Based on the notion of "arithmetic triangles," arithmetic...

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**James R.C. Leitzel Lecture**

by Robert F. Witte

rfwitte@aol.com

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

rfb@math.ucla.edu, jgirolo@calpoly.edu

Given a continuous function *f* : *X* → *X*, can *f* be deformed to a continuous function *g* : *X* → *X* 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

smith@math.sfsu.edu

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 answerÂ—only *trivial* ones provable just from *aa* = *bb* = *cc* = I Â—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.

**Trigonometries**

by John McCleary

mccleary@Vassar.edu

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

brown@math.vt.edu, btmyers@orion.ncsc.mil

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**

**Notes**

**Solution of the Direct Problem of Uniform Circular Motion In Non-Euclidean Geometry**

by Robert L. Lamphere

Robert.Lamphere@kctcs.net

**Fibonacci Numbers and Cotangent Sequences**

by M. J. Jamieson

mjj@dcs.glasgow.ac.uk

**An Equidistribution Phenomenon: Is There a Principle Behind It?**

by Iosif Pinelis

ipinelis@mtu.edu

**On a Question of Kaplansky**

by P. G. Walsh

gwalsh@mathstat.uottawa.ca

**A Theorem of Touchard on the Form of Odd Perfect Numbers**

by Judy Holdener

holdenerj@Kenyon.edu

**Reviews**

**Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being**

by George Lakoff and Rafael Núñez

Reviewed by Jeffrey Nunemacher jlnunema@cc.owu.edu

**An Invitation to Algebraic Geometry**

by K. Smith, L. Kahanpää, P. Kekäläinen, and W. Traves

Reviewed by Mark Green mlg@math.ucla.edu

**Telegraphic Reviews**