3.1. Basic analysis

The results in the previous section are intriguing but not very satisfying, because they are based on a radial city model in which travel distance to radial streets or arteries is ignored and on a grid-city model with no commuting arteries. This section develops similar across-model comparisons with more realistic models. These models incorporate commuting arteries, which were introduced by [5] and which appear in [4, 9, 11, 12], and [13].

This paper compares three types of cities with commuting arteries. The first, which I call “Circle City,” builds on the transportation network and associated urban model in the classic article by Anas and Moses [9]. This network consists of circular streets and commuting arteries, which could be either highways or commuter rail. This type of network is also considered by [12] and [13].

As shown in Fig 1, this network is essentially the design for Black Rock City, although in that case the “arteries” are not distinguished by relatively high commuting speeds [14]. (Indeed, the speed limit for cars in Black Rock City on streets and “arteries” is only 5 MPH [1].) Black Rock City does have a center, dominated by a large statue called The Man, where numerous galleries and activities attract the city’s inhabitants. However, the “market” for land in Black Rock City is unusual. “In the early 2000s theme camps—elaborate enclaves built by groups of longtime Burners [i.e. Burning Man Festival participants]—filled the innermost of the inhabited circles, with the best views of the Man. A 2005 rezoning spread them out along the radial streets” [15]. Both of these allocation methods hint at the logic of an urban model, as groups that have invested the most are placed in the sites with the best access to the center. Nevertheless, the analysis in this paper is intended to shed light on cities with the street network in Fig 1, not on Black Rock City itself.

[9] uses this setup to study the impact of mode choice on household location. This step is accomplished by adding the equivalent of Ray City, that is, a second transportation mode using unspecified secondary streets that allow commuters to travel radially to the CBD, at least as a first approximation. [9] then determines the boundaries between the residential areas in which commuters use the arteries and those in which they use the secondary streets. To facilitate comparisons across cities with different transportation networks, this paper focuses on a “Circle City” in which every commuter uses a radial artery. I start with a transportation network that has with four equally spaced radial arteries originating in the CBD. This is case b in Anas and Moses. The impact of additional arteries is considered in Section 3.3.

The iso-cost line for the Anas/Moses transportation network that goes through a point on an artery u miles from CBD is defined by: (2) In this equation, t_a is per-mile transportation costs along the artery, t_s is the comparable cost along a circular street, ρ is distance along a circular street, and θ is this distance expressed in radians. This equation identifies locations where the cost of traveling along a circular street to reach a commuting artery at location uʹ equals the cost of traveling along the artery from u to uʹ.

These concepts are illustrated in Fig 2, which depicts the positive quadrant of a city with this Anas/Moses transportation system. The axes in this figure correspond to two of the city’s four arteries. Note that this city is characterized by a commuting-shed boundary; households living above this boundary commute to the vertical artery and households living below it commute to the horizontal artery. In addition, the circular streets are incomplete at many distances from the CBD, as illustrated by the narrowing of the illustrative streets in Fig 2, because, in this model and others, the residential area extends the farthest directly along the paths of the relatively high-speed arteries. In this figure, commuting costs on the artery are assumed to be half as large as commuting costs along the circular streets.

The first step in understanding Circle City is to derive the shape and length of an iso-cost line, that is, a set of locations with equal transportation costs. A segment of the iso-cost line is defined by the distance from point A to point B in Fig 2. The length of the iso-cost line that intersects the vertical artery u miles from the CBD, I_Circle{u}, equals this integral evaluated from the shed boundary to the y-axis. With four arteries, this integral must be multiplied by eight, because this formula applies to one of eight segments that make up the city. This integral is spelled out in the S1 Appendix.

Arteries also have been added to urban models with a street grid by [8, 11, 13], and [16]. This paper focuses on arteries that go to the CBD. Other arrangements are possible, of course. See the plan for Washington, D.C. [17] and models of arteries that do not go to the CBD in [5] and [8]. Consider first a transportation network with a street grid that is lined up with four commuting arteries to the CBD. See Fig 3. I call this type of city “Grid City.” Assuming that the cost of travel is the same on both vertical and horizontal streets, the iso-cost line through u for this transportation system in the positive quadrant is defined by:

(3) where and x and y are coordinates. Hence an iso-cost line is a straight line with slope . Moreover, the shed boundary is defined by y = x. These concepts are illustrated in Fig 3. The area of the city is defined by the triangle [(0,0), (0,u), (x*,y*)] when u = . The length of the outer iso-cost line is the hypotenuse of this triangle. In both cases the result must multiplied by the number of segments (eight in this case). The S1 Appendix derives these results and proves that I_Grid{u} = (ϕ_Grid)u, where ϕ_Grid is a constant.

