The Mathematics of ConfigurationThis is the way the generic city internalises economic and social processes, and builds them into a complex but highly efficient and, it has been argued, sustainable network.1 But where does the distinctive pattern of linearity in the scale-free grid come from in the first place? Topologically, the city is a network of linear spaces created by clumps of outward-facing buildings a very simple generator that creates complexity only with growth. It has been known since the 1970s that this topology emerges from a restricted random process of cellular aggregation made up of cells with open spaces attached that accumulate by joining their spaces randomly onto one already in the system. This generative process leads to very plausible forms at the scale of the hamlet or village, however at the urban scale the forms that emerge manifest the topology of the city, but not its geometry. The blocks are too irregular, so the space pattern is far too complex. How and why, then, does geometry intervene in this basic generative process?

We must introduce two more factors into the city-creating process: spatial laws and the human minds intuitive knowledge of them. If we think of spaces not just in terms of their shape and scale, but in terms of the relations between all the points that make up the space, then space no longer behaves in the normal metric way, but in a characteristically configurational way. For example, if we take a linear rectangle made up of eight notional spaces, each one with a person in its centre, and move a partition successively from centre to edge, the total metric area of the two sides remains the same, but the number of people who can see each other the total intervisibility increases as the partition goes from centre to edge. This simple mathematical principle implies, among other things, that an object placed near the corner of a bounded space will increase mean intervisibility (and also decrease mean distance to some degree) between points in the ambient space more than if the object is placed in the centre, that a square form will reduce intervisibility (and increase distance) less than a rectangular object of equal area, and that a long line and a short line have more intervisibility (and less mean distance between points) than two lines of equal length and the same total.

Bill Hillier, A linear system of eight notional cells right: Moving a partition from centre to edge increases intervisibility between points while keeping total area constant.

Bill Hillier, The restricted random generative process of cells with attached spacestop: The process generates the topology, but not the geometry, of the city.

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