Original version belong to this story appear in Quanta Magazine.
In the 2020 Georgia gubernatorial election, some voters in Atlanta waited more than 10 hours to vote. One reason for the long queues is virtually 10% of polling places in Georgia are closed in the previous seven years, despite about 2 million voters turning out. These closures are disproportionately concentrated in predominantly black areas that tend to vote Democratic.
But pinpointing the locations of “voting deserts” is not as simple as it seems. Sometimes the lack of capacity is reflected in long waits at the polls, but other times the problem is the distance to the nearest polling place. Combining these elements systematically is difficult.
in one The paper will be published this summer in the magazine SIAM Review, Mason Porter, a mathematician at the University of California, Los Angeles, and his students used tools from topology to do just that. Abigail Hickok, one of the paper’s co-authors, came up with the idea after seeing images of long lines in Atlanta. “Voting is always on my mind, in part because this is a particularly anxiety-provoking election,” she said.
Topologists study the fundamental properties and spatial relationships of transforming geometric shapes. Two shapes are considered structurally equivalent if one can deform into the other through continuous movements without tearing, gluing, or creating new holes.
At first glance, the topology seems inconsistent with the issue of polling place placement. The topology involves continuous shapes and polling places located in discrete locations. But in recent years, topologists have adapted their tools to work on discrete data by creating graphs of points connected by lines and then analyzing the properties of those charts. Hickok said these techniques are useful not only for understanding the distribution of polling places, but also for studying who has better access to hospitals, grocery stores and parks.
That’s where topology begins.
Imagine creating small circles around each point on the graph. The circles start with a radius of 0, but they grow larger over time. Specifically, when the time exceeds the waiting time at a certain polling place, the circle will begin to expand. As a result, places with shorter wait times will have larger circles—they start to grow first—and places with longer wait times will have smaller circles.
Some circles will eventually touch each other. When this happens, draw a straight line between the points at their centers. If there are multiple overlapping circles, connect all those points into “simple shapes,” which is just a general term meaning shapes like triangles (2 simplex) and tetrahedra (3 simplex).
Courtesy of Merrill Sherman/Quanta Magazine
