A Continuous Transformation Useful for Districting

Abstract: Imagine that one could stretch a geographical map so that areas with many people would appear large, and areas with few people would appear small. If such a map could be made one would expect all voting districts to be the same size for they should contain equal numbers of people. Drawing on such maps should simplify the process of creating district boundaries. The construction of maps of the requisite type is shown to require the simultaneous solution of a pair of nonlinear partial differential equations, for which an iterative computer solution procedure has been devised. An experimental attempt to district using this method is described. Suppose that one could stretch a geographical map so that areas containing many people would appear large, and areas containing few people would appear small. On a rubber map, for example, every person might be represented by an inked dot. We now imagine the rubber sheet to be stretched so that all the dots are at an equal distance from each other (see Ruston 1971). If such a map could be constructed, then all perfect political districts should be the same size, for they should contain equal numbers of people. Alternately, one might wish to construct district boundaries by drawing them as hexagons on such a map. These general notions are made more precise and given mathematical definition in the paragraphs that follow. The political problem of districting is related to a classical theoretical problem in the field of geography. The location-theoretic problem of positioning