MAA Honors Three Exemplars of Mathematical Exposition

Scrawl a shape on a piece of paper. Really. Any shape. Now draw a curve tightly around your shape, as close as you can. Do this again. And again. Do you notice anything as you draw more outlines? Is the doodle getting more circular?

Will this always happen, no matter what shape you start with?

This is the question Stanford mathematician Ravi Vakil tackles in the American Mathematical Monthly paper that has garnered him the 2014 Chauvenet Prize, awarded by the Mathematical Association of America (MAA) at the Joint Mathematics Meetings in Baltimore. Named for a professor of mathematics at the United States Naval Academy, the Chauvenet Prize recognizes outstanding articles on mathematical topics.

Vakil’s paper, titled “The Mathematics of Doodling,” begins with the question posed above and then devotes 13 pages to “the mathematics that flows inevitably from it.”

The award citation calls the paper “an enticing illustration of how mathematical curiosity can lead us from gentle musings to sophisticated, interconnected, and deep ideas.”

Vakil has presented an MAA Distinguished Lecture on this topic. He is also the author of the book A Mathematical Mosaic: Patterns & Problem Solving (Brendan Kelly, revised edition, 2008).

While “The Mathematics of Doodling” grew out of the youthful curiosity of its author, another paper honored at the Joint Mathematics Meetings was inspired by a question Martin Gardner featured in his Scientific American column in the 1970s: Can the plane be tiled with the integer squares?

For their treatment of this problem, the father-son duo of Frederick and James Henle have won the David P. Robbins Prize. Awarded every three years, the prize goes to a paper that “has a significant experimental component and is on a topic which is broadly accessible.”

The award citation praises the cleverness of “Squaring the Plane,” which appeared in the January 2008 issue of the American Mathematical Monthly: The authors’ argument “does not bring in any ‘big guns’ to settle the problem, rather it uses ‘big ingenuity,’ which is always preferable.”

Ingenuity figures a bit differently into the third MAA publication to receive recognition in Baltimore. In the introduction to the book A Historian Looks Back: The Calculus as Algebra and Selected Writings, Pitzer College mathematician Judith Grabiner writes: “Mathematics is incredibly rich and mathematicians have been unpredictably ingenious. Therefore the history of mathematics is not rationally reconstructible. It must be the subject of empirical investigation.”

A Historian Looks Back, in which Grabiner offers a selection of writings that span her illustrious career as an expositor of mathematics, is the winner of the 2014 Beckenbach Book Prize, which honors “a distinguished, innovative book published by the MAA.”

Upon learning that she had received the award, Grabiner gave her students credit: “They’ve taught me most of what I know about being clear and presenting material in ways that interest others beside myself,” she said.

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MAA Mathematical Communication (mathcomm.org) is a developing collection of resources for engaging students in writing and speaking about mathematics. The site originated in the MIT Department of Mathematics and was expanded through support from an NSF grant.