Geographical Compactness in Shape Assessment

Geographic shape has long been an intriguing feature of observed and defined facets of an area or region. Compactness reflects a critical element of shape with important practical and policy implications. It may suggest characteristics of urban/regional form, efficiency in trade and service provision, fairness in political representation and distributional qualities of the physical environment, among others. While there has been much study of compactness and a wealth of measures and metrics derived to reflect nuances of geographic form, there are questions that remain about their ability to characterize shape in a meaningful manner. Given this, exploration of relationships between various categories of methods for quantifying compactness is critical. Further, recent developments of, advances in and access to physics based spatial measures of compactness suggest an opportunity for better theoretical understanding. Assessment of 388 districts is carried out. Significant correlation is demonstrated between contemporary measures, opening the door for research advancements associated with the compactness of spatial shapes. This work is interesting, important, and of current relevance because compactness measures are given serious consideration in management, planning, and policy, but also are regularly relied upon in legal proceedings. Further, compactness measures continue to drive automated and semi-automated approaches in districting and redistricting, often embedded in optimization approaches.