American Mathematical Monthly -January 2008

January 2008

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Squaring the Plane
By: Frederick V. Henle and James M. Henle
[email protected], [email protected]
The authors describe how to tile the plane using exactly one square of each positive, integral side-length. They discuss the history of the task and pose a range of open problems.

A Novel Way to Generate Fractals
By: Michelle Previte and Sean Yang
[email protected], [email protected]
Have you ever wanted to build your own fractal? This article will describe a procedure called a vertex replacement rule that can be used to construct fractals. We also show how one can easily compute the topological and box dimensions of the fractals resulting from vertex replacements.

Ultrafilters
By: Péter Komjáth and Vilmos Totik
[email protected], [email protected]
Ultrafilters are useful tools in various branches of mathematics. In this article they appear in the infinite analogues of three finitary problems with similar conclusions: the hunter problem, the 3-way switch-lamp problem, and Arrow's social choice theorem. The conclusions of the finitary problems represent the fact that on a finite set every ultrafilter is generated by a single element. We solve the general ultrafilter forms of these problems, and also show how ultrafilters can be used to define Banach-limits in an elegant way.

A Lost Theorem: Definite Integrals in Asymptotic Setting
By: Ray Cavalcante and Todor D. Todorov
[email protected], [email protected]
We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using Riemann sums. In our axiomatic approach even the proof of the existence of the definite integral (which does use Riemann sums) becomes slightly more elegant than the conventional one. The method presented in this article has a long and interesting history. The reader might be surprised to learn that practically all textbooks on calculus and its applications that were written in the period between Leibniz and Riemann motivate, define, and set up integrals by methods very similar to the method presented in this article, although disguised in the language of infinitesimals. The method (among other treasures) was lost in the history of calculus after infinitesimals were abolished at the end of the 19th century. This explains the choice of the title "A Lost Theorem....'' The article is written for readers who teach calculus and its applications. It might be accessible to students under a teacher's supervision and suitable for senior projects on calculus, real analysis, or history of mathematics.

Notes

A Simple Polynomial for a Simple Transposition
By: Greg Martin
[email protected]

Covers for Angleworms
By: Banyat Sroysang, John E. Wetzel, and Wacharin Wichiramala
[email protected], [email protected], [email protected]

A Note on Tarski's Note
By: Stefanie Ucsnay
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Another Simple Proof of the High Girth, High Chromatic Number Theorem
By: Simon Marshall
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Reviews

Fearless Symmetry: Exposing the Hidden Patterns of Numbers
By: Avner Ash and Robert Gross
Reviewed by: Jeffrey Nunemacher
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Symmetry and the Monster
By: Mark Ronan
Reviewed by: Ron Solomon
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