I’ve discussed before the topic of missed connections on subway systems, both here and on City Metric. I’ve for the most part taken it for granted that on a rapid transit network, it’s important to ensure that whenever two lines intersect, they offer a transfer. This seems like common sense. The point of this post is not to argue for this principle, but to distinguish two different kinds of missed connections: city center misses, and outlying misses. Both are bad; if I had to say which is worse I’d say it’s the city center miss, but city center misses and outlying misses are bad for distinct reasons.

A useful principle is that every pair of rapid transit lines should intersect, unless one is a shuttle, or both are circumferential. If the city is so large that it has multiple circular lines at different radii (Beijing has two, and London vaguely has two as well depending on how one counts the Overground), then they shouldn’t intersect, but rapid transit networks should be radial, and every radial line should connect to every other line, with all radial-radial transfers ideally located within the center. City center misses weaken the network by making some radials not connect, or perhaps connect at an inconvenient spot. Outlying misses often permit more central transfers, and their problem is that they make it harder to transfer to the better or less crowded radial on the way to the center. London supplies a wealth of examples of the latter without the former.

What counts as a missed connection?

Fundamentally, the following picture is a missed subway connection:

The red and blue lines intersect without a transfer. Even if a few stations later there is a transfer, this is a miss. In contrast, the following picture is not a missed connection:

It might be faster for riders to transfer between the southern and western leg if there were a station at the exact physical intersection point, but as long as the next station on the red line has a transfer to the blue line it counts, even if the blue line has one (or more) stations in the middle. Washington supplies an example of this non-miss: it frustrates riders that there’s no connection between Farragut West and Farragut North, but at the next station south from the intersection on the Red Line, Metro Center, there is a transfer to the Blue and Orange Lines. London supplies another pair of examples: the Northern line and the Waterloo and City line appear to intersect the District line without a transfer, but their next station north from the physical intersection point, Bank, has an in-system transfer to Monument on the District.

There are still a few judgment calls in this system. One is what to do at the end of the line. In this case, I rule it a missed connection if the terminal clearly has an intersection without a transfer; if the terminal is roughly between the two stations on the through-line, it doesn’t count. Another is what to do about two lines that intersect twice in close succession, such as the Bakerloo and Hammersmith and City lines in London, and Metro Lines 4 and 10 in Paris. In such cases, I rule that, if there’s just one station on the wrong side (Paddington on Bakerloo, Mabillon on M10) then I rule it a single intersection and allow transfers at the next station over, by which standard London has a missed connection (Edgware Road has no Bakerloo/H&C transfer) and Paris doesn’t (Odeon has an M4/M10 transfer).

How many missed connections are there?

In Paris, there are three missed connections on the Metro there is one missed connection on the Metro (update: see comments below): M9/M12, M5/M14, M9/M14. As I discuss on City Metric, it’s no coincidence that two of these misses involve this miss involves Line 14, which has wide stop spacing. Narrow stop spacing makes it easier to connect within line-dense city centers, and Paris famously has the densest stop spacing of any major metro system. M9/M12 and M9/M14 morally should connect at Saint-Augustin and Saint-Lazare, but in fact there is no in-system transfer. M5/M14 should connect at Gare de Lyon, but when M5 was built it was not possible to get the line to the station underground and then have it cross the Seine above-ground, so instead it meets M1 at Bastille, while M14 doesn’t serve since it expresses from Gare de Lyon to Chatelet. A fourth missed connection is under construction: the extension of M14 to the north misses M2 at Rome, prioritizing long stop spacing over the connection to the M2/M6 circumferential.

