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MONTHLY, March 2004

**Yeuh-Gin Gung and Dr. Charles Y. Hu Award to T. Christine Stevens for Distinguished Service to Mathematics**

by Robert E. Megginson

meggin@msri.org

**Pascal Matrices**

by Alan Edelman and Gilbert Strang

edelman@math.mit.edu, gs@math.mit.edu

Put the famous Pascal triangle into a matrix. It could go into a lower triangular *L* or its transpose *L'* or a symmetric matrix *S*. The entries are binomial coefficients (*i*/*j*) , and (*j*/*i*), and (*i + j*/*i*) for the symmetric matrix S.

The amazing thing is that *L* times *L'* equals* S* (known!). Since det *L* = 1 it follows that *S* has determinant 1. The matrices have other unexpected properties too, that give beautiful examples in teaching linear algebra. (Combinatorics too: The rows of the "hypercube matrix" *L*2 count corners and edges and faces in n-dimensional cubes.)

*L* *L'* = *S* is proved four ways, we don’t know which you will prefer:

1. By an identity for the binomial coefficients in these matrices.

2. By recognizing that each matrix counts paths on a directed graph.

Gluing the graphs for *L* and *L'* multiplies the matrices and gives *S*.

3. By induction: elimination on each matrix gives one size smaller.

* 4. By applying both sides to the column vector [ 1 *x*^{x2 x3 ... ]'.}

This last way suggests a fifth proof, not complete, that would find a representation of all Mobius transformations by a matrix group including *L* and *L'* and *S*. Those matrices would represent the maps to *x*/(1-*x*) and 1+*x* and 1/(1-*x*). Then the composition of these transformations gives *L L'* = *S*. But the cube of the map from *x* to 1/(1-*x*) is the identity and we certainly doubt that *S*^{3} = I !

**Interesting Dynamics and Inverse Limits in a Family of One-Dimensional Maps**

by William T. Ingram and William S. Mahavier

ingram@umr.edu, wms@mathcs.emory.edu

In this article attention is restricted to the study of a simple family of piecewise linear unimodal maps from [0,1] onto [0,1]. The chaotic behavior of maps in this family was investigated by Susan Bassein in a paper in this MONTHLY in 1998. These same maps are used to give an elementary introduction to the study of inverse limits and to show that there are remarkable correlations between the chaotic nature of the maps in this family and the complexity of the inverse limits of the maps.

**Fermat and the Quadrature of the Folium of Descartes**

by Jaume Paradís, Josep PLa, and Pelegrí Viader

jaume.paradis@upf.edu, pla@mat.ub.es, pelegri.viader@upf.edu

Do you think you know everything about elementary integration? Try to find the area inside the loop that the folium of Descartes, the curve described by the cubic *x*^{3} + *y*^{3} = *xy*, bounds in the first quadrant. You will undoubtedly succeed, but only after some effort and with quite heavy artillery at your disposal: the integral calculus. Perhaps a generalization of the folium proposed in 1917 will be more of a challenge: *x*^{2q+1} + *y*^{2q+1} = (2*q*+1)*x*^{q}* y*^{q}. But, what can you do about the generalization we propose in this article: *x*^{2q+1 }+ *y*^{2q+1}= *xy*? Fermat, with a method of his own based on elementary means was able, in the parlance of his time, to square the loop of the folium of DescartesÂ—quite an achievement in the mid-seventeenth century. We will see how Fermat’s method works and how can it be applied to attack the more formidable generalizations just indicated.

**The Spiral of Theodorus**

by Detlef Gronau

gronau@uni-graz.at

The discrete spiral of Theodorus can be represented in the complex plane by *z*_{0} = 1 and *z*_{n} + 1, (1 + *i*/√*n* +)*z*_{n} n = 0,1,2,Â… (see Philip J. Davis, *Spirals from Theodorus to Chaos*, A. K. Peters, 1993). Davis proposes the "Theodorus function" *T*, defined by an infinite product, as an interpolating function of the discrete spiral of Theodorus. The function *T* satisfies a first order difference equation. Davis asked for a (geometrical) characterization of the Theodurus function among the various solutions of this difference equation. In our paper we give some characterizations of this function. We give a new formula for the Theodorus function that converges faster than the infinite product of Davis. We also provide pictures of some different spirals and spiral-like curves.

**Putting Fractions in Their Place**

by Leslie Blackwell Galen

leslie@integretechpub.com

Authors, students, and editors should learn to recognize that mathematics has a typography all its own, and should feel comfortable reading, writing, and typesetting mathematical expressions. This article deals with fractions. Fractions are explained as typographical entities, rather than mathematical functions. The four kinds of fractions—case, special, shilling, and built-up—are discussed, as well as where and how they should be used. Examples of misuse, as well as tips for proper use and setting of different sorts of fractions, are shown with examples and discussion.

**Problems and Solutions**

**Notes**

**On Prime Factors of ***A*^{n} - 1

by Tsuneo Ishikawa, Nobuhiko Ishida, and Yoshito Yukimoto

ishikawa@ge.oit.ac.jp, ishida@an.email.ne.jp

**An Algorithmic Proof of the Motzkin-Rabin Theorem on Monochrome Lines **

by Lou M. Pretorius and Konrad J. Swanepoel

lpretor@scientia.up.ac.za, swanekj@unisa.ac.za

**A Note on Isometric Embeddings of Surfaces of Revolution**

by Martin Engman

um_mengman@suagm.edu

**On Subgroups of Prime Index**

by T. Y. Lam

lam@math.berkeley.edu

**Reviews**

**A ***Singular *Introduction to Commutative Algebra

by Gert-Martin Greuel and Gerhard Pfister

Reviewed by John B. Little

little@mathcs.holycross.edu

**Asymptotology: Ideas, Methods, and Applications **

by I. V. Andrianov and L. I. Manevitch

Reviewed by Stephen A. Fulling

fulling@math.tamu.edu

**Telegraphic Reviews**