Don’t Judge a District by Its Shape

Following the 2020 United States Census, each State is required to re-draw its House districts to re-ensure equal population amongst them. With the increased attention given to the dangers of gerrymandering in recent years, people are quick to point at odd district shapes as evidence of foul play.

A map of Arizona’s Congressional districts in the 2001–2011 decade. The second district is practically discontiguous.
Arizona’s 2001–2011 Congressional Districts. District 2 looks suspicious, no?

However, districts can take strange shapes for a variety of reasons, and we shouldn’t confine ourselves to a naïve notion like compactness (defined many ways, but generally maximizing area while minimizing diameter) to examine the district shapes. Indeed, compactness is a geometric property — often defined as a ratio of areas and/or perimeters — and as such should take into account the political geography of the state. As I mentioned in a previous article on geometric transformations, the assumption that every unit of land should be treated equally is one of the oldest misconceptions in the book.

Communities of interest can be strangely shaped, noncompact, and even noncontiguous. Sometimes these occur for natural reasons: a plot of population density in Phoenix shows obvious holes where South Mountain and the Mountain Preserve (Piestewa) are, and to the north, the serpentine route the 89A takes between Prescott and Flagstaff yields a curved string of settlements.

Population density heatmaps for regions of Arizona. There are holes in the population distribution in Phoenix, and a curved line of higher population between Prescott and Flagstaff.
Plots of population density of census block groups in regions of Arizona. Yellow=more dense. (Source: Census)

Odd shapes can also occur for artificial reasons: the Native American reservations and historical redlining clearly define communities in ways that made no consideration for compactness; the shapes of the Navajo and Hopi reservations in northeast Arizona are perfect examples of this. Finally, odd shapes can be made just by being the complements of nice shapes! Suburbs, being regions of moderate density that are not city centers, necessarily have messy borders.

The Navajo reservation entirely surrounds the Hopi reservation.
Map of Arizona Indian Reservations, note the shapes and locations of the Navajo and Hopi reservations (Source)

Districts represent people, not land. Our first priority when evaluating drawn lines is whether or not they are likely to yield representation that aligns with the population of the state. People deserve to have their voices heard regardless of where they live and regardless of where their ideological allies and opponents live. If proportionate outcomes can be achieved by nice-looking districts, then that’s a cherry on top, but we should not forget what’s important.

This requirement for proportionality requires active thought when designing districts. It is common for people to naively advocate for algorithmic redistricting — this is a mistake. Not only is the assumption of algorithmic neutrality extremely problematic (algorithms inherit the biases of their designer), but just because an algorithm is not intended to produce unfair maps does not mean that it will not do so! We can use the examples of Wisconsin and Connecticut to drive this point home.

Wisconsin is incredibly partisanly polarized, with a majority of Democratic votes coming from the greater Madison and Milwaukee areas, while the rest of the state has evenly distributed Democratic-leaning towns among the strongly-Republican rural areas. As such, any map that attempts to keep Madison and Milwaukee whole — which compactness prioritizes — will result in Republicans winning 6 of the 8 districts in a state that split nearly evenly in the 2020 Presidential election.

A map of Wisconsin, showing that half the population lives in Madison, Milwaukee, and assorted small towns.
The Yellow and Blue regions have nearly equal populations, yet voted opposite each other in the 2020 Presidential election, both decided by about 25 points. (Source: Districtr)

In Connecticut, however, the tables are turned. Only the most rural areas of Connecticut are Republican-leaning, causing maps drawn by both Democrats and Republicans in the state to feature a Democratic majority in all five districts, despite Republicans receiving about 40% of the statewide vote in the 2018 Senate and 2020 Presidential elections. Allowing for non-compact shapes, however, it is possible to draw a district that Republicans won in 2018.

Two maps of Connecticut, showing how district lines can be drawn (none of them are pretty).
Proposed Connecticut map on the right; its most competitive seat was about 54% Democratic in the 2018 Senate race. On the left, a sample map, where the blue district was 53% Republican in 2018. (Source: 538 and Districtr)

Allowing the political geography to dictate the district shapes is only fair to the people living in these states. Arizona broached this issue correctly in 2001, where it produced a rather hideous 2nd Congressional District (pictured at the top of this article). The story here is not unfair gerrymandering, but rather of political reality. The two major tribes in the northeast, the Navajo and the Hopi, have long not gotten along and as such were drawn into different districts — the Hopi feared that sharing a district with the much larger Navajo reservation would stifle their political voice. Under strict adherence to compactness rules accommodating this request would have been impossible, as the Hopi reservation lies fully enclosed by the Navajo reservation, but Arizona’s districts were drawn by an independent commission willing to take the political reality into account.



I’m a mathematician and strategy gamer who enjoys looking for patterns in data and investigating what those patterns mean.

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Patrick Martin

I’m a mathematician and strategy gamer who enjoys looking for patterns in data and investigating what those patterns mean.