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

The Gerrymandering Jumble: Map Projections Permute Districts' Compactness Scores