The mathematical definition of a city

20th century professional urbanism is the story of a war on complexity in order to control urbanization.

The modernists rebelled against the “mess” of the city. They put everything in their place. In this square shall be the houses. In that square the offices. In that square the stores. In some form of another, this system, called zoning, is in force over 99% of the American continent. Its main advantage is that it is incredibly lazy.

For more than half a century, the in-between, what is not really a house, a shop or an office, has had no place. The “boomtowns” of today are endless grids of single-purpose zones.

They call this urban planning. They took control of the city’s future by destroying it. But they didn’t really know what they were destroying.

A city is not reducible to parts. A city is a mesh of relationships between spaces. It begins once a space is built to provide a specialized function that is not fulfilled by another existing space, and the two spaces are linked together by a communication system. Let’s call this first space a and the new space b. Once a and b form relationship a-b, the city X is born. X is a set which contains relationships.

When a and b are deficient in some manner, a third space, c, is added to the set. X then becomes (a-b, c-b). Then space d may be added to form the set (a-b, c-b, c-d). This process continues as more spaces are created and new relationships are formed. The city becomes a very complex mesh, or semi-lattice. You cannot isolate any part of this mesh from the rest.

Sim City 4 pictures the shape of these relationships when you click on the commutes for any building. Each building has a web extending out through the city, and these webs overlap and interweave each other as a single system.

The relationships do not split up into group. You cannot define “sub-cities,” groups of relationships independent from each other. You cannot say that city A is made up of building set B and C. Inevitably some buildings in either group will need to form relationships to each other. But this is exactly what zoning is meant to prevent! In doing so, zoning destroys many forms of exchange and holds back the complexity of the city.

What exactly are such relationships? Any reason you might have to get out of the house. It could be going to the bakery. Your house d would form a relationship with bakery f, d-f. The bakery would have many customers in the neighborhood, and they would form relationships f-g, f-h, f-i and so on, even though you may never meet any of them. These people will have jobs that will form relationships g-m, h-n, i-n. All of you, together, create the life of the city, though you may never run into each other. Without business m, bakery f may not have enough customers to continue, and then you would no longer have access to a bakery.

Sometimes a space will lose all of its relationships and will be destroyed, but all the other relationships will remain part of the set. The continuous mesh of relationships is itself fully permanent. This is why cities have names that last through millennia, such as London and Paris, even though every building that made them up at their beginning has long since been removed and forgotten. The set of relationships is still exactly where it has always been. It has been transformed and developed, but never destroyed. At every point in time the set exists even though spaces flow in and out of it, much like a river is not defined as a lump of water molecules but the flow of them.

It is only relationships and not the individual spaces that form a city. A block of identical row houses will not form relationships. Relationships will only form between spaces that are complementary, that is to say spaces that are differently adapted to their own specific functions. It thus makes no sense to create zoning codes for identical houses as there is no reason for these houses to be near each other. However, it does make sense to create multiple houses around a playground, as these houses will form a relationship with the playground.

Defining a city as relationships allows us to differentiate cities which are alive and growing from cities which are dead or dying. When the number of relationships in a city is increasing or stable, the city is alive. When the number of relationships in a city is shrinking or zero, the city is dead, despite the fact that there may still be buildings there! A ghost town does not have relationships.

Good urbanism is the creation of support systems for building relationships. Streets, public spaces, transportation networks and building codes achieve this. Zoning kills them.

The best support systems, the best urbanism, will permit the greatest density of relationships (not density of people), implying the greatest spacial complexity and diversity achievable.

Reference:

Alexander, Christopher. A City is not a Tree.

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One response to “The mathematical definition of a city

  1. Pingback: Architecture + Urbanism « Life is a Design Thesis

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