Geometry of Gerrymandering
Learning Objectives
Understand what gerrymandering is and how it is used
Evaluate various compactness scores that measure gerrymandering (Polsby-Popper, Schwartzberg, Reock, Convex Hull, Length-Width, X-Symmetry)
Understand examples and implications of gerrymandering in real life
Common Core Standards and Mathematical Practices
7th Grade: Geometry
7.G.A.1: Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.A.2: Draw (freehand, with ruler and protractor, and with technology) two-dimensional geometric shapes with given conditions.
7th Grade: Ratios and Proportional Relationships
7 RP.A.1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units.
High School Geometry: Modeling with Mathematics
HSG.MG.A.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
HSG.MG.A.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Mathematical Practices
MP.2 Reason abstractly and quantitatively.
MP.4 Modeling with Mathematics
Vocabulary
Gerrymandering: The practice of redrawing electoral districts to gain an electoral advantage for a political party
Compactness: How "reasonably shaped" a district is
Compactness score: A measurement used to evaluate the compactness of a district; can be used to determine whether a district is gerrymandered
Voting Districts
States are divided into voting districts, with one U.S. Representative per district, to represent groups of people with similar political interests, priorities, and goals.
These voting districts sometimes have odd shapes (see Figures 1 and 2).
Though these districts are not uniform, they group together people who share similar concerns—Hispanics in Chicago and coastal dwellers in California—so it makes sense for them to share a U.S. Representative.
Figure 1. Chicago (Hispanic community)
Figure 2. California (coastal community)
Gerrymandering was named after Senator Elbridge Gerry from Massachusetts.
In 1812, he drew the above district that looked like a salamander, which was dubbed a "gerrymander."
What Is Gerrymandering?
After every U.S. Census, changes in population are considered and districts are reevaluated and redrawn by their states to reflect new demographics.
The power of states to redrawn districts can be abused. Using the redrawing of district maps for political purposes is called gerrymandering.
The redrawing of districts by people already in office is often criticized as representatives choosing their voters, instead of voters choosing who represents them.
But remember: as shown above, districts that look odd are not always drawn with political motives.
What Are Compactness Scores?
Compactness scores are methods of quantifying if and how much a district is gerrymandered. They are vital to proving the occurrence of gerrymandering, especially in a court of law, where a judge will look for concrete evidence of gerrymandering. Different compactness scores use different mathematical approaches to draw conclusions about gerrymandering.
Polsby-Popper Compactness Score
The Polsby-Popper score is the ratio of the area of the district to the area of a circle which has a circumference equal to the perimeter of the original district.
Polsby-Popper score = 4π * (area of district / perimeter of district ²)
PP(S) is always between 0 and 1 because of the Isoperimetric Inequality. A score closer to 0 means that the area is relatively small when compared to the perimeter that encloses it. This can be thought of as stretching the perimeter into a circle and noticing that the area of the circle is much greater than the area of the region. When looking at districts, this indicates that gerrymandering might have taken place.
Sample Problems
Sample Problem: Rectangle
Say we have a district, Rectangle. This district has a perimeter of 14 feet and an area of 12 squared feet. What would the Polsby-Popper score be?
Answer: Rectangle Problem
Let A = area and P = perimeter
PP(S)= 4pi(A/P^2) = 4pi(12/14^2) = 0.76937
Given what we know about Polsby-Popper scores, what does this indicate about District Rectangle?
We know that if there is too much perimeter for the area, Polsby-Popper is closer to 0, and if there is a small amount of perimeter for the area, Polsby-Popper is closer to 1.
In this example, we can conclude that there is a small amount of perimeter for the area, because .8 is closer to 1 than 0. This district is not gerrymandered!
Sample Problem: Triangle
Say we have a district, Triangle. This district has a perimeter of 20 feet and an area of 22.5 squared feet. What would the Polsby-Popper score be?
Answer: Triangle Problem
Let A = area and P = perimeter
PP(S)= 4pi(A/P^2) = 4pi(22.5/20^2) = 0.70686
Given what we know about Polsby-Popper scores, what does this indicate about District Triangle?
We know that if there is too much perimeter for the area, Polsby-Popper is closer to 0, and if there is a small amount of perimeter for the area, Polsby-Popper is closer to 1.
In this example, we can conclude that there is a small amount of perimeter for the area, because .7 is closer to 1 than 0. This district is not gerrymandered!
Sample Problem: E
Say we have a district, E. This district has a perimeter of 414 centimeters and an area of 2,590 squared centimeters. What would the Polsby-Popper score be? How does it compare to the Polsby-Popper score of a triangle?
Perimeter of Figure:
= 12 + 50 + 14 + 50 + 14 + 25 + 14 + 25 + 14 + 50 + 14 + 50 + 12 + 70
= 414 cm
Area of Figure:
= Area of A + Area of B + Area of C + Area of D
= (70 × 12) + (50 × 14) + (25 × 14) + (50 × 14)
= 840 + 700 + 350 + 700
= 2590 cm2
Answer: E Problem
Let A = area and P = perimeter
PP(S)= 4pi(A/P^2) = 4pi(2590/414^2) = 0.18989
Given what we know about Polsby-Popper scores, what does this indicate about District Star?
