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How Math Can Help Detect Gerrymandering

Learn how geometry can be used to detect gerrymandering.

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How Math Can Help Detect Gerrymandering

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American elections are a tricky business. While the act of voting to elect an official may seem simple enough, the way in which votes are counted isn’t so simple…sometimes where you live matters more than who you vote for. If you live in a district that has been drawn specifically to ensure the victory of a certain political party, your vote may not matter as much as it should.

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Can Math Help Detect Gerrymandering?

This week the Supreme Court heard a case about gerrymandering in Wisconsin, and mathematicians have been watching closely. Here’s why.

Posted by NOVA l PBS on Thursday, October 5, 2017

Congressional districts are drawn based on data released by the Census Bureau at the start of a decade. Usually it’s the state legislature that conducts the process of redistricting. They’re supposed to ensure that each district has very nearly the same population and that racial minorities have an opportunity to elect a candidate of their choice.

However, political parties can try to use the process of redistricting to give themselves an advantage. When they are drawing the districts, politicians can safely assume that certain communities will vote in a predictable manner, thanks to increasingly detailed voter data. While some bias in drawing districts is legal and unavoidable, politicians can exploit this legal grey area to benefit their political party. Politicians can target specific communities and draw district lines in ways that make it harder for those communities to be represented during elections.When congressional districts are extremely manipulated with the aim to favor one party or candidate over another, this is called gerrymandering.

The two main tools of the gerrymanderer are called cracking and packing.  Cracking is when you draw the lines so that opposing voters are scattered across several districts, so that the power of their vote is watered down and it becomes harder for them to achieve representation.


Packing is when you try to contain opposing voters in as few districts as possible. By creating a small number of districts where opposing voters make up vast majorities, you free up the other districts to be won by your party.


Both of these methods can give a party an unfair advantage in elections by creating many ‘wasted’ votes for the other side — votes that have no effect on the outcome of an election. Wasted votes can be cast either for a losing candidate or for a candidate that is winning so comfortably that additional votes make little difference.

In October 2017, the Supreme Court heard the case Gill v. Whitford . The plaintiffs argued that Wisconsin’s 2012 State Assembly districts were drawn in a way that created lopsided wastage: Democrats wasted far more votes than Republicans because of how the state was broken up.

Mathematicians like Tufts University’s Moon Duchin think that besides looking at the numerical distributions, another key to detecting gerrymandering may lie in geometry. Duchin is searching for new ideas of what kinds of district shapes make for fair maps.  For instance, by drawing lines across a state based on population density, and then folding along those lines as if the state were a piece of origami, she’s exploring whether gerrymandering may be revealed by the “curvature” of the resulting shape.

It is a long-standing idea that funny-looking, sprawling regions with long, winding perimeters are hallmarks of unfair mapmaking. But the political scientists have come up with over thirty different scores that can be used to measure how bad or inefficient a shape is!

That’s where curvature might be helpful.  In pure math, there are three types of curvature: positive, zero, and negative. A sphere is an example of a shape with positive curvature: all of the curves along its surface bend in the same direction. Negative curvature, however, occurs when a shape curves in two different directions (think of a saddle).

In Duchin’s research specialty, there is a family of results saying that negative curvature produces sprawling regions with high perimeters, both classical symptoms of gerrymandering. In other words, negative curvature might be an underlying cause that creates many of the different telltale signs of a gerrymander.

The hope is that mathematics be used both to detect gerrymandered maps and to propose better ones!


Guide further class discussion with the following questions. Students may need to do research to answer some of the questions.

  • How are congressional districts formed? What are the traditional districting principles?
  • What is gerrymandering? How do the two main methods of gerrymandering work to provide people who draw district lines with an advantage during elections?
  • Does your state have any of its own additional laws for redistricting?
  • What are two different ways that mathematics might be used to study gerrymandering?
  • What are other examples of how big data has been manipulated to provide certain groups with an unfair advantage?
  • What are examples of when mathematics can be used to prevent bias?