the original version of this story appeared in Quanta Magazine.
In the 2020 Georgia gubernatorial election, some voters in Atlanta waited more than 10 hours to cast a vote. One of the reasons for the long lines was that almost 10 percent of Georgia polling stations had closed during the previous seven years, despite an influx of around 2 million voters. These closures were disproportionately concentrated in predominantly black areas that tended to vote Democratic.
But identifying the location of “voting deserts” is not as simple as it might seem. Sometimes the lack of capacity is reflected in long waits at polling stations, but other times the problem is the distance to the nearest polling station. Combining these factors systematically is complicated.
in a article to be published this summer In the diary SIAM Review, janitor bricklayer, a mathematician at the University of California, Los Angeles, and his students used topology tools to do just that. Abigail Hickok, one of the paper’s co-authors, conceived the idea after seeing images of long lines in Atlanta. “I kept the vote on my mind a lot, in part because it was a particularly nerve-wracking election,” she said.
Topologists study the underlying properties and spatial relationships of geometric shapes under transformation. Two shapes are considered topologically equivalent if one can be deformed into the other by continuous movements without tearing, gluing, or introducing new holes.
At first glance, the topology would seem to be a poor fit for the voting site location problem. The topology deals with continuous shapes and the voting sites are in discrete locations. But in recent years, topologists have adapted their tools to work with discrete data by creating graphs of points connected by lines and then analyzing the properties of those graphs. Hickok said these techniques are useful not only for understanding the layout of polling places but also for studying who has better access to hospitals, grocery stores and parks.
That’s where the topology begins.
Imagine creating small circles around each point on the graph. The circles start with a radius of zero, but grow over time. Specifically, when time exceeds the wait time at a certain polling location, the circle will begin to expand. As a result, locations with shorter wait times will have larger circles (they start growing first) and locations with longer wait times will have smaller circles.
Some circles will eventually touch each other. When this happens, draw a line between the points at their centers. If multiple circles overlap, connect all those points into “symplices,” which is just a general term meaning shapes like triangles (2-simplex) and tetrahedra (3-simplex).
Courtesy of Merrill Sherman/Quanta Magazine