In Tokyo, there are many misses. I am not sure why this is, but judging by line layout, Tokyo Metro and Toei try to stick to major roads whenever possible, to avoid tunneling under private property, and this constrains the ability of newer lines to hit station locations on older lines. If I understand this map correctly, there are 19 missed connections: Ginza/Hibiya (Toranomon and Kasumigaseki should connect), Ginza/Mita, Ginza/Yurakucho, Ginza/Shinjuku, Marunouchi/Mita (Ginza and Hibiya should connect), Marunouchi/Yurakucho, Asakusa/Yurakucho, Asakusa/Hanzomon, Hibiya/Namboku, Hibiya/Yurakucho (Tsukiji and Shintomicho should connect), Hibiya/Hanzomon, Hibiya/Shinjuku, Hibiya/Oedo, Tozai/Oedo, Tozai/Fukutoshin, Mita/Oedo, Chiyoda/Oedo twice, and Oedo/Fukutoshin. Oedo is particularly notable for being a circumferential line that misses a large number of transfers.

In New York, there are even more misses. Here the culprit is clear: the two older layers of the subway, the IRT and BMT, have just two missed connections. One, 3/L at Junius Street and Livonia Avenue, is an outlying miss. The other is central: Bowling Green on the 4-5 and Whitehall on the R-W should connect. But the newer layer, the IND, was built to drive the IRT and BMT into bankruptcy through competition rather than to complement them, and has a brutal number of misses: ABCD/2-3, ACE/1-2-3, AC-F/2-3-4-5, AC-G/2-3-4-5-BQ-DNR, BD/NQRW, BDFM/NQRW, BD/JZ, E/1, E/F, M/NW, R/7, F/BD-NQ, F/NRW, F-Q/4-5-6, F/NW, G/7, G/JMZ. Counting individual track pairs, this is 46 misses, for a total of 48 including the two IRT/BMT misses; I’m excluding local-only transfers, such as Columbus Circle and 53rd/Lex, and counting the 42nd Street Shuttle as an express version of the 7, so it doesn’t miss the BDFM transfer.

Finally, London only has eight misses. In Central London there are three: the Metropolitan or Hammersmith and City line misses the Bakerloo line as discussed above, and also the Victoria line and Charing Cross branch of the Northern line at Euston. The other five are outlying: the Central line misses the Hammersmith and City line at Wood Lane/White City, and its branches miss the Piccadilly line’s Uxbridge branch three times; the fifth miss is Metropolitan/Bakerloo. But one more miss is under construction: the Battersea extension of the Northern line is going to intersect the Victoria line without a transfer.

The difference between the two kinds of miss

Many misses are located just a few stations away from a transfer. In New York, some misses are just a station away from a transfer, including the G/7 miss in Long Island City, the E/1 miss between 50th Street and 59th Street, and several more are a few stations away, such as the various BDFM/NQRW misses. In London, these include two of the three Central London transfers: there is an H&C/Bakerloo transfer at Baker Street and an H&C-Met/Victoria transfer at King’s Cross-St. Pancras. As a result, not counting the Waterloo and City line, only two trunk lines in the system do not have any transfer: the Charing Cross branch of the Northern line and the Metropolitan/H&C line.

On a radial network, if two lines don’t have any transfer, then the network is degraded, since passengers can’t easily connect. In New York, this is a huge problem: some station pairs even within the inner networks require two transfers, or even three counting a cross-platform local/express transfer. My interest in subway networks and how they function came about when I lived in Morningside Heights on the 1 and tried socializing with bloggers in Williamsburg near the JMZ.

In Paris the three misses are also a problem. Line 4 is the only with a transfer to every other main line. Line 9 intersects every other line, and Line 14 will when its northern extension opens, but both miss connections, requiring some passengers to take three-seat rides, in a city infamous for its labyrinthine transfer stations. Fundamentally, the problem is that the Paris Metro is less radial than it should be: some lines are laid out as grid routes, including Lines 3, 5, and 10; moreover, Lines 8, 9, 12, and 13 are radial but oriented toward a different center from Lines 1, 4, 7, and 11.

In London, in contrast, there is almost no pair of stations that require a three-seat ride. The Charing Cross branch of the Northern line doesn’t make any stop that passengers from the H&C or Met line can’t get to from another line with one interchange (Goodge Street is walking distance to Warren Street). A bigger problem is the lack of interchange to the Central line on the west, which makes the connections between the H&C stations on the west and some Central line stations awkward, but it’s still only a small number of stations on each line. So the problem in London is not network robustness.

