# American Mathematical Monthly -May 2006

## May 2006

A $1 Problem by Michael J. Mossinghoff mjm@member.ams.org Suppose you need to design a new$1 coin with a polygonal shape, fixed diameter, and maximal area or maximal perimeter. Are regular polygons optimal? Does the answer depend on the number of sides? We investigate these two isodiametric problems for polygons and describe how to construct polygons that are optimal, or very nearly so, in each case.

What Can be Approximated by Polynomials with Integer Coefficients
by Le Baron O. Ferguson
ferguson@math.ucr.edu
A well-known result of Weierstrass states that any continuous function on a closed bounded interval of the real line can be uniformly approximated by polynomials. If we restrict ourselves to polynomials whose coefficients are all integers can anything interesting be said? The answer is yes.

Periodicity and Predictability in Chaotic Systems
by Marcelo Sobottka and Luiz P.L. de Oliveira
sobottka@dim.uchile.cl, luna@exatas.unisinos.br
In this paper, we present a simple chaotic system (satisfying Devaney’s definition) that is periodic and computationally predictable under a symbolic representation scheme. The system consists of the restriction of the tent map to the rational numbers of its original domain. The example contradicts the usual belief that chaotic systems are necessarily nonperiodic and nonpredictable. A general discussion on the concept of computational predictability and its relationship with the existence of periodic orbits is included.

The Simplest Example of a Normal Asymptotic Expansion
by José Antonio Adell and Alberto Lekuona
The central limit theorem has been described as one of the most important results in mathematics, mainly due to its proven application well beyond its own field. The investigation of the rates of convergence in this theorem, taking the form of normal asymptotic expansions, has great interest both from theoretical and practical points of view. However, proofs of these kinds of results are generally intricate, no matter what method is used. Being guided by the principle of "the general is embodied in the concrete," we provide a very simple example of a normal asymptotic expansion in which the technical complexity is reduced to a minimum. This example has the additional advantage of making clear the connections between some familiar notions and tools from probability theory and mathematical analysis. The content is accessible to a nonexpert looking for the "what" and "why" of this amazing research area.

Notes

The Arbitrariness of the Cevian Triangle
by Mowaffaq Hajja

Curiosities Concerning Weak Topology in Hilbert Space
by Gilbert Helmberg
gilbert.helmberg@telering.at

More Formulas for π
by Hei-Chi Chan
chan.hei-chi@uis.edu

On Gauss’s Entry from January 6, 1809
by Detlef Gröger
groeger.d@t-online.de

Problems and Solutions

Reviews

Prime Obsession
by John Derbyshire
Reviewed by Jeffrey Nunemacher
jlnunema@cc.owu.edu

Stalking the Riemann Hypothesis
by Dan Rockmore
Reviewed by Jeffrey Nunemacher
jlnunema@cc.owu.edu