Polsby–Popper test

The Polsby–Popper test is a mathematical compactness measure of a shape developed to quantify the degree of gerrymandering of political districts. The method was developed by lawyers Daniel D. Polsby and Robert Popper,[1] though it had earlier been introduced in the field of paleontology by E.P. Cox.[2] The formula for calculating a district's Polsby–Popper score is ${\displaystyle PP(D)={\frac {4\pi A(D)}{P(D)^{2}}}}$, where ${\displaystyle D}$ is the district, ${\displaystyle P(D)}$ is the perimeter of the district, and ${\displaystyle A(D)}$ is the area of the district.[3] A district's Polsby–Popper score will always fall within the interval of ${\displaystyle [0,1]}$, with a score of ${\displaystyle 0}$ indicating complete lack of compactness and a score of ${\displaystyle 1}$ indicating maximal compactness.[4] Compared to other measures that use dispersion to measure gerrymandering, the Polsby–Popper test is very sensitive to both physical geography (for instance, convoluted coastal borders) and map resolution.[5] The method was chosen by Arizona's redistricting commission in 2000.[6]