I’m reading A City is not a Tree, a brilliant article by Christopher Alexander, about urban planning. It reminds of, but doesn’t mention Jane Jacobs’s The Death and Life of Great American Cities. A very brief summary of what Alexander writes:

There are “natural” and “artificial cities,” the former having “arisen more or less spontaneously over many, many years,”, whereas the latter were “deliberately created by designers and planners.”

Artificial cities “always organized to form a tree.” Natural cities are semilattices. “Both the tree and the semilattice are,” Alexander writes, “ways of thinking about how a large collection of many small systems goes to make up a large and complex system.” Axioms:

A collection of sets form a semilattice if and only if, when two overlapping sets belong to the collection, the set of elements common to both also belongs to the collection. [–––]

A collection of sets forms a tree if and only if, for any two sets that belong to the collection either one is wholly contained in the other, or else they are wholly disjoint.

Point being:

[T]he semilattice is potentially a much more complex and subtle structure than a tree. We may see just how much more complex a semilattice can be than a tree in the following fact: a tree based on 20 elements can contain at most 19 further subsets of the 20, while a semilattice based on the same 20 elements can contain more than 1,000,000 different subsets.

This enormously greater variet is an index of the great structural complexity a semilattice can have when compared with the structural simplicity of a tree. It is this lack of structural complexity, characteristic of trees, which is crippling our conceptions of the city.

In other words, urban planning can’t be carried out as if cities were trees. This is about zoning, about separating the different uses of a city into geographically distinct areas. Without overlap. Overlap is key. Overlap in sets of humans, but also in sets of all entites found in cities – or systems. Alexander: “When the elements of a set belong together because they co-operate or work together somehow, we call the set of elements a system.”