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Gerrymandering and Convexity
Award: George Pólya
Year of Award: 2011
Publication Information: College Mathematics Journal, vol. 41, no. 4, September 2010, pp. 312-324.
This article integrates a variety of mathematical topics to give the reader insight into the vagaries of gerrymandering, in which each state re-carves itself into often highly non-convex shapes or districts which have roughly the same population but are designed to give unfair advantage to the party in power. Beginning with several simple examples to show how this works, the authors then give a clear overview of gerrymandering and measures of shape compactness and convexity that have been introduced over the past several decades. They then proceed to develop their own measure, the 'convexity coefficient' of a planar region, which is the probability that the line segment connecting two random points of the region is itself entirely contained within the region. They use Monte Carlo methods to approximate the convexity coefficient for each of the 435 congressional districts in the country, which allows the reader to compare their state to others.
But is this measure fair, comparing highly non-convex states to highly convex states? Delaware with only one district only gets a 0.855 rating. The authors are up to the challenge, modifying their measure to not only take into account the natural boundary of the state, but also non-uniform population distributions. To illustrate the subtle complexity of reapportionment, the paper ends with an example to show that gerrymandering can even occur if the boundaries of the districts look extremely regular and the population is distributed evenly across the geographic district.
About the Authors (From the MathFest2011 Prizes and Awards Booklet)
Jonathan Hodge is an Associate Professor of Mathematics at Grand Valley State University. He is a graduate of Calvin College (B.S., 1998) and Western Michigan University (Ph.D., 2002). Jonathan is proud that the bulk of his research has been conducted with undergraduates. He is currently the co-PI of GVSU‟s Summer Mathematics REU, and he looks forward to many more successful undergraduate research projects in the future. Apart from his professional activities, Jonathan enjoys traveling and spending time with his wife, Melissa, and his two sons, Caden (5) and Asher (1).
Emily Marshall received her bachelor‟s degree from Dartmouth College in 2009 and is currently a graduate student in mathematics at Vanderbilt University. Her enthusiasm for mathematics flourished during the Budapest study abroad program in the fall of 2007 and has been growing ever since. At Vanderbilt, she studies graph theory and teaches calculus to eager freshmen. Emily is excited about beginning research soon. In her free time she enjoys running, playing cards, and exploring the city of Nashville.
Geoff Patterson, raised in Haslett, Michigan, has loved mathematics since an early age. After originally pursuing a degree in film and video at Grand Valley State University, he could no longer suppress his desire to do advanced mathematics. After changing his major, Geoff graduated in 2009 with a Bachelor of Science degree in mathematics and a minor in statistics. During his time at GVSU, Geoff participated in several research projects, both in theoretical and applied mathematics. Some of Geoff's favorite research interests include logic, computability, control theory, and topology.
Geoff is now pursuing his Ph.D. in Mathematics at the University of Hawaii at Manoa, where he is starting work on an exciting project collaborating with the Hawaii Institute for Astronomy.