Table 1 presents the indicators’ values for each of the cities studied. We find that Η o and Η w are very strongly correlated (Pearson product-moment correlation coefficient r > 0.99, p < 0.001) and thus provide essentially redundant statistical information about these networks. Therefore, the remainder of these findings focus on Η o unless otherwise explicitly stated. Three American cities (Chicago, Miami, and Minneapolis) have the lowest orientation entropies of all the cities studied, indicating that their street networks are the most ordered. In fact, all 16 cities with the lowest entropies are in the US and Canada. Outside of the US/Canada, Mogadishu, Kyoto, and Melbourne have the lowest orientation entropies. Surprisingly, the city with the highest entropy, Charlotte, is also in the US. São Paulo and Rome immediately follow it as the next highest cities. Chicago, the most ordered city, has a φ of 0.90, while Charlotte, the most disordered, has a φ of 0.002. Recall that a φ of 0 indicates a uniform distribution of streets in every direction and a φ of 1 indicates a single perfectly-ordered grid. Charlotte’s and São Paulo’s street orientations are nearly perfectly disordered.
Table 1 Resulting indicators for the 100 study sites Full size table
Venice, Mogadishu, Helsinki, Jerusalem, and Casablanca have the shortest median street segment lengths (indicating fine-grained networks) while Kiev, Moscow, Pyongyang, Beijing, and Shanghai have the longest (indicating coarse-grained networks). Due to their straight gridded streets, Buenos Aires, Detroit, and Chicago have the least circuitous networks (only 1.1%–1.6% more circuitous than straight-line distances), while Caracas, Hong Kong, and Sarajevo have the most circuitous networks (13.3%–14.8% more circuitous than straight-line distances) due largely to topography. Helsinki and Bangkok have the lowest average node degrees, each with fewer than 2.4 streets per node. Buenos Aires and Manhattan have the greatest average node degrees, both over 3.5 streets per node. Buenos Aires and Manhattan similarly have the largest proportions of four-way intersections and the smallest proportions of dead-end nodes.
Figure 1 and Table 2 aggregate these results by world region (though note that the regional aggregation sample sizes are relatively small and thus the usual caveats apply). On average, the US/Canadian cities exhibit the lowest street orientation entropy, circuity, and proportions of dead-ends as well as the highest median street segment lengths, average node degrees, and proportions of four-way intersections. They are also by far the most grid-like in terms of φ. On average, the European cities exhibit the highest street orientation entropy and proportion of dead-ends as well as the lowest average node degrees. They are the least grid-like in terms of φ.
Fig. 1 Probability densities of cities’ φ, Η o , and Η w , by region, estimated with kernel density estimation. The area under each curve equals 1 Full size image
Table 2 Mean values of indicators aggregated by world region Full size table
To illustrate the geography of these order/entropy trends, Fig. 2 maps the 100 study sites by φ terciles. As expected, most of the sites in the US and Canada fall in the highest tercile (i.e., they have low entropy and highly-ordered, grid-like street orientations), but the notable exceptions of high-entropy Charlotte, Boston, and Pittsburgh fall in the lowest tercile. Most of the sites in Europe fall in the lowest tercile (i.e., they have high entropy and disordered street orientations). Most of the sites across the Middle East and South Asia fall in the middle tercile.
Fig. 2 Map of study sites in terciles of orientation-order, φ Full size image
To better visualize spatial order and entropy, we plot polar histograms of each city’s street orientations. Each polar histogram contains 36 bins, matching the description in the methods section. Each histogram bar’s direction represents the compass bearings of the streets (in that histogram bin) and its length represents the relative frequency of streets with those bearings. The two examples in Fig. 3 demonstrate this. On the left, Manhattan’s 29° angled grid originates from the New York Commissioners’ Plan of 1811, which laid out its iconic 800-ft × 200-ft blocks (Ballon 2012; Koeppel 2015). Broadway weaves diagonally across it, revealing the path dependence of the old Wickquasgeck Trail’s vestiges, by which Native Americans traversed the island long before the first Dutch colonists arrived (Holloway 2013). On the right, Boston features a grid in some neighborhoods like the Back Bay and South Boston, but they tend to not align with one another, resulting in the polar histogram’s jumble of competing orientations. Furthermore, the grids are not ubiquitous and Boston’s other streets wind in various directions, resulting from its age (old by American standards), terrain (relatively hilly), and historical annexation of various independent towns with their own pre-existing street networks.
Fig. 3 Street networks and corresponding polar histograms for Manhattan and Boston Full size image
Figures 4 and 5 visualize each city’s street orientations as a polar histogram. Figure 4 presents them alphabetically to correspond with Table 1 while Fig. 5 presents them in descending order of φ values to better illustrate the connection between entropy, griddedness, and statistical dispersion. The plots exhibit perfect 180° rotational symmetry and, typically, approximate 90° rotational symmetry as well. About half of these cities (49%) have an at least approximate north-south-east-west orientation trend (i.e., 0°-90°-180°-270° are their most common four street bearing bins). Another 14% have the adjacent orientations (i.e., 10°-100°-190°-280° or 80°-170°-260°-350°) as their most common. Thus, even cities without a strong grid orientation often still demonstrate an overall tendency favoring north-south-east-west orientation (e.g., as seen in Berlin, Hanoi, Istanbul, and Jerusalem).