Rather, the problem in London is severe capacity limitations on some lines. Without good outlying interchanges, passengers who want to get between two lines need to ride all the way to the center. Most likely, passengers between the Piccadilly and Central line branches to the west end up driving, as car ownership in West London is relatively high. Passengers without a car have to instead overload the Central line trunk.

The same problem applies to misses that are strictly speaking not missed connections because the two lines do not actually intersect. In Paris, this occurs on Line 7, which swings by the Opera but doesn’t go far enough west to meet Lines 12 and 13. In London, the best example is Hammersmith station: the H&C and District lines have separate stations without an interchange, but they do not intersect since it’s the terminus of the H&C line and therefore I don’t count it as a miss. But morally it’s an outlying miss, preventing District line riders from changing to the H&C line to reach key destinations like Euston, King’s Cross, and Moorgate without overloading the Victoria or Northern line.

In New York this problem is much less acute. The only outlying misses are the 3/L and the ABCD/2-3; the 3/L connects two very low-ridership tails, so the only serious miss is on the Upper West Side. There, passengers originating in Harlem can walk to either line, since the two trunks are two long blocks apart, and passengers originating in Washington Heights can transfer from the A-C to the 1 at 168th Street; at the other end, passengers bound for Midtown can transfer at Columbus Circle, using the underfull 1 rather than the overcrowded 2 and 3.

The role of circumferential lines

Outlying transfers are useful in distributing passengers better to avoid capacity crunches, but they are incidental. They occur when formerly competing suburban lines get shoehorned into the same subway network, or when two straight roads intersect, as in Queens. But the task of distributing passengers between radial lines remains important and requires good connections between as many pairs of radials as possible.

The usual solution to this is a circumferential line. In Moscow, there are several missed connections in the center (Lines 3/6, 3/7, 6/9) and one more planned (8/9), but the Circle Line helps tie in nearly all the radii together, with just one missed connection (to Line 10 to the north) and one more under construction (to Line 8 to the west). The point of the Circle Line is to allow riders to connect between two outlying legs without congesting the center. This is especially important in the context of Moscow, where there are only a handful of interchange stations in the center, most of which connect more than two lines.

In London, the Overground is supposed to play this role. However, the connections between the Underground and Overground are weak. From Highbury and Islington clockwise, the Overground misses connections to the Central line, the Victoria line, the main line of the District line, the Piccadilly line, the Hammersmith and City line, both branches of the Northern line, and the Piccadilly line (it also misses the Metropolitan line, but that’s on a four-track stretch where it is express and local service is provided by the Jubilee line, with which there is a transfer). Much of this is an unforced error, since the Underground lines are often above-ground this far out, and stations could be moved to be better located for transfers.

In New York, the only circumferential line is the G train, which has uniquely bad transfers, legacy of the IND’s unwillingness to build a system working together with the older subways. Triboro RX (in the original version, not the more recent version) would play this role better: with very little tunneling, it could connect to every subway line going counterclockwise from the R in Bay Ridge to the B-D and 4 at Yankee Stadium. On the way, it would connect to some major intermediate centers, including Brooklyn College and Jackson Heights, but the point is not just to connect to these destinations in the circumferential direction but also to facilitate transfers between different lines.

Going forward, cities with large metro network should aim to construct transfers where feasible. In New York there are perennial proposals to connect the 3 and L trains; these should be implemented. In London, the missed outlying transfers involve above-ground stations, which can be moved. The most important miss, White City/Wood Lane, is already indicated as an interchange on the map, but does not to my understanding have an in-system transfer; this should be fixed.

Moreover, it is especially important to have transfers from the radial lines to the circumferential ones. These improve network connectivity by allowing passengers to change direction (from radial to circumferential, e.g. from east-west to north-south within Queens), but also help passengers avoid congested city centers like outlying radial-radial transfers. Where circumferential lines don’t exist, they should be constructed, including Triboro in New York and Line 15 in Paris; where they do, it’s important to ensure they don’t miss connections the way the Overground does.

