The city is not a tree - it's a graph

Where I grew up there weren't many trees
Where there was we'd tear them down
And use them on our enemies
They say that what you mock
Will surely overtake you

And you become a monster
So the monster will not break you

(U2, Peace on Earth)

The way we organize the world around us is a major concern when designing architectural products, whether it might be a house or an apartment, a building, a street or even larger urban components.

Easy to use architectural products are ones that fit with the way we perceive the world, ones that can be seamlessly added to our mental system of connections. The problem is that this pattern of connections is quite hard to define. It's rarely linear and it has a hierarchy, but there aren't any clear rules other than that. So how do we construct our environment? How do we connect the different parts into a comprehensible system?
When we design small scale architectural products we tend to do it intuitively, the connections are few and all the details are comprehensible, but how do we design lager architectural products? how do we design an urban tissue? a city?

Christopher Alexander had used, in his 1965 article "the city is not a tree", the mathematical structure of a tree to demonstrate the problems with the zoning based modernist concepts of a city as structure that originates from a certain point and expands in a clear direction.

Instead Christopher Alexander had chosen the mathematical structure of a semi lattice to describe the city, a fact that seems to indicate that he hadn't read enough in his discrete math book, which seems right enough as he was busy fighting modernism.

Although Alexander was quite right to point out that what defines cities is the overlapping of zones, the mixture of systems, where the natural conjunction and density creates a multi centered city, he seemed to have missed the fact that the city is not composed of zones, as modern architecture wants us to believe.

The city is composed of urban functions which are combined together to create an interesting environment, rich with multi sensory stimulations. The city is not a reserved old man; it's more like a hyperactive child – an ever surprising and creative creature that can never fit into any traditional system.

A city can't and shouldn't be tamed into strict patterns but rather follow predetermined guidelines. In order to do that we need a much more flexible structure which we can build our cities around, the skeleton of the city to which urban tissues will be attached.

Such a structure needs to have two main attributes: the ability to express hierarchy and the ability to define connections in a non linear fashion. These two attributes are the basis for the creation of intricate systems that are seemingly complex and yet can be separated into clear sub-systems.

The ability to define for each component it's hierarchy level and to which elements it is connected enables us to create a system that can grow to be complex and yet clearly defined and easy to understand.

The structure which provides us with both complexity and hierarchy is the mathematical graph. I'll skip the technical detail (what is wikipedia for?) and just do with the basic definition that graph is a data structure which is composed of two types of components: nodes and edges, when each two nodes are connected by an edge.

Each node can be assigned a value, which can be used to place node along the hierarchy scale, and when we connect all the nodes together we can get a spatial skeleton of our system.

Further more, when considered in an urban context the system gets more meanings. For example, a node with many edges connected to it has relatively more intensity. Some features of the system are byproducts of other attributes and it is almost impossible to control them as the system grows, much like in real life cities.

The question is what exactly the nodes represent? it's clear that each one is an urban function but that hardly a value, and barely constructs any kind of hierarchy.

In order to find out the units of the values, we must first define the scale of the hierarchy, what is at the top of the scale and what is in the bottom? Does a shopping center is ranked higher than a public park and where it stands in relation to housing or offices?

I suggest a definition of what can be called an urban mass, a term which represents the relative effect an area has on the urban fabric. The unit would consist of a combination between density (and how the measure that is another topic) and land area, or more correctly volume. Such a scale that recognizes the fact the large low density areas has an effect on a city as much as high density areas. Take for example the Central Park in Manhattan and Time Square, both of New York's landmarks with completely different nature.

When the urban mass of a function and its intensity (the number of edges) are combined we are able to define urban centers and the relations between them, and combine them into a complex yet coherent system, which is the basis of designing urban tissues.