Fig 4 compares the iso-cost lines for Circle City and Grid City. The iso-cost line segments in Circle City are curved, whereas the iso-cost line in Grid City is a series of straight-line segments. The formula for I_Circle{u}, which is derived in the S1 Appendix, is a complicated function of u, whereas I_Grid{u} = (ϕ_Grid)u, where ϕ_Grid is a constant. Numerical simulation reveals, however, that I_Circle{u} closely approximates this form; that is, I_Circle{u} ≈ (ϕ_Circle)u, With four arteries and = 0.5, for example, (ϕ_Circle) exceeds ϕ_Grid) by close to 8 percent for all u. As discussed below, this property of the iso-cost lines is useful for determining city populations.

Another possibility, also considered by Yinger [8], is that the commuting arteries are not lined up with the grid. The plan for Washington, D.C., which is largely based on the 1791 plan by Pierre Charles L’Enfant (see [17]), defines the U.S. Capitol as the city center and the Mall as one of the “arteries” emanating from this center, most of which are not aligned with the street grid. Many arteries in this plan go to other centers, of course, but only one center is considered here. This arrangement leads us to consider “Diag City,” which has arteries at a forty-five-degree angle from the grid. See Fig 5, in which the diagonal arteries are treated as the (tilted) x and y axes. One segment of Diag City has the area of the triangle [(0, 0), (0, u), (x*, y*)] evaluated at u = . Moreover, the length of an iso-cost segment is the hypotenuse of the triangle [(0, y*), (0, ), (x*, y*)], The formulas for these quantities and for I_Diag{u} = (ϕ_Diag)u are derived in the S1 Appendix.

One key feature of Diag City is that its shape could be a multi-pointed star, as in Fig 5, or a regular polygon. [8] shows that with four arteries an octagon shape arises if . The S1 Appendix derives a more general formula, which indicates a decline in the minimum that leads to a polygon as the number of arteries goes up. With sixteen arteries, for example, the polygon shape arises whenever . Thus, a polygon shape is more likely to arise in cities with more arteries or with relatively low-cost travel on streets, all else equal.

The objective of this paper is to compare the areas, populations, and average population densities of Circle City, Grid City, and Diag City. The areas are the spaces on a map enclosed by the outermost iso-cost lines, that is, the ones associated with . The relevant formulas are in the S1 Appendix. The calculations in this paper set t_a = 1.5 and alter by altering the value of t_s. Changing t_a obviously would alter the characteristics of a city, but it would have little or no impact on inter-city comparisons.

The population of a city can be determined using a theorem in Yinger [8], namely that Eq (1) determines city population for a city with any road network, so long as the land supply at location u is proportional to u, that is, if I{u} = ϕ^*u, where ϕ^* is a constant. As shown earlier, this formulation applies to the three cities in the paper. The application of this theorem to these cities must reflect the balance between two factors. First, lowering usually increases the length of a city’s iso-cost lines, which leads to an increase in population. The second factor, which I call the “squish” factor, is that lowering reduces a city’s area and thereby reduces its population. Formulas to account for these two factors are described in the S1 Appendix. Average population density is, of course, the ratio of total population to total area.

3.2. Comparisons that account for relative commuting costs on different routes

The relative commuting costs along arteries and streets may vary considerably across cities. This section shows how changes in this relative speed affect urban form and how they affect a comparison of the three city types in this paper.

Consider first the case of area. The first four columns of Table 1 and the left-most points in Fig 6 describe cities with four arteries. As increases, that is, as the cost of travel on streets moves closer to the cost of travel on the artery, the area of a city increases regardless of its transportation network design. An increase in represents an increase in transportation speed on streets (equivalent here to a decrease in transportation costs), which encourages households to locate farther away from arteries and thereby to increase the physical size of the city. These four columns reveal that Grid City has the largest area among the three cities when is small, Grid City and Diag City have the same area when = 0.414, and Diag City has the largest area at higher values of . Circle City does not have the largest area in any of these columns.

A different pattern emerges with population. With four arteries, Diag City has the largest population when travel on streets is relatively fast and the city takes the shape of an octagon, but it has the smallest population when street travel is relatively slow. Circle City and Grid City have similar populations, but Circle City’s population is higher regardless of the value of . These results also appear in Fig 7, where the four-artery case appears at the left edge of the graph.

These results lead to clear statements about average population density with four arteries. Circle City has the highest density, Grid City has density only slightly lower than that of Circle City, and Diag City has the lowest density. Moreover, the densities of the two cities with grids do not vary with the relative cost of commuting on arteries versus streets.

Finally, the changes in needed to bring all cities up to the same population are not large. The largest change in the first four columns of Table 1 is the change required to bring the population of Diag City up to the population of Circle City when = 0.414. In this case the value of in Diag City would have to increase by 1.22 miles or 1.22/30 = 4.1 percent.

3.3. Comparisons that account for the number of arteries

A second key dimension for analyzing these three cities is the number of arteries. Models with more than four arteries are discussed by [8, 9], and [13]. Actual cities are unlikely to have only four arteries, of course. The portion of Washington, D.C. around the U.S. Capitol, for example, has ten arteries, including the Mall. The 2019 design for Black Rock City contained seventeen arteries in two-thirds of a circle [14].

