Urban Compression Patterns: Fractals and Non-Euclidean Geometries - Inventory and Prospect

Urban growth and fractality is a topic that opens an entrance for a range of radical ideas: from the theoretical to the practical, and back again. We begin with a brief inventory of related ideas from the past, and proceed to one specific application of fractals in the non-Euclidean geometry of Manhattan space. We initialize our discussion by inventorying selected existing knowledge about fractals and urban areas, and then presenting empirical evidence about the geometry of and movement in physical urban space.Selected empirical analyses of minimum path distances between places in urban space indicate that its metric is best described by a general Minkowskian one whose parameters are between those for Manhattan and Euclidean space. Separate analyses relate these results to the fractal dimensions of the underlying physical spaces. One principal implication is that theoretical, as well as applied, ideas based upon fractals and the Manhattan distance metric should be illuminating in a variety of contexts. These specific analyses are the focus of this paper, leading a reader through analytical approaches to fractal metrics in Manhattan geometry. Consequently, they suggest metrics for evaluating urban network densities as these represent compression of human activity. Because geodesics are not unique in Manhattan geometry, that geometry offers a better fit to human activity than do Euclidean tools with their unique geodesic activities: human activity often moves along different paths to get from one place to another.Real-world evidence motivates our specific application, although an interested reader may find the subsequent "prospect" section of value in suggesting a variety of future research topics that are currently in progress. Does "network science" embrace tools such as these for network compression as it might link to urban function and form? Stay tuned for forthcoming work in Geographical Analysis.