# The Mathematics of Doodling

Mathematics can be found in almost every aspect of daily life—even doodling. Ravi Vakil of Stanford University unveiled the surprisingly rich mathematics of this common pastime in a recent presentation at the MAA's Carriage House Conference Center.

Start, for example, by writing the word "hi" on a whiteboard (or piece of paper). Then outline the letters, drawing what look like successively larger contours around the word. Vakil posed the question: If you continue to outline the word "hi," does the resulting envelope of curves become more circular? He demonstrated mathematically that this is, indeed, the case. Further investigations of this particular doodle and several variants  highlighted the relevance of such mathematical notions as the convexity of polygons, the nature of polyhedrons, and the role of derivatives in characterizing various aspects of the resulting figures. Properties of shadows and winding numbers became part of the discussion as Vakil encouraged the audience to "work on the simplest possible doodle that you're confused by."

The mathematics of doodling can lead to some unusual (and often hard) puzzles. In the "Russian train problem," for example, a train company has a rule that you are not allowed to transport rectangular boxes whose sum of dimensions (length plus width plus height) is more than one meter. Is it possible to cheat by putting an illegal box inside a larger box that is legal?  Vakil's conclusion, supported by doodle math, was that, while it's not possible to sneak an illegal box onto the train in a similarly shaped legal box, the mathematics used to prove this fact turns out to be "sneaky and not what you would expect." Audio.

Another of Vakil's puzzles considered what would happen if a string were wrapped tightly around Earth's equator and somebody inserted an extra meter of string then raised the lengthened string above Earth's surface to a constant height. Vakil asked the packed house how high off the ground the string would be, giving the audience three choices: Would the height be closest to 1 millimeter, 1 micrometer, or 1 nanometer? The answer of considerably more than 1 millimeter came as a surprise to some in the crowd and seemed to defy common sense. Vakil then used some doodles to show how you could arrive at an elegant solution to the problem.

Vakil also discussed Buffon's needle problem, which asks to find the probability that, when randomly dropped, a needle of a certain length will land on a line, given a surface with equally spaced parallel lines a certain distance apart. The problem was first posed by the French naturalist Georges Buffon in 1733. Vakil doodled some (roughly) parallel lines on an overhead, then dropped a pen cap randomly on the projector to illustrate the concept to the crowd and, again, show the relevance of doodle math for estimating the requisite probability.

One of Vakil's goals is to attract high school and undergraduate students to the mathematical sciences, and his lectures on the mathematics behind doodling are an exciting way to expose young people to interesting and accessible mathematics. The audience at the Carriage House left delighted as Vakil had shown them that the patterns, shapes, and numbers involved in simple doodles can lead to both fun and sophisticated mathematics.

Vakil is author of A Mathematical Mosaic: Patterns & Problem Solving, available at the MAA Bookstore.—Ryan Miller

This MAA Distinguished Lecture was funded by the National Security Agency.