Because of the symmetry provided by circular streets, the addition of arteries to Circle City is straightforward. In the case of sixteen arteries, for example, the city can be divided into thirty-two identical segments, and the maximum distance between arteries at the outer edge of the city is 11.8 miles, compared to 47.1 miles with four arteries. Because a city extends farther along high-speed arteries than along streets, adding arteries leads to “suburbanization,” that is, to an outward shift in the distribution of a city’s population by distance from the CBD. This shift takes the form of more locations that extend out to along arteries and smaller indentations in the city boundary between arteries.

This link between arterial highways and an outward shift in population is not quite the same thing as the suburbanization studied by [13] and [18]. This analysis is based on closed urban models, which assume that highway changes arise in all cities so that a city’s population does not change. This approach is appropriate for studying a national phenomenon. My analysis is based on open urban models, which assume that highway changes arise in only one city in a system, so that these changes may lead to inter-city migration and population change. This approach is appropriate when one city adds arteries but other cities do not.

As shown by [8], additional arteries can also be added to a city with a standard grid, but the geometry of this case is complicated. An alternative is to consider cities with grids that are oriented around arteries. This approach leads to an extension of Grid City in which each artery is surrounded by streets that are perpendicular to the artery within the artery’s commuting shed and therefore provide access to it. This extension also includes streets that are parallel to the artery but have no role in commuting to the CBD. This arrangement of the street grid continues to the edge of the artery’s commuting shed and then rotates to line up with the next artery. The number of arteries in Diag City can be increased by arranging the street grid so that streets intersect each artery at a 45^o angle. This design, which has not appeared in the literature, also requires a rotation in the grid at each commuting shed boundary.

Results for the three city types with more than four arteries are provided in Table 1 and in Figs 6 through 8. The associated derivations are presented in the S1 Appendix. The last four columns of Table 1 provide results for cities with sixteen arteries. Each figure shows how one feature of each city changes as the number of arteries changes. Results are given for = 0.25 and = 0.95. These figures present results based on a doubling of arteries from the previous case. However, the formulas in the S1 Appendix can be applied to any number of evenly spaced arteries. To maintain contact with real cities, these figures stop at thirty-two arteries, which implies a distance of 5.9 miles between arteries at the outer edge (assumed to be 30 miles) of the city.

One key difference between the last four and the first four columns of Table 1 is not surprising: at any given level of , adding arteries boosts the area of every city. With a high value for , this effect is larger for Circle City and Grid City than for Diag City, and the areas of the three cities converge as the number of arteries grows. See Fig 6. Moreover, Table 1 shows that regardless of the number of arteries, the areas of Grid City and Diag City are the same when = 0.414. In all three cities, the areas slowly converge to the maximum possible area, π(30)² = 2,827 square miles, as the number of arteries grows. Even with sixteen arteries and = 0.5, however, the areas for Circle, Grid, and Diag cities all fall below 75 percent of the area of the basic Ray City with the same value of .

Additional arteries also boost the population in all three cities, but at different rates. See the “Population” panel of Table 1 and Fig 7. When is low, additional arteries have a particularly large impact on the population of Grid City, and when is high, additional arteries have little impact on Diag City’s population. None of the cities expands far from the arteries when is low, so arteries fill in spaces that fall far toward the CBD and therefore boost population. This effect is particularly strong in Grid City, where, as shown in Fig 4, the iso-cost lines are not “squished” as far toward the CBD as are the iso-cost lines for Circle City. When is high, however, these spaces are much smaller, particularly with the polygon shape of Diag City, so adding arteries does not have as large an impact. In fact, with thirty-two arteries the area of Grid City with small is almost as large as the area of Diag City when is large.

The population density results with many arteries in Table 1 and Fig 8 reinforce the density results with four arteries. Both Grid City and Diag City have constant densities, that is, densities unaffected by or the number of arteries, with a higher density in Grid City than in Diag City. In these two cities the impact of higher or a higher number of arteries on area is exactly offset by their impact on population. (These constant densities are somewhat difficult to see in Fig 8 because the curves for high and low overlap.) The density in Circle city is not constant, however. Instead, density declines as the number of arteries increases, and this decline is greater when the value of is low. In fact, Circle City has the highest density with a high value of , regardless of the number of arteries, but its density drops below that of Grid City (but not that of Diag City) when is low and the number of arteries is above twelve.

Increasing the number of arteries not only increases the population of all three cities, it also increases the changes in that are required to bring all three cities up to the same population. When = 0.25 and the cities have sixteen arteries, for example, the longest commuting distance in Diag City would have to increase by 1.61 miles (or 5.4 percent) to bring its population up to that of Circle City. See fourth panel of Table 1. Alternatively, one can compare the size requirements needed to reach a given population with each city design. See the last two panels of Table 1. These panels show that the increase in artery length required for a population change from 100,000 to 500,000 is about 12 miles for all city types and for either 4 or 16 arteries. Moreover, adding arteries cuts the value of needed to reach a given population by considerably more when is small than when it is large. This effect is particularly striking for Diag City, which has the smallest impact of arteries on the required when is large and the largest impact when is small.