Geometry Reveals the Tricks behind Gerrymandering

Voting districts are deliberately manipulated to favor one political party or another—a feature that played a role in the recent midterms. Geometry is a crucial element in the process of gerrymandering. Earlier this year, the Missouri Supreme Court unanimously ruled to halt a congressional district map backed by President Donald Trump from being implemented for the November elections.

The court also proposed a statewide vote on the new map's adoption. In July 2025, the Trump administration encouraged Republican leaders in Missouri to redraw the boundaries of electoral districts. The primary target was the fifth congressional district, encompassing Kansas City and typically considered a Democratic stronghold. The plan was enacted in September 2025, with the aim of using the new boundaries in the November 2026 midterms.

However, the proposal faced opposition in court. Fair district design is a complex issue that has been debated for decades, even among mathematicians, who employ sophisticated computing methods to address the challenge. Gerrymandering exploits geometry and computational techniques to detect partisan district maps. While few people realized that calculating the area of a triangle or the volume of a prism in high school could be used to influence an election outcome, geometry can indeed be a potent tool in shaping election results—especially in plurality voting systems.

Designing the perfect election system for multiple parties is theoretically impossible, even with mathematical tools. However, given that the U.S. primarily operates with two dominant parties, the shape of voting districts can significantly impact the results. If designed skillfully, a party that is losing in terms of overall votes can still secure a majority of representatives—an issue that was evident in the recent U.S. midterms.

Math plays a significant role in determining election outcomes, particularly for the 435 seats in the House of Representatives. Cleverly chosen boundaries of a congressional district can enable a party's candidate to win even if the vote count does not accurately represent the voters' sentiment. To illustrate this, imagine a state with 50 voters, 20 of whom support the blue party and 30 support the red party.

If the voters are arranged in a grid pattern, like some areas of Manhattan, with 10 north-south avenues and five east-west streets, the red voters reside on the first two avenues (the westernmost ones), while the blue voters live on the remaining three avenues. The objective is to divide the voters into five electoral districts of equal size.

One approach is to draw five vertical boundaries, resulting in two districts with only red voters and three with only blue voters. Consequently, the votes in those districts would produce three representatives from the blue party and two from the red party, accurately reflecting voters' preferences. However, if the blue party were to redraw the boundaries horizontally, all districts would have a similar composition of four red voters and six blue voters, allowing the blue party to win all five representatives.

A similar scenario occurred in New York state in 2012, where 58 percent of voters supported the Democrats, but the party secured 21 out of 27 seats—five more than if the electoral districts had been drawn fairly. State legislatures and commissions responsible for redrawing district lines often create more complex partitions. They can pack most blue voters into two districts, giving the red party a majority in the remaining three districts, where the Democrats would have a majority—despite a higher number of overall votes for the blue party.

This deliberate redrawing of districts to gain a majority advantage is known as gerrymandering, a combination of "gerry" and "salamander." The term originated from Elbridge Gerry, the governor of Massachusetts in the early 19th century, who approved oddly shaped voting districts that gave his party an advantage. Even today, in most U.S. states, legislatures redistrict every 10 years (following the release of the new census data).

Time and again, incumbent parties are accused of using redistricting to their advantage. This often manifests in unusual-shaped electoral districts, reminiscent of the salamander district in Massachusetts during the early 19th century. The term "gerrymandering" was coined by a cartoonist who noticed the resemblance between one of the districts and a salamander.

Redistricting has led to numerous legal challenges. In 1986, the U.S. Supreme Court ruled that intentional gerrymandering is illegal, but its impact on election districts has been limited. Setting rules for fair districting remains a complex task. Even mathematicians are grappling with the question and leveraging extensive computing power to tackle the problem.

The identification of gerrymandering can be observed by examining a Maryland district, as depicted above. The irregular shape may prompt suspicions of manipulation.