This is my third post about scale variance in transit planning; see parts 1 and 2. In part 1, I discussed how good bus networks exist at a certain scale, which can’t easily be replicated at larger scale (where the slowness of city buses makes them less useful). In part 2, I went over a subway planning feature, especially common in the communist bloc, that again works only at a specific scale, namely cities with enough population for 3-4 subway lines; it gets more complex in larger cities, and cannot be imported to bus networks with 3-4 lines. In this post, I will focus on one scale-variant feature of surface transit: the grid.

The grid works only for surface transit and not for rapid transit, and only at a specific scale, so constrained as to never be maximally useful in an entire city, only in a section of a city. This contrasts with what Jarrett Walker claims about grids. Per Jarrett, grids are the perfect form of a transit network and are for the most part scale-invariant (except in very small networks). One of the impetuses for this post is to push back against this: grids are the most useful at the scale of part of a transit city.

Grid Networks Versus Radial Networks

I’ve written a few posts exhorting subway planners to build their networks in a certain way, which, in the most perfect form, is radial. In particular, tangential subway lines, such as the G train in New York (especially when it ran to Forest Hills), Line 10 in Paris, and Lines 3 and 6 in Shanghai, are weak. When the G train was running to Forest Hills, most local passengers would switch from it to the next Manhattan-bound train, leading New York City Transit to send more Manhattan-bound local subways to Forest Hills and eventually to cut back the G to Long Island City. Based on these examples, I contend that on a subway network, every line should be either radial, serving the CBD, or circumferential, going around the CBD.

My post about New York light rail proposes a network with some lines that are neither: in the Bronx, my proposal is essentially a grid, with north-south routes (Grand Concourse, Webster, 3rd) and east-west ones (161st, Tremont, Fordham) and one that combines both (145th-Southern). Regular commenter NewtonMARunner criticized me for this on Twitter. I answered that the lines in my proposal are based on the busiest buses in the Bronx, but this simply shifts the locus of the question to the existing network: if transit lines should be radial or circumferential, then why are the tangential Bx19 bus (145th-Southern) or the Bx40/42 and Bx36 (Tremont, with a long radial eastern tail) so successful?

To answer this requires thinking more carefully about the role of circumferential routes, which by definition don’t serve the most intensely-used nodes. In Paris, Lines 2 and 6 form a ring that misses five out of six train stations and passes just outside the CBD, and yet they are both busy lines, ranking fourth and fifth in ridership per km. The reason is that they are useful for connecting to radial Metro lines and to some RER lines (namely, the RER A and the southern half of the RER B). Tangential lines miss connections much more easily: in the west, Line 10 here has a decent transfer to Line 9 and a somewhat decent one to Line 8, but to Lines 12 and 13 it’s already not very direct. The G train in New York has the same problem to the south – few connections to lines that actually do go into Manhattan.

Consider the following three possible networks:

The radial network is a typical subway network. The full grid lets you go from everywhere to everywhere with just one transfer, at the cost of having far more route length than the radial network. The partial grid no longer lets you go from everywhere to everywhere easily, and has the outer two lines in each service direction missing city center, but still has more overall route-length than the radial network. The principle here is that a grid plan is useful only if the grid can be complete.

The scale, then, is that rapid transit is so expensive that there’s no money for a complete grid, making a radial plan more appropriate. But surface transit, especially by bus, can be spread across a grid more readily. The Bronx’s size, density, and bus ridership patterns are such that a mostly complete grid is feasible within the western two-thirds of the borough, supplemented by the subway. In this environment, a tangential route is fine because it hits all the radial routes it could, and could provide useful two-seat rides to a large variety of destinations.

Are Grids Really Grids?

Chicago has a relentless bus grid. The three busiest north-south routes are the tangential 8 (Halsted), 9 (Ashland), and 49 (Western), which are 22, 29, and 26 km long respectively. None enters the Loop; Halsted, the easternmost, is at the closest approach 800 meters from the Loop, across a freeway. The two busiest east-west routes, the 77 (Belmont) and 79 (79th), are also far from the Loop.

