# 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.