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Transportist: Circular Cities
Do metropolitan areas become more circular as their population increases?
Geographically, a circle minimizes the distance from the edge to the center. As urban areas expand, land value and the economic cost of irregular shapes increase, prompting city-builders (governments, network builders, land developers, residents) to fill in irregularities. For example, residents may choose home locations that decrease their commuting distance while maximising land area.1 Studies on urban scaling laws indicate that larger cities become more efficient in other aspects and increasingly interconnected.2 Bigger cities also have more chances for "reversion to the mean."
However, a larger metropolitan area may encounter natural features that hinder smooth expansion, and coastlines and harbours historically have been the locations where many urban areas began. These natural features (e.g., bodies of water) may offer unique development opportunities, such as trade or picturesque views. Moreover, these natural features may have been the rationale for the city in the first place.
We should also consider reverse causality, as cities with more potential for circular growth (i.e., more developable area within a given radius from the city center) tend to grow larger due to reduced expansion costs compared to geographically constrained cities, which face higher overall transport costs.
The "coastline paradox" is also a concern, as a geographical feature, such as a coastline or metropolitan area, measured or mapped with increasing precision, would yield lengths much greater than those from lower resolution maps or images, or from census-defined boundaries in a GIS file.3 We hope that the US Census boundary definition files construction rules result in a fair comparison at a consistent spatial definition level.
This analysis uses population and urbanised area data from the 487 US Census-defined metropolitan areas in the United States in 2010, and a QGIS-developed urbanised area perimeter measure kindly provided by John Ottensmann.4
One issue with using the downloaded geographic boundary files for Urbanized Areas from the US Census is that some comprise two or more separate features that require dissolving before perimeter calculation.
To investigate this hypothesis, we explore various measurements.5
The first test uses the perimeter-to-area (P/A) ratio to measure circularity. Assuming all else is equal, a more circular city will have a P/A ratio closer to the circle's theoretical minimum.
A circle has the lowest possible P/A ratio. Recall the perimeter (circumference) of a circle is given by
and the area by
Thus, for a circle,
where P is the perimeter, A is the area, and r is the radius.
While we test the P/A ratio, note it changes with size. Perimeter is measured in units of length, and area in units of length squared. As the area enlarges, the area increases more rapidly than the perimeter (and the circumference and radius for a circle), resulting in a smaller P/A ratio. Consequently, if two urban areas have the same circularity degree but different sizes, the larger one would have a lower P/A ratio. This implies that a negative relationship between the ratio and size does not necessarily demonstrate a negative relationship between the ratio and circularity degree.6
We use population, not physical area, as the explanatory variable, but we note that population and area are typically highly correlated (with an R2 of 0.82 for US cities in this case). Thus, we also test the ratio of the radii implied by circles with the given perimeter and area to account for the natural increase in P/A as the metropolitan area grows:
The minimum possible ratio is 1.0, which would occur for a perfectly circular urban area.
We first performed a regression analysis of the natural log of the perimeter-to-area ratio (ln(P/A)) against the natural log of the 2010 urbanized area population (ln(pop10)).
Table 1 displays the regression results, and Figure 1 presents the observed and model results graphically. For each 1% increase in the population, the perimeter-to-area ratio decreases by 0.318%, strongly supporting the hypothesis that a more populous metropolitan area has a higher P/A ratio.
To account for the fact that the P/A ratio naturally decreases for circular areas, we regressed the natural log of the radii ratio (ln(rP/rA), with results shown in Table 2 and illustrated in Figure 2. For each 1% increase in population, the radii ratio increases by 0.119%, strongly refuting the hypothesis that more populous cities are more circular.
Future research should strive to examine this relationship over time, especially assessing whether this connection, which may have been historically significant, remains pertinent as cities become more dispersed following decades of decentralization, a declining share of CBD employment, increased suburb-to-suburb commuting, and the growing prominence of remote work and other remote activities.
Table 1: OLS Regression Results, Dependent Variable: ln(P/A) Coefficients Standard Err t Stat P-value Intercept -2.942 0.121 -24.2 1.86E-85 ln(Pop10) -0.318 0.01 -31.7 1.60E-120 Regression Statistics Adjusted RSq 0.675 Observations 486 Table 2: OLS Regression Results, Dependent Variable: ln(r_P/r_A) Coefficients Standard Err t Stat P-value Intercept 0.153 0.153 0.998 0.319 ln(Pop10) 0.119 0.013 9.443 0 Regression Statistics Adjusted RSq 0.153 Observations 486
Levinson, D., & El-Geneidy, A. (2009). The minimum circuity frontier and the journey to work. Regional Science and Urban Economics, 39(6), 732-738.
Ribeiro, F. L., & Rybski, D. (2023). Mathematical models to explain the origin of urban scaling laws. Physics Reports, 1012, 1-39.
Levinson, D. (2012). Network structure and city size. PloS One, 7(1), e29721.
Steinhaus, H. (1954). Length, shape and area. In Colloquium mathematicum (Vol. 3, No. 1, pp. 1-13). Polska Akademia Nauk. Instytut Matematyczny PAN.
Ottensmann uses the perimeter of Urbanized Areas, focusing on perimeter length as an irregularity measure. The irregularity index relates directly to the perimeter/area ratio, being 1 minus the ratio of 2 times the square root of the area divided by the perimeter. It is essentially an inverse of a form of the perimeter/area ratio subtracted from 1. Results for Urbanized Areas reveal somewhat higher values for the 50 largest cities.
Ottensmann, J. R. (2021). The measurement of the irregularity of urbanized areas. Available at SSRN 3838296.
Ottensman does not specifically analyze the perimeter and area, using census tract data rather than Urbanized Areas. However, a proximity index compares mean distances to the Central Business District (CBD) if the area were circular to the actual mean distances of tracts to the CBD, providing one form of hypothesis testing (a smaller ratio indicates less circularity). A very weak negative correlation exists between land area and proximity; greater land area corresponds to smaller proximity and less circularity. Though the correlations were present each year, they were consistently not statistically significant. Many cities, including some larger ones, are situated along coastlines with off-center CBDs, reducing proximity to the CBD compared to a geographic center.
Selecting an appropriate measure of circularity for urban areas is challenging. The perimeter/area ratio not only indicates circularity but also irregularity in the boundary. For instance, a pure circular boundary and a circular boundary with many zig-zags may have significantly different perimeters. One alternative is a proximity index. Another option is drawing a circle around the CBD or geographical center (which may differ greatly) with the same area as the urban area, then determining the proportion of the urban area inside the circle. However, this method depends on center placement, which can lead to other issues, particularly with more asymmetric cities.
Analyzing changes over time (whether individual cities become more circular as they grow) presents additional empirical challenges. Ottensmann created a proximity index over time using consistent census tract data for urban areas. However, extending the analysis further back is difficult due to discrepancies in the 2000 Urbanized Area shapefiles. Although manual editing could address this issue, differences in Urbanized Areas definitions from 2000 to 2010 would still make comparisons questionable, as would attempts to go earlier.
Ottensmann, John R., (2021) The Shapes of Large Urban Areas in the U.S., 1950-2010: Patterns, Causes, and Consequences. Available at SSRN 3799728 Ottensmann, J. R. (2021). Measuring the shape of urban areas. Available at SSRN 3786248.
Maceachren, Alan M. 1985. Compactness of geographic shape: comparison and evaluation of measures. Geografiska Annaler. Series B, Human Geography 67, 1:53-67.