Anthropologists, archaeologists, economists, sociologists, geographers, historians, and many other scholars have long been fascinated with the way humans organize themselves in space, especially with regard to the many shapes that cities take. Some are crooked and curvy while others are relentlessly regular. Several are small and mazelike while still others are grand, orderly, and geometric. With this immense variation, can one say anything in general about the shape of cities?
The answer requires flipping the question from form to function. What is the role of a city or neighborhood? One must treat cities as complex systems, always in a state of flux as they grow and change — both causing and reacting to shifts in use and structure by their inhabitants. At the infrastructure level, a city’s transportation network serves as a means of connection; one must be able to get “from here to there” within a city, regardless of where “here” and “there” are physically located.
Mathematically, this description is not at all about shape or street patterns. Instead, it is about connectivity and accessibility. Topology describes relationships of this kind, as well as their various manifestations in space. For example, a city’s transportation network—the collection of streets, roads, paths, rail lines, etc.—must be path connected: given any two points on the network, a continuous path from point \(A\) to point \(B\) that does not leave the network must exist.
In mathematical analysis, the building blocks of space are based on distance; a neighborhood of a point in space is the collection of all points within a fixed distance epsilon from the point itself. In topology, we relax this definition by removing distance altogether. A neighborhood is thus whatever we say it is, provided our choices behave in appropriate ways under unions and intersections. This lack of focus on measurements allows us to stretch and deform one space into another as long as we preserve certain fundamental relationships.
The essential topological element of a city’s street network is the city block. City blocks are two-dimensional spaces, typically filled with buildings and surrounded by streets on all sides. To appreciate the topology of a city, take its street map and punch a bunch of holes in it — one for each city block. Imagine moving these holes around, making them smaller or larger and therefore changing the shape of streets at will, so long as you keep the number of blocks the same. You can turn this fabric into a lot of rectangular blocks, as in New York or Chicago, or you can make each a little different—some big and others much smaller—and end up with sections like Rome or Cairo, all with the same number of city blocks. This means there is a topologically invariant number expressing the equivalence of all of these street plans; this is the Euler characteristic of a city with \(b\) blocks, which is just \(\chi =1-b\) independent of shape or geometry.
Figure 1. Collaborative mapping that combines building footprints with street networks has increased in recent years. Image courtesy of Geoff Boeing.
Upon further inspection, we have just made a critical underlying assumption about accessibility within a city — an assumption whose absence fundamentally changes both form and function of the city itself. We assumed that every location within a city block has direct access to the transportation network. It turns out that real cities are not this simple and the properties we invoked do not yet describe many fast-developing cities, particularly where people occupy land informally even as services are not yet present. Informal settlements are often “slums,” places that lack proper streets despite the presence of houses, shops, temples, and other buildings. Such places consequently have no addresses, meaning that there is no way to deliver services, especially those like water, sanitation, fire protection, and ambulance pickup — all of which require physical infrastructure.
The lack of an accessible transportation network becomes a kind of poverty and development trap, restricting these places and their residents from full integration into the fabric of the city. This is far from an exceptional occurrence, as almost any fast-growing city has experienced this kind of informal settlement structure at some point throughout its history. In some of the world’s largest cities, like Mumbai or Lagos, between 50 and 70 percent of the population lives this way. The lack of access to a street network is then a signal of an incipient city, a set of latent material and socioeconomic relationships that are absent, weak, or not yet formed.
We can thus look to mathematically model the lack of connectivity between places and the street network with a topological viewpoint. We can take any city block and diagnose its degree of inaccessibility to each building from the transportation network using measures from graph theory and topology. On a larger scale, we can scan an entire city to identify blocks within which some buildings lack access. Once equipped with this diagnostic, we may turn the problem around and inquire about minimal additions to the existing street network that will solve this access problem, creating the topology of a universally accessible city. This problem is a type of constrained optimization search with an objective function dictated by the topological features of its city blocks.
Using formal mathematical tools to solve the problems of physical access to slums may seem excessive — can’t urban planners make proposals that are quicker and more suitable than an algorithm? This turns out to be true only in very simple cases. Many of the neighborhoods in large, dense, rapidly-developing cities have hundreds or thousands of places. Proposing a viable but minimal street network becomes an enormous puzzle whose computational complexity quickly explodes. Furthermore, too often a single “best” solution is not sufficient; residents and local authorities must be able to discuss different possibilities that may be more convenient for local reasons. Quickly creating many good solutions can dramatically accelerate politics and promote productive discussions and multi-stakeholder resolutions.
There are currently about a million neighborhoods in cities throughout the world that require dramatic improvements in their access infrastructure. These places are currently being mapped—many for the first time in history—in great detail so as to account for every building and service. Recent humanitarian emergencies—such as earthquakes in Nepal and Haiti, or the Ebola crisis in West Africa—have accelerated collaborative mapping in platforms like OpenStreetMap, thus revealing vast informal settlements that require additional accesses and services (see Figure 1). All of this mapping will soon be complete. The next chapter—during which we must reimagine our cities for a sustainable and prosperous future, producing places rich in history and local flavor—will then be more necessary than ever.
This is one instance where mathematicians can really impact the world by fundamentally understanding the essential nature of the spatial fabric of our lives and creating faster and better computational tools that enhance our collective imagination. Such developments will open new avenues for fast and sustainable global human development.