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American Mathematical Monthly -February 2009

February 2009

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Who Was Miss Mullikin?
By: Thomas L. Bartlow and David E. Zitarelli,
In 1946 R. L. Moore closed a letter with a personal note to "Please remember me to Miss Mullikin." Who was Miss Mullikin? And why did someone of Moore’s stature wish to convey his best wishes to her? This paper answers these questions by describing the role this virtually unknown, early-20th-century mathematician played in the evolution of connected sets by examining her influence on international communication between Polish and American mathematicians and by providing biographical details.


A New Look at the So-Called Trammel of Archimedes
By: Tom M. Apostol and Mamikon A. Mnatsakanian,
The paper begins with an elementary treatment of a standard trammel (trammel of Archimedes), a line segment of fixed length whose ends slide along two perpendicular axes. During the motion, points on the trammel trace ellipses, and the trammel produces an astroid as an envelope that is also the envelope of the family of traced ellipses. Two generalizations are introduced: a zigzag trammel, obtained by dividing a standard trammel into several hinged pieces, and a flexible trammel whose length may vary during the motion. All properties regarding traces and envelopes of a standard trammel are extended to these more general trammels. Applications of zigzag trammels are given to problems involving folding doors. Flexible trammels provide not only a deeper understanding of the standard trammel but also a new solution of a classical problem of determining the envelope of a family of straight lines. They also reveal unexpected connections between various classical curves; for example, the cycloid and the quadratrix of Hippias, curves known from antiquity.


Manifolds with Density and Perelman's Proof of the Poincaré Conjecture
By: Frank Morgan
Perelman’s 2003 proof of the Poincaré conjecture considers a manifold with density (the same kind of density as in freshman physics or calculus). Manifolds with density long have appeared in mathematics, but their systematic study is in its infancy, with some seminal contributions by undergraduates. This article begins with the premier example—Euclidean space with Gaussian density—and describes the surprising solution to the isoperimetric problem. It then explores appropriate generalizations of different kinds of curvature, concluding with the generalized Ricci curvature and its use in Perelman’s paper.


Shape Distortion by Analytic Functions
By: Joseph Bak and Pisheng Ding,
There are numerous examples of analytic functions which map many lines or rays into other lines or rays. On the other hand, except for linear polynomials, no functions analytic on a closed triangle can map all three of its sides into three other lines. This is one of the lemmas which help us show that linear polynomials are the only analytic functions which ever map one closed polygon onto another. Thus, unless it is a linear polynomial, a function analytic on a closed polygon never preserves its shape, despite the fact that every analytic function is locally conformal, or shape-preserving, at all noncritical points. Looking at it another way, there is no analytic mapping between any two closed dissimilar polygons, despite the fact that the Riemann Mapping Theorem guarantees the existence of many conformal equivalences between the interiors of any two polygons. The proof rests on some special arguments for the rectangle case and the triangle case, and proceeds somewhat unexpectedly by induction on the number of sides of the polygon.


Lattice Polygons and the Number 2i + 7
By: Christian Haase and Josef Schicho,
In this note we classify all triples (a,b,i) such that there is a convex lattice polygon P with area a which has b and i lattice points on the boundary and in the interior, respectively. The crucial lemma for the classification is the necessity of b ≤ 2i + 7. We sketch three proofs of this fact: the original one by Scott, an elementary one, and one using algebraic geometry. As a refinement, we introduce an onion skin parameter l: how many nested polygons does P contain? and give sharper bounds.



Approximations to π Derived from Integrals with Nonnegative Integrands
By: Stephen K. Lucas
An intriguing definite integral due to Dalzell equals 22/7 - π where the integrand is nonnegative, and can be used to derive an infinite series for π. Here we extend Dalzell's results in two ways. First we look at a new family of integrals leading to series for π that converge arbitrarily fast. Then we show how integrals with nonnegative integrands can be found that equal z - π or π - z for any real z.

New Proofs of Euclid's and Euler's Theorems
By: Juan Pablo Pinasco
Yet another proof of the infinitude of prime numbers, together with a proof of divergence of the series of their reciprocals. They are based on a connection between the inclusion-exclusion principle and the infinite product of Euler.

A Note on Covering a Square of Side Length 2 + ε with Unit Squares
By: Janusz Januszewski
It is impossible to cover a square of side length greater than 2 with five unit squares.


Introduction to Calculus and Classical Analysis
By: Omar Hijab
Reviewed by: V. S. Varadarajan