We know that if there is too much perimeter for the area, Polsby-Popper is closer to 0, and if there is a small amount of perimeter for the area, Polsby-Popper is closer to 1.
In this example, we can conclude that there is a large amount of perimeter for the area, because .2 is closer to 0 than 1. This district might be gerrymandered!
Sample Problem: Star
Say we have a district, Star. This district has a perimeter of 25 feet and an area of 13.3 squared feet. What would the Polsby-Popper score be?
Answer: Star Problem
Let A = area and P = perimeter
PP(S)= 4pi(A/P^2) = 4pi(13.3/25^2) = 0.26741
Given what we know about Polsby-Popper scores, what does this indicate about District Star?
We know that if there is too much perimeter for the area, Polsby-Popper is closer to 0, and if there is a small amount of perimeter for the area, Polsby-Popper is closer to 1.
In this example, we can conclude that there is a large amount of perimeter for the area, because .3 is closer to 0 than 1. This district might be gerrymandered!
Polsby-Popper Score Visuals
Below are different shapes and their Polsby-Popper scores. The circle is the most "compact," which loosely means that it has the most "reasonable" shape, with the Polsby-Popper score of 1. The less compact the shape is, the closer to zero the score.
Four regions, each with the same perimeter, are shown below from left to right in order of increasing area. The region with the largest possible area relative to the fixed perimeter—the circle—is deemed the most compact by Polsby-Popper scoring.
Gerrymandering in Court Cases
Pennsylvania
The Polsby-Popper score is widely used in court cases as the dominant compactness score. For example, in the PA-07 congressional district (infamous for being shaped like Goofy kicking Donald Duck), the Polsby-Popper score was cited numerous times in the court documents—and the PA-07 district was court-ordered to be redrawn.
Florida
In 2002, a Florida mayor sued then-Governor Jeb Bush for the state's new districts, which he argued were racially gerrymandered.
Dick Engstrom, a political science professor, explained the significance of compactness scores in measuring gerrymandering:
"'Compactness,' unlike contiguity, is a continuous concept that concerns the geographical shapes of districts. There is no bright line test that determines whether a district is or is not compact, but districts may be considered more or less compact. While numerous quantitative measures of compactness have been proposed for this purpose, the two measures that are now referenced the most are a dispersion measure known as the Reock measure and a perimeter measure known as the Polsby-Popper measure."
Discussion Question Checkpoint:
What are some gerrymandering court cases in your state? How can mathematics help solve these cases?
Other Compactness Scores
Federal guidelines state that districts should be compact. The Polsby-Popper score is used most widely, but compactness is also measured in other ways.
Schwartzberg
The Schwartzberg score is the ratio of the perimeter of a district to the circumference of a circle which has the same area as the district.
Schwartzberg score = perimeter of district / circumference of circle
Reock
The Reock score is the ratio of the area of a district to the area of the smallest possible enclosing, or bounding, circle surrounding the district.
Reock score = area of district / area of smallest bounding ciricle
X- Symmetry
Flip a district over a horizontal line that runs through the middle of the district. The x-symmetry score is the ratio of the overlapping area of the original district and the flipped district to the total area of the whole district.
X-symmetry score = overlapping area / total area of district
Convex Hull
The convex hull score is the ratio of an area of a district to the area of the smallest shape containing the district in which any 2 points can be connected by a line segment contained in the shape (namely, the convex hull of the district).
Convex Hull score = area of district / area of convex hull
Length - Width
The length-width score is the ratio of the length to the width of the smallest possible bounding rectangle containing the district.
Length-width score = length of smallest bounding rectangle/ width of smallest bounding rectangle
Discussion Questions Checkpoint:
What can you find out about the compactness scores for some of the districts in your state?
Compactness Scores Study Guide
Downloadable version here.
Discussion Question Checkpoint:
After exploring the different ways to quantify gerrymandering, how gerrymandered do you think your state is? Your congressional district? Do you think this is politically motivated or strictly for districting purposes?
Further Resources
Other Materials & Real Life Applications
Washington Post: How Gerrymandered Is Your Congressional District?
Compactness scores by state and which are most gerrymandered
Solving the gerrymandering problem with graph theory (Tufts University): Metric Geometry and Gerrymandering Group (MGGG)
Solving the gerrymandering problem with independent commissions: Alaska, Arizona, Arkansas, California, Colorado, Hawaii, Idaho, Iowa, Michigan, Missouri, Montana, New Jersey, New York, Ohio, Pennsylvania, Utah, Vermont, and Washington use independent redistricting commissions.
Federal gerrymandering legislation: For The People Act (H.R. 1/S 1), 2021
Videos: Measuring Gerrymandering with Polsby-Popper and Gerrymandering, explained
Kiera Barstad | Bryn Flanigan | Olivia LaRoche | Macy Lipkin