Population landscape: a geometric approach to studying spatial patterns of the US urban hierarchy

We present a geometric and graphic approach to studying spatial patterns of urban hierarchy in the US. The multiplicatively weighted Voronoi diagram is found to be effective for visualizing theoretical regions delineated by socioeconomic variables. The population landscape of the continental US demonstrates overall and stepwise patterns reflecting population, neighborhood and distance, with overwhelming influence from huge metropolitan areas. Stepwise exploration and cluster analysis of the spatial pattern reveal an urban hierarchy. Attributes and arrangement are the two important factors of urban hierarchy, with attribute having a stronger local influence and arrangement having a stronger global influence. The study also presents a variation of Zipf's law to visualize the rank-size distribution from tabular and statistical space to map space.