However, I contend that these routes don’t really form a grid, at least not in the sense that passengers ride between two arbitrary points in Chicago by riding a north-south bus and connecting to an east-west bus. Instead, their outer ends form tails, which people ride to the L, while their inner ends are standard circumferentials, linking two L branches. The L in turn is purely radial and doesn’t follow the Chicago grid, with the Blue Line’s O’Hare Branch, the Orange Line, and the Brown Line all running diagonally.

Vancouver is similar. The north-south routes are radial, veering to enter Downtown. The east-west ones are more circumferential than tangential: they connect the Expo and Canada Lines, and most also connect to UBC. The Broadway buses (9 and 99) pass so close to Downtown Vancouver they’re more tangential, but they also offer the shortest path between the Expo and Canada Lines (making them a strong circumferential) while at the same time serving high job density on Central Broadway (giving them some characteristics of a radial).

In the absence of a radial rail network to connect to, long grid routes are less useful. Cities have a center and a periphery, and the center will always get more ridership, especially transit ridership. The outermost grid routes are often so weak that they should be pruned, but then they weaken the lines they connect to, making it necessary to prune even more lines until the grid is broken.

The Optimal Scale for a Grid

A strong transit grid will not form in a city too small for it. There needs to be a large enough center with enough demand for transit ridership to justify more than a purely radial bus network with a timed transfer. At the same time, the city cannot be too big, or else the arterial buses are too slow to be useful for ordinary work and leisure trips, as in Los Angeles.

What’s more, there is no Goldilocks zone, just right for a grid. Chicago is already too big for a bus grid without the radial rail layer. It’s also too big for what Jarrett calls grid accelerators – that is, rapid transit routes that replace bus grid lines: the Red Line is plausibly a grid accelerator, but the other lines in Chicago are not, and if there were L lines only at grid points, then the Red Line and the one east-west route would get overcrowded heading toward the Loop. Even Vancouver, a compact metro area hemmed by mountains and the ocean, relies on the diagonal Expo Line to serve Downtown and doesn’t really have a grid beyond city limits. A less dense city in the same land area could have a grid, but without much traffic or a strong CBD, cars would always beat transit on time and only the poor would ride the bus.

The scale in which grid networks work more or less on their own seems to be that of Vancouver proper, or that of the Bronx. Vancouver is 115 km^2 and the Bronx is 110 km^2; Vancouver’s bus grid spills over to Metrotown and the Bronx’s to Upper Manhattan, but in both cases these are small increases in the relevant land area.

Tellingly, Vancouver still relies on the bus network to feed SkyTrain; the Canada Line is a grid accelerator, but the Expo Line is not. The Bronx is the more interesting case, because it is not a city or even the center of the city, but rather a dense outlying portion of the city with an internal arterial grid. In both cases, the grid is supplementary to the radial rail core, even if the routes that use it have a lot of independent utility (Metro Vancouver has higher bus ridership than rail ridership, and the Bronx buses combined have slightly more ridership than the combined number of boardings on the Bronx subway stations).

Geographical constraints matter as well. The Bronx and Upper Manhattan are hemmed by water and by the administrative border of the city (which also includes a sharp density gradient), and Vancouver is hemmed by water and by a density gradient in the east. This makes it easier to equip both with grids that are close enough to the complete grid in the middle image above rather than the incomplete one in the third image. The Bronx’s lower-density eastern tails happen to meet up with those of Queens, forming circumferential routes, and also have enough north-south subway lines to feed that they remain useful.

In a transit city, the grid cannot come first. Even if there is a street grid, the spine of the network has to be radial as soon as there is demand for more than two rapid transit lines. The role of surface transit remains feeding rapid transit. Grids look attractive, but the optimal scale for them is awkward: large-scale surface transit grids are too slow, forcing the city to have a rapid transit backbone, and if the city is too small for that then the arterial grid provides too good auto access for public transit to be useful.