The Mathematical Association of America has selected David A. Cox (Amherst College), Graham Everest (University of East Anglia) and Thomas Ward, Peter Sarnak (Princeton University), and Ravi Vakil (Stanford University) as the 2012 winners of the Lester R. Ford Award.

 

Ravi Vakil (Stanford University)

The Mathematics of Doodling,” American Mathematical Monthly, 118:2 (2011), p. 116-129.

This article, based on the first of Vakil’s Hedrick lectures at the 2009 MathFest, is a wonderful example of expository mathematics. A doodle involves starting with a shape (for example a “W”) on a piece of paper, and then drawing a curve around it, roughly the set of points within a small constant "distance" from the W. Now repeat the procedure starting with the curve obtained and keep repeating. Do the successive doodles get more and more "circular?" The author began with this simple mathematical question, one that a seventh grader might ask, and "just followed where it took us."

Definitions and questions are made more precise, and eventually lead to reworded precise and satisfying answers. Just following “where it took us” inspires a sequence of natural generalizations and, inevitably, to more sophisticated topics. Vakil touches on the relevance of these investigations to elementary, and not so elementary, well-known and not so well-known problems in geometry. Along the way the reader gets an informal introduction to linear invariants, winding numbers, differential geometry, Hilbert’s third problem, and current research in algebraic and hyperbolic geometry. As Vakil concludes: "In some sense our journey is a metaphor for mathematical exploration in general."