Fig. 4 Polar histograms of 100 world cities’ street orientations, sorted alphabetically corresponding with Table 1 Full size image
Fig. 5 Polar histograms from Fig. 4, resorted by descending φ from most to least grid-like (equivalent to least to greatest entropy) Full size image
Straightforward orthogonal grids can be seen in the histograms of Chicago, Miami, and others. Detroit presents an interesting case, as it primarily comprises two separate orthogonal grids, one a slight rotation of the other. While Seattle’s histogram looks fairly grid-like, it is not fully so: most of Seattle is indeed on a north-south-east-west grid, but its downtown rotates by both 32° and 49° (Speidel 1967). Accordingly, there are observations in all of its bins and its Η o = 2.54 and φ = 0.72, whereas a perfect grid would have Η o = 1.39 and φ = 1. Thus, it is about 72% of the way between perfect disorder and a single perfect grid. However, its rotated downtown comprises a relatively small number of streets such that the rest of the city’s much larger volume swamps the histogram’s relative frequencies. The same effects are true of similar cites, such as Denver and Minneapolis, that have downtown grids at an offset from the rest of the city (Goodstein 1994). If an entire city is on a grid except for one relatively small district, the primary grid tends to overwhelm the fewer offset streets (cf. Detroit, with its two distinct and more evenly-sized separate grids).
Figures 4 and 5 put Chicago’s low entropy and Charlotte’s high entropy in perspective. Of these 100 cities, Chicago exhibits the closest approximation of a single perfect grid with the majority of its streets falling into just four bins centered on 0°, 90°, 180°, and 270°. Its φ = 0.90, suggesting it is 90% of the way between perfect disorder and a single perfect grid, somewhat remarkable for such a large city. Most American cities’ polar histograms similarly tend to cluster in at least a rough, approximate way. Charlotte, Rome, and São Paulo, meanwhile, have nearly uniform distributions of street orientations around the compass. Rather than one or two primary orthogonal grids organizing city circulation, their streets run more evenly in every direction.
As discussed earlier, orientation entropy and weighted orientation entropy are strongly correlated. Additionally, φ moderately and negatively correlates with average circuity (r(φ, ς) = − 0.432, p < 0.001) and the proportion of dead-ends (r(φ, P de ) = − 0.376, p < 0.001), and moderately and positively correlates with the average node degree (r(φ, k̅) = 0.518, p < 0.001) and proportion of four-way intersections (r(φ, P 4w ) = 0.634, p < 0.001). As hypothesized, cities with more grid-like street orientations tend to also have more streets per node, more four-way junctions, fewer winding street patterns, and fewer dead-ends. Besides these relationships, φ also has a weak but significant correlation with median street segment length (r(φ, ĩ) = 0.27, p < 0.01), concurring with previous findings examining the UK alone (Gudmundsson and Mohajeri 2013). Average circuity moderately strongly and negatively correlates with the average node degree (r(ς, k̅) = − 0.672, p < 0.001) and the proportion of four-way intersections (r(ς, P 4w ) = − 0.689, p < 0.001). Cities with more winding street patterns tend to have fewer streets per node and fewer grid-like four-way junctions.
Figure 6 presents the dendrogram obtained from the cluster analysis, allowing us to systematically explore cities that are more- or less-similar to each other. The dendrogram’s structure suggests three high-level superclusters of cities, but for further analysis, we cut its tree at an intermediate level (eight clusters) for better external validity and more nuanced insight into those larger structures. To visualize these clusters another way, we map their four-dimensional feature space to two dimensions using t-SNE, a manifold learning approach for nonlinear dimensionality reduction that is well-suited for embedding higher-dimensional data in a plane for visualization (van der Maaten and Hinton 2008). Figure 7 scatterplots the cities in these two dimensions: the t-SNE projection preserves their cluster structure relatively well despite inherent information loss, but, given the global density-equalizing nature of the algorithm, the relative distances within and between clusters are not preserved in the embedding and should not be interpreted otherwise.
Fig. 6 Cluster analysis dendrogram. Cluster colors correspond to Fig. 7 Full size image
Fig. 7 Scatterplot of cities in two dimensions via t-SNE. Cluster colors correspond to Fig. 6. Triangles represent US/Canadian cities and circles represent other cities Full size image
Most of the North American cities lie near each other in three adjacent clusters (red, orange, and blue), which contain grid-like—and almost exclusively North American—cities. The orange cluster represents relatively dense, gridded cities like Chicago, Portland, Vancouver, and Manhattan. The blue cluster contains less-perfectly gridded US cities, typified by San Francisco and Washington (plus, interestingly, Buenos Aires). The red cluster represents sprawling but relatively low-entropy cities like Los Angeles, Phoenix, and Las Vegas. Sprawling, high-entropy Charlotte is in a separate cluster (alongside Honolulu) dominated by cities that developed at least in part under the auspices of twentieth century communism, including Moscow, Kiev, Warsaw, Prague, Berlin, Kabul, Pyongyang, and Ulaanbaatar. Beijing and Shanghai are alone in their own cluster, more dissimilar from the other study sites. The dark gray cluster comprises the three cities with the most circuitous networks: Caracas, Hong Kong, and Sarajevo. While the US cities tend to group together in the red, orange, and blue clusters, the other world regions’ cities tend to distribute more evenly across the green, purple, and light gray clusters.