Electoral Geometry
Trinity professor, student using math to identify gerrymandering
Tuesday, May 28, 2019
cabral drawing on whiteboard for student

There’s no “silver bullet” to fight gerrymandering, the practice of manipulating electoral districts for partisan advantage.

But thanks to nonpartisan mathematics, Trinity mathematics professor E. Cabral Balreira and senior Alice Von Ende-Becker ’19 are getting pretty close.

Von Ende-Becker, a mathematics, art history and economics triple major, spent the past semester working on a research project with Balreira to examine a potential link between gerrymandering and Texas elections. The pair produced a project, “Who Chooses Who,” an examination of electoral data from 36 U.S. Congressional races in Texas in years 2012, 2014, 2016, and 2018. Using a combination of tools including programming software MATLAB, map ensembles, and concepts such as the efficiency gap, Von Ende-Becker and Balreira have already yielded results:

“There’s something interesting going on in Texas,” Balreira says. “In 2012, 2014, and 2016, where there was relatively low turnout, one party controlled most of the elections. That’s expected. But after voter turnout rose significantly in 2018, there was almost no change in representation, compared to what the numbers would project.”

This research, first launched in fall 2018, is partially funded through a grant from the Metric Geometry and Gerrymandering Group (MGGG). This nonpartisan, Boston-based initiative, led by researchers from Tufts University and MIT, aims to apply geometry and computing to U.S. redistricting, with the belief that gerrymandering of all kinds is a fundamental threat to U.S. democracy.

Through the MGGG grant, Balreira is also training to become an expert witness for future gerrymandering court cases, which involves a heavy slate of research. So, when word broke in the Trinitonian that Balreira had gotten the grant—and was looking for students to join his project—Von Ende-Becker answered the call.

“Until I saw this article in the Trinitonian, I’d never heard of math being applied to the gerrymandering problem,” Von Ende-Becker says. “That’s the moment I was excited I was a math major, because that was the moment I knew I could do something about this problem.”

Alice Von Ende-Becker and E. Cabral Balreira at whiteboard


With Balreira, Von Ende-Becker used a dizzying array of complex mathematical and geometric concepts to start sifting through Texas’ electoral data. The significance—and irony—of this type of research, Balreira adds, is that courts have typically been hesitant to use mathematics to prove gerrymandering simply because of the huge variety of mathematical techniques related to the subject: There have been more than 40 mathematical concepts introduced in U.S. gerrymandering court cases to date, just dealing with the issue of “compactness”, or the shapes of districts, alone. This makes courts hesitant to settle on any one method of proving gerrymandering’s existence in the first place.

“Some of these formulas, these methods, will give you different results when you’re talking about gerrymandering,” Balreira says.

Ultimately, Balreira and Von Ende-Becker have settled on the “efficiency gap” as a launching point for future research. This method uses a count of “wasted votes” (votes “wasted” on a winner who didn’t need them, or on a loser that could have been spent in a more competitive district) to produce a comparative rating for individual districts. The higher the efficiency gap, the more gerrymandered the district.

“This method isn’t perfect, but it’s a start,” Balreira says. “This gives you a number, a rating—the way we rate quarterbacks in football, for example—and if we’re being honest, courts are made up of people, and people like these types of ‘ratings.’”

Using this comparative data, Von Ende-Becker and Balreira compiled a “gerrymandering index” for Texas’ 2012, 2014, 2016, and 2018 elections. This index is essentially a series of box plots that group districts in order from left to right based on how much they favored one party to the other. The boxes that are in the middle represent the competitive races.. This gives viewers an easy examination of how many of Texas’ races were expected to be competitive based on statewide turnout, compared to the actual results of those elections.

If that sounds confusing, imagine learning this from scratch in one semester.

“This was challenging,” Von Ende-Becker laughs, but with a trace element of trauma that any exhausted student researcher will recognize. “There were definitely meetings where I walked out where I was stressed and concerned about whether I was going to be able to get my work done. But that’s what made this so enjoyable, were those moments when we were able to make a breakthrough, to get a result. That made this work worth it.”

Alice Von Ende-Becker at whiteboard


And while Von Ende-Becker had the math chops to handle this type of project, Balreira says the research will also have a huge effect on her interdisciplinary experience moving forward.

“We focus on interdisciplinarity at Trinity. I do geometry, I do dynamical systems, I focus on the pure math of this research. We’re looking at maps and seeing grids, blocks, statistics,” Balreira says. “But with Alice, she’s taking more than just math classes. She can think beyond the numbers to see the applications of this research. She comes with a different background, a different point of view. So if Alice learns enough math from me, and she’s taking political science classes, economics, the humanities: The student at Trinity becomes this ‘communication’ between the classical fields. And that’s where this research, at Trinity, can really move forward.”

In the fall, for example, Balreira hopes to take the numbers from Von Ende-Becker’s project and launch the next phase of his research with political science professor Katsuo Nishikawa, an expert on voter turnout.

“Through this research, Alice becomes the translation between Katsuo and me,” Balreira says. “It’s exciting to know that through Alice, we’ve created a person who can go out into the world and take on these complex problems.”

And as stressful as this gerrymandering project was for Von Ende-Becker, she’s just as excited to have made an impact on the political world from within the cozy confines of a math lab in Marrs McLean Hall.

“I never thought something like gerrymandering would overlap with math,” Von Ende-Becker says. “This is the type of opportunity—to work on something like this—I never imagined I’d get when I came to Trinity.”


The MGGG, which is funding Balreira’s training to become an expert witness, is a nonpartisan group dedicated to promoting fair electoral practices. While the group may participate in cases brought or defended by partisan entities, affiliation with this group should not be understood as aligning with a particular political party, cause, or organization.

Jeremiah Gerlach is the brand journalist for Trinity University Strategic Communications and Marketing.

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