After every new US census, states have to redraw their congressional districts to divide up their populations fairly. But in practice, these districts don’t always end up equal: Federal judges recently ordered Wisconsin lawmakers to redraw maps of the state’s legislative districts, after finding the districts had been shaped to favor Republican candidates. Allegations of gerrymandering are also playing out in states like Texas and North Carolina. So what does a gerrymandered district even look like on a map? More like a carved-out jigsaw piece than a rounded blot, as it turns out. But as Tufts University mathematician Moon Duchin explains, gerrymandering can be difficult to prove, even when something about a district’s shape seems fishy. “We’ve had justices saying that, ‘We know a bizarre, irrational shape when we see it, but we don’t know what precisely should the threshold be which makes a shape too tortured, or irregular, or unreasonable,’” she says. (Take a closer look at district shapes across the US.)
She says the problem courts face often boils down to a single word that defies simple definition. “The word ‘compactness’ appears a lot — it appears in many state constitutions,” she says. “It’s not in the US Constitution, but it’s become a traditional districting principle, so courts look to see if the shape is ‘compact.’ Well, they would, except the problem is, nobody knows what it means.”
In the end, she says, there are “dozens of competing possible definitions” for compactness. “And that cacophony of definitions has been one of the problems.”
But complex shapes are Duchin’s area of expertise — she’s a geometer. She’s also the founder of a working group, the Metric Geometry and Gerrymandering Group, which is planning to train mathematicians around the country to act as expert witnesses in gerrymandering cases.