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American Mathematical Monthly - August/September 2002


James R.C. Leitzel Lecture
by Robert F. Witte

In the 2002 James R. C. Leitzel Lecture, retired Exxon Education Foundation Senior Program Officer Bob Witte explains why the Foundation has granted over $3 Million to the MAA. The author shares his experiences with the MAA and with mathematics education and tells why he believes that mathematics holds potential educational productivity for schools and students that goes well beyond its content. He shows why working on mathematics education is the right thing for schools and students. Learn the inside story about how the Foundation came to provide funds for Project NExT. The MAA's important accomplishments, the author argues, demand additional national leadership of the Association and its members.


Isolating Fixed Points
by Robert F. Brown and Jack E. Girolo,

Given a continuous function f : XX, can f be deformed to a continuous function g : XX such that g is close to f and the fixed points of g are isolated? We show that if X is a space that is homeomorphic to a poyhedron, then the answer is yes. We consider the class of maps of polyhedra called simplicial maps and characterize those that have isolated fixed points. We also consider the more general class of absolute neighborhood retracts (ANRS) and discuss the progress that has been made, with regard to the question, for maps of these spaces.


Thomsen's Equation
by James T. Smith

This paper is about the reflections across lines a, b, and c in plane Euclidean geometry. Notation is simplified by letting these letters denote the reflections, too. Compositions abc and bca are glide reflections, so their squares are translations. The equation that says that the squares commute can be rewritten (cancelling aa, bb, and cc) as a composition of twenty-two reflections equaling the identity I. Gerhard Thomsen asked in 1931 whether any shorter such equations hold for all triangles. The present paper describes in detail Hellmuth Kneser's elegant answer—only trivial ones provable just from aa = bb = cc = I —and his method for deriving all such generally valid equations. It employs the notion of algebraic independence of complex numbers, and makes striking use of the familiar hexagonal "honeycomb" lattice. The paper considers analogous higher-dimensional problems, and relates the history of these methods.


by John McCleary

One of the main goals of the originators of non-Euclidean geometry was the proof of the trigonometric relations that hold on the non-Euclidean plane. The model for results of this sort was the set of relations that make up spherical trigonometry. The aim of this paper is to find unified proofs of these relations for both spherical and non-Euclidean trigonometry. The basis for the proofs is a description of the sphere and of the non-Euclidean plane given by Beltrami consisting of a subset of the ordinary plane together with a metric. This metric has a parameter in it that corresponds to the curvature of the model. The integrals associated to arc lengths in this model have nice properties that lead to the desired trigonometric relations. From this point of view, I can prove some classical results including the relation between area and angle defect, the concurrence of medians in triangles and the Bolyai-Lobachevsky theorem. I also give a proof of the irrationality of π and e using the same argument.


Elliptic Curves from Mordell to Diophantus and Back
by Ezra Brown and Bruce T. Myers,

A reading of L.J. Mordell's "Diophantine Equations" raised several questions (to one of us) about solutions of cubic equations in two variables. Many years later, we found answers to these and other questions on a path that took us to Diophantus' "Arithmetica" and back to Mordell. In this paper, we tell about the rank of an elliptic curve and prove that in a certain family of elliptic curves (one of which Diophantus studied), there are infinitely many with rank at least 3. We also tell about a big surprise we found on returning to Mordell's book.


Problems and Solutions


Solution of the Direct Problem of Uniform Circular Motion In Non-Euclidean Geometry
by Robert L. Lamphere

Fibonacci Numbers and Cotangent Sequences
by M. J. Jamieson

An Equidistribution Phenomenon: Is There a Principle Behind It?
by Iosif Pinelis

On a Question of Kaplansky
by P. G. Walsh

A Theorem of Touchard on the Form of Odd Perfect Numbers
by Judy Holdener


Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being
by George Lakoff and Rafael Núñez
Reviewed by Jeffrey Nunemacher

An Invitation to Algebraic Geometry
by K. Smith, L. Kahanpää, P. Kekäläinen, and W. Traves
Reviewed by Mark Green

Telegraphic Reviews