# The Mathematics of Doodling

by Ravi Vakil

Year of Award: 2014

Award: Chauvenet Prize

Publication Information: American Mathematical Monthly, 118 (2011), no. 2, 116-129.

Summary: (Adapted from the Joint Mathematics Meetings 2014 Prizes and Awards Booklet) In this article we learn about the radius $r$ neighborhood $N_r(X)$ of a set $X$ in the plane and how $N_r(X)$ becomes more disk-like as $r$ increases. We see how the perimeter of $N_r(X)$ is related to the area of $X$, first when $X$ is a convex polygon, then when $X$ is any convex set, then when $X$ is arbitrary. We see how the winding number and the Euler characteristic account for the changes in the resulting formulas. We move to three dimensions and encounter Hilbert's Third Problem and the Dehn invariant, and to $n$ dimensions and meet other dissection invariants. Finally, our tour culminates in a brief visit to the moduli space of curves.

About the Author: (From the Joint Mathematics Meetings 2014 Prizes and Awards Booklet)

Ravi Vakil is professor of mathematics and the Packard University Fellow at Stanford University. He is an algebraic geometer whose work touches on topology, string theory, applied mathematics, combinatorics, number theory, and more. He was a four-time Putnam Fellow while at the University of Toronto. He received his Ph.D. from Harvard, and taught at Princeton and MIT before moving to Stanford. He has received the Dean's Award for Distinguished Teaching, the American Mathematical Society Centennial Fellowship, the Terman Fellowship, a Sloan Research Fellowship, the NSF CAREER grant, and the Presidential Early Career Award for Scientists and Engineers. He has also received the Coxeter- James Prize from the Canadian Mathematical Society and the André-Aisenstadt Prize. He was the 2009 Hedrick Lecturer at MathFest, and is currently an MAA Pólya Lecturer. He is a director of the entity running the website mathoverflow, and the director of a potential new school in San Francisco called the "Proof School." He works extensively with talented younger mathematicians at all levels, from high school through recent Ph.D.s.

Publication Date:
Tuesday, January 21, 2014