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AUGUST-SEPTEMBER 2003

**On the Sums** |
**Σ** |
**∞** |
**(4***k*+1)^{- n} |

**k - ∞** |

by Noam D. Elkies

[email protected]

The sum in the title is a rational multiple of *π*^{n} for all integers *n*=2,3,4 ... for which the sum converges absolutely. This is equivalent to a celebrated theorem of Euler. Of the many proofs that have appeared since Euler, a simple one was discovered only recently by Calabi: the sum is written as a definite integral over the unit cube, then transformed into the volume of a polytope *Π*_{n} in *R*^{n} whose vertices' coordinates are rational multiples of *π*. We review Calabi's proof, and give two further interpretations. First we define a simple linear operator *T* on *L*^{2}(0,*π*/2) , and show that *T* is self-adjoint and compact, and that Vol(Π_{n}) is the trace of *T*^{n}. We find that the spectrum of *T* is {1/(4*k*+1): *k* ∈ *Z*} , with each eigenvalue 1/(4*k*+1) occurring with multiplicity 1; thus Vol(Π_{n}) is the sum of the *n*th powers of these eigenvalues. We also interpret Vol(Π_{n}) combinatorially in terms of the number of alternating permutations of *n*+1 letters, and if *n* is even also in terms of the number of cyclically alternating permutations of *n* letters. We thus relate these numbers with S(*n*) without the intervention of Bernoulli and Euler numbers.

**Which Functor is the Projective Line?**

by Daniel K. Biss

[email protected]

The goal of this article is to describe the rudiments of category theory in a way that focuses on examples and applications. The first several sections are devoted to defining these concepts and explaining how they provide a common generalization of most areas of mathematics. This exposition culminates with the statement of the Yoneda lemma, which demonstrates that one can study a category by analyzing the set of functors on it. Then, to allay the reader's fears that this discussion is too hopelessly general to allow for any content, we introduce the category of finitely generated commutative algebras over the complex numbers, explain how its objects can be understood geometrically, and then use the language developed in the first half of the article to describe the complex projective line. This construction hints at a deep marriage between geometry and algebra that is best understood at the level of categories and functors.

**Curvature in the Calculus Curriculum**

by Jerry Lodder

[email protected]

A particular drawback with the instruction of many topics in the undergraduate mathematics curriculum is the lack of context and direction for the material, with reliance on opaque definitions and mechanical formulas instead. The definition of curvature as the magnitude of the rate of change of the unit tangent with respect to arclength and the resulting formula is an example of such. The paper offers an alternative form of instruction based on original historical sources, outlining the development of curvature through Christiaan Huygens's discovery of the isochronous pendulum, continuing with Leonhard Euler's work on the curvature of surfaces, and concluding with Sophie Germain's analysis of elastic force in terms of mean curvature. The material is organized into written projects for use in a multivariable calculus course, with the projects offering descriptions of the discoveries or actual excerpts from the original work of Huygens, Euler, and Germain, along with a sequence of student exercises designed to illuminate the ground-breaking efforts of these pioneers.

**Polygons Whose Vertex Triangles Have Equal Area**

by Guershon Harel and Jeffrey M. Rabin

[email protected], [email protected]

We describe the set of planar *N*-gons such that the triangles formed by any three consecutive vertices have the same area. Modulo the action of the affine group, this set is an algebraic variety of dimension *N*-5 for *N* > 5. It has several connected components corresponding to the possible signs by which the *oriented* areas of the triangles can differ. We give explicit equations for it in terms of polynomials related to continued fractions. Examples of these polygons and their degenerations are given.

**Problems and Solutions**

**Notes**

**The Fundamental Theorem of Algebra and Linear Algebra**

by Harm Derksen

[email protected]

**Projective Generalizations of Two Points of Concurrence on the Nine-Point Circle**

by Charles Thas

[email protected]

**On the Lagrange Remainder of the Taylor Formula**

by Ulrich Abel

[email protected]

**A Proof of the Mazur-Ulam Theorem**

by Jussi Väisälä

[email protected]

**Reviews**

**Indra's Pearls: The Vision of Felix Klein**

by David Mumford, Caroline Series, and David Wright

Reviewed by John H. Hubbard

[email protected]

**Conversations with a Mathematician: Math, Art, Science and the Limits of Reason**

by Gregory J. Chaitin

Reviewed by Marion D. Cohen

[email protected]

**Quantum Calculus**

by Victor Kac and Pokman Cheung

Reviewed by Ranjan Roy

**Telegraphic Reviews**