"All happy families are alike; each unhappy family is unhappy in its own way."

- Leo Tolstoy, Anna Karenina

While Tolstoy's insight into human nature applies specifically to families, it may also describe the state of many of the world's cities and the division between areas that have been formally developed, where people have easy access to obtaining life's necessities, and areas that have been informally developed, which are often defined by their lack the same kind of access.

At least, that's one takeaway from a new study by Christa Brelsford, Taylor Martin, Joe Hand and Luís M. A. Bettencourt, who found that when they applied the mathematics of topology to the layout of developed neighborhoods and undeveloped slums found in many of today's cities around the world.

Topology is the branch of mathematics that studies the properties of an object that are preserved when it is morphed into very different shapes without either tearing the object or fusing portions of the object back into itself. Think of a ball of clay, which could be shaped into a cube, a bowl or even into a leafy tree, where a wide variety of complex geometries can be grouped into families described by one simple base shape because they share certain unique mathematical characteristics. Because they share those characteristics, it becomes possible to compare very different or complex geometries and arrive at valid conclusions that would then apply to all potential members of the same topological family. Here's how they authors applied the math of topology to the study of urban areas:

The physical volume of all paths, streets, and roads in a city is a connected two-dimensional surface: Any point on this surface can be reached from any other point on the same surface. This surface ends where buildings begin and at external city boundaries. Thus, the urban access network surface, U, of any city has a number of internal boundaries, b, one for each city block and another for city limits. Mathematically, such an access system is topologically equivalent to a disk with b punctures or "holes" (or a sphere with b + 1 disks removed). In this way, all urban access systems with the same number b of city blocks (i) are topologically equivalent and (ii) share an invariant number, the Euler characteristic χ(U) = 1 - b, which is independent of geometry.

The authors found they could identify similarly developed but geometrically and geographically distinct regions as mid-town Manhattan, the Las Vegas suburb of Summerlin, Nevada, and the Dharavi neighborhood of Mumbai, India, and directly link them so long as they shared the same number of blocks, as shown in the following video.

They found that all these formally developed areas were all similar in their characteristics to one another, but the undeveloped areas either within or on the periphery of these cities were quite different from these "happy" communities, where "happy" means "having adequate access to roads, water and sanitation services". Moreover, their approach in mathematically describing slum areas without such similar access to life's necessities may lead to their effective development, with minimal cost and disruption to lives of the one billion people who live in these communities around the world today.

Here's how the authors foresee their new topographical tools being used to address the problem of bringing necessities into today's impoverished urban areas.

Over the next couple of decades, it is estimated that infrastructure investments will need to exceed \$1 trillion/year in developing nations to meet international development goals, with the majority in poor areas of developing cities. Slum upgrading is a key strategy for achieving these goals, with infrastructure costs accounting for about 50% of the total. Efficient reblocking is an essential part of these transformations because the most important determinant of the cost of building or upgrading urban infrastructure is the existence and layout of the access network. By identifying and formalizing the essence of the spatial transformations necessary for neighborhood evolution, the methods proposed here increase the benefit-cost ratio for infrastructure provision [currently ~3 for water and sanitation] and markedly accelerate — from months to minutes — most technical aspects of creating viable reblocking plans. This enables nontechnical stakeholders to focus their time and effort on the socioeconomic tradeoffs of alternative layouts [leading to savings of up to 30%] and creates precise digital maps that can formalize land uses and property records, facilitating political and civic coordination and further local development.

In other words, the problem of how to transform impoverished areas into developed ones will stop being a technical challenge and will instead becomes purely a political problem. As for its potential, imagine unlocking the what Hernando de Soto described as the problem of "dead capital" in bringing formal property rights to these undeveloped urban areas and the one billion people who live in them.

### References

Brelsford, Christa. Martin, Taylor. Hand, Joe., Bettencourt, Luís M.A. Toward cities without slums: Topology and the spatial evolution of neighborhoods. Science Advances. Vol. 4., No. 8., eaar4644. DOI: 10.1126/sciadv.aar4644. 29 August 2018.