environment, we need to adapt in order to survive, and we need to broaden the role and mission of our departments.


A Mechanical Derivation of the Area of the Sphere
by David Garber and Boaz Tsaban,
We use a medieval proof technique in order to derive the area of the sphere and its horizontal sections. The basic argument is mechanically oriented, and thus can be grasped intuitively by non-professionals. We also provide a mathematical formalization of the medieval proof technique as well as of our generalization.


Reorganizing Large Web Sites
by David Aldous
When looking within a large web site, sometimes I must click on many links until I find the right page. There must be some optimal trade-off between number of links per page and the number of clicks needed; this is related to classical coding/data compression, but the issue is complicated by the requirement that links be "human-interpretable" categories. We describe some mathematical theory to show how a given site can be reorganized to approximately optimize the trade-off.


Birkhoff's Theorem for Panstochastic Matrices
by Dean Alvis and Michael Kinyon,
Birkhoff's theorem for doubly stochastic matrices states that any such matrix is a convex combination of permutation matrices. Does an analogous result hold for panstochastic matrices; that is, is every panstochastic matrix a convex combination of panstochastic permutation matrices? (A square matrix with nonnegative entries is panstochastic if the sum of the entries along any row, column, upward or downward diagonal, either broken or unbroken is equal to 1.) We answer this question completely using methods from linear algebra and matrix theory.


Compass and Straightedge in the Poincaré Disk
by Chaim Goodman-Strauss
For over 120 years, tesselations of the hyperbolic plane have appeared in mathematical journals and books, in the art of M.C. Escher, and as interesting graphic images on posters, buttons, etc. Though dozens of books have been published on the analysis underlying such tesselations, it appears that there has not been one single article describing the synthetic techniques actually used to make them. We show how a "hyperbolic compass and straightedge" can be constructed with the usual Euclidean compass and straightedge; that is, given any two points in the Poincaré disk model (or the other well-known models), one can give a simple Euclidean construction for the geodesic through the two points, or for the hyperbolic circle centered at one, passing through the other. Moreover, there is a beautiful duality between geodesic/points inside the disk and points/lines outside the disk. This duality is extremely helpful in the construction of tesselations; it appears to have been discovered many times over the last 120 years, and was certainly the basis for such images as M.C. Escher's Circle Limit prints.


Constructivism is Difficult
by Eric Schechter
Learning most mathematical subjects merely involves adding to one's knowledge, but learning constructivism involves modifying all aspects of one's knowledge: Theorems, methods of reasoning, technical vocabulary, and even the use of everyday words that do not seem technical, such as "or". This article discusses (in the language of mainstream mathematicians) some of those modifications. Perhaps newcomers to constructivism will not be so overwhelmed by it if they know what kinds of difficulties to expect.



The Smith College Diploma Problem
by Ira M. Gessel

Cochran's Theorem for Rings
by William C. Waterhouse

The Tumbling Tetrahedron
by Andrew Simoson

Tangent Spheres and Triangle Centers
by David Eppstein

Acyclic and Totally Cyclic Orientations in Plane Graphs
by Marc Noy

The Hölder-McCarthy and the Young Inequalities Are Equivalent for Hibert Space Operators
by Takayuki Furuta

the evolution of ... Modular Miracles
by John Stillwell

Problems and Solutions


Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture.
By David Bressoud

Reviewed by Doron Zeilberger

Stephen Smale: The Mathematician who Broke the Dimension Barrier.
By Steve Batterson

Reviewed by John Guckenheimer

Calculus Mysteries and Thrillers.
By R. Grant Woods

How to Ace Calculus: The Streetwise Guide
By Colin Adams, Joel Hass, and Abigail Thompson

Reviewed by William G. McCallum

Telegraphic Reviews