How to measure legislative district compactness if you only know it when you see it

To deter gerrymandering, many US state constitutions require legislative districts to be geographically “compact” (and a similar requirement holds explicitly or implicitly for numerous political jurisdictions around the world). Yet, the law offers few precise definitions other than “you know it when you see it,” which effectively implies a common understanding of the concept. In contrast, academics have shown that compactness has multiple dimensions and have generated many conflicting measures. We hypothesize that both are correct — that compactness is complex and multidimensional, but a single common understanding exists across people. We develop a survey to elicit this understanding, with high reliability (in data where the standard paired comparisons approach fails). We then create a statistical model that predicts, with high accuracy, solely from the geometric features of the district, compactness evaluations by judges and public officials responsible for redistricting, among many others. We also offer compactness data from our validated measure for 20,160 state legislative and congressional districts, software to compute this measure from any district, and discussion about how to measure other concepts you only know when you see.