Similar to countless natural phenomena, cities have inherent orders that can be properly captured and expressed through a complex analysis of their components. Using Geographic Information Systems (GIS), this work offers a ring-buffer fractal approach to analyze the spatial characteristics of the components of an urban system. This approach was applied to road length, number of intersections, population+employment, and building gross floor area for the city of Chicago. The complex nature of these four components manifested itself in power-law relationships and represented by their fractal dimensions. Results showed that road length and number of intersections were closely related, albeit their fractal patterns followed slightly different trends. Additionally, population+employment and building gross floor area are significantly similar and one can explain the other. Moreover, the method developed in this study was able to identify the boundary of the old city of Chicago, highlighting its ability to capture hidden characteristics of an urban system. The proposed method could further be used to correlate complex properties of urban transportation systems to other relevant measures, including connectivity, accessibility, and mobility to name a few.