# The Gerrymandering Jumble: Map Projections Permute Districts' Compactness Scores

## TL;DR

This paper demonstrates mathematically and empirically that common map projections can alter the perceived compactness rankings of electoral districts, impacting fairness assessments in redistricting.

## Contribution

It proves that all sphere-to-plane map projections can reverse compactness score orderings and shows the significant impact of projections on Reock score rankings.

## Key findings

- Map projections can invert compactness score orderings.
- The Cartesian latitude-longitude projection significantly affects Reock scores.
- Mathematical proof applies to multiple compactness measures.

## Abstract

In political redistricting, the compactness of a district is used as a quantitative proxy for its fairness. Several well-established, yet competing, notions of geographic compactness are commonly used to evaluate the shapes of regions, including the Polsby-Popper score, the convex hull score, and the Reock score, and these scores are used to compare two or more districts or plans. In this paper, we prove mathematically that any map projection from the sphere to the plane reverses the ordering of the scores of some pair of regions for all three of these scores. Empirically, we demonstrate that the effect of using the Cartesian latitude-longitude projection on the order of Reock scores is quite dramatic.