# Mathematical Underpinnings Of Good Urban Design

Urban design principles can be found to have mathematical underpinnings. Good urban design is defined generally as what is pleasing, healthful, and harmonious to people. Good urban design follows rules and patterns that are mathematical in nature. The mathematical theory of the urban web describes some rules underlying connections between human activities in a city. A pattern measure is a formula for calculating the degree of organized complexity of a two-dimensional design. The content of this paper is derived from the papers on architecture, patterns and urbanism of Nikos A. Salingaros, professor of mathematics at the University of Texas at San Antonio.

The mathematical theory of the urban web is structurally based upon three principles. These are nodes, connections, and hierarchy. The nodes serve as anchors in the urban web and are defined by places of human activity, such as home, store, church, etc. Interconnections between the nodes are the connections. They arise strongest between complementary nodes like home and work and less strongly between similar nodes like home and neighbor. Connections can not be supported when nodes are too far apart on the landscape. The hierarchy of the urban web consists of networks at different levels of scale that are self-organized. The hierarchy includes footpaths, collector streets, main roads, highways, etc. A complex hierarchy is difficult to create all at once and tends to arise spontaneously to fit the needs of the nodes and connections. The terms for these three principles were first given by Kevin Lynch in his book, The Image of the City, but here they are more specific.

The connections in urban design join together three distinct types of elements, which are natural elements, human activity nodes, and architectural elements. Natural elements are those which are part of the environment, such trees, hills, rocks, etc. Human activity nodes are places defined by places where people do things, such as work, play, eat, etc. Architectural elements are built forms such as buildings, monuments, roads, etc. These usually reinforce the human activity nodes and natural elements. However, if they do not, then they become isolated in the urban web. An example would be the never used benches and sidewalks at the edges of strip malls. Without a well-defined activity, an architectural element remains unused.

When urban planners try to create an urban web, they usually fail because the try to enforce a high degree or geometric regularity. While this looks appealing visually from a birds-eye view, it is insensitive to the three elements of the urban web as just described. Highly complex patterns of human activity can not fit within neat, simple, geometric forms of a city. Indeed, the urban web may look organized yet be disconnected, and conversely, may look disorganized yet be highly connected and functional. Take, for example, the irregular but well-used streets of old-world European towns.

The connections of the urban web should be multiple and irregular. In mathematics, there is a theorem which says that two points can only be connected in one way by a straight line, but in infinite ways by curved lines. Therefore, in order to increase the number of connections between points, curved connections must be used. A principle of physics states that the interaction between two objects can be written as the sum of interactions over all possible paths between them. Similarly, in urban design, we wish to increase the interaction between two objects by increasing the number of connections and strength of the connections between them. It is important to note that all streets do not have to be curved, but there should be diagonals in any grid to provide better connections. Multiple connections serve a functional role too because they reduce overloading on singular connections.

Suppose there are N nodes laid out on a plane. If we add connections between random pairs of nodes incrementally, at some point we will have every node connected to every other node by at least one path. The point at which this happens can be viewed as a phase transition of the system from a disorganized state to an organized state. The urban design lesson to be drawn from this is that connecting nodes incrementally will result in an urban web with that suddenly becomes highly organized at some point. The result will be a coalesced form that makes sense as a whole to observers in the city.

A good urban web is characterized by organized complexity as opposed to empty purity. Complexity arises when there are multiple processes occurring together. When the processes are not organized, the situation becomes chaotic. Therefore, the urban web requires organized complexity. When planners try to make a design which is visually simple, it curtails the human processes which incrementally lead to organized complexity. What results is a kind of empty purity, where the information inherent in the system is lost. One can measure the complexity of a system by the ratio between the number of connections and the number of nodes. Organization is much harder to measure, and can easily be confused for visual purity.

The application of the three principles of nodes, connections, and hierarchy gives rules for how to build better neighborhoods. In order for connections between nodes to be used, there must be complementary uses for the nodes. This comes from the physics principle that electrical or fluid flows only between points of differing potential. The nodes have to have a sufficient density too. Multiple paths for walking will be created naturally between complementary nodes when there is enough density. Multiple paths between like nodes will become amalgamated. The important point is that uses must be mixed in order for the connective process to begin.

Another application is that pedestrian paths should be short enough for people to agree to walk on them. If they are too long, too ill-defined, or too exposed, then they will not be used. Simple geometry tells us that the shortest distance between two points is a straight line. Therefore, since people always want to walk the shortest distance, the connections between nodes should be straight. This doesn't contradict with the previous statements that connections should be curved, because another mathematical result states that any global curve is locally straight in the limit of small measure. In other words, the path may be straight in the short distance between two complementary nodes, but curved when looked at from a distance.

Having many intermediate nodes and placing paths at the edges of regions helps to strengthen the web structure. In large malls there are many smaller stores located between anchors, and this makes the whole space lively. The design criteria to be drawn from a mall is that there is a maximum distance for pedestrian activity, but not a minimum distance. In addition to connectivity and length limitations, placement of paths should be done with care. Paths should not go through areas, but along the edges of areas. Both the edge and path reinforce each other when they are together. However, a path through the middle of an area creates an artificial divide through the area. In addition, walking through the middle of spaces makes people uncomfortable. All these things show that having many intermediate nodes and placing paths at the edges of regions results in better urban design.

Given the hierarchy of the urban web, it is necessary for the networks at different scales to connect to each other. However, they do not have to coincide or be joined together. Cross-connectivity results in a stronger web and also eases congestion when compared to a system with only one network. Pedestrian paths should have first priority when determining this connectivity. This should be the first step in creating a design. Following this, bicycle paths and roadways can be laid out. Footpaths and streets can coincide, but not if the road is too heavily traveled or not separated enough from the footpath. When the roads are laid out, they should be made so that they are the appropriate size for the traffic they are meant to carry. A similar amount of design effort should be given for the foot and bicycle paths as to the road paths. The pieces of this puzzle will be simple by themselves, but will make a complex system when they are united. Therefore, they must be designed dynamically and thoughtfully, not simplistically and quickly, because that would leave out many of the needed connections.

Nodes in the city which would hurt each other should be separated by barriers. For example, a highway and an apartment building should have separation as should a factory and a park. Natural features, such as rivers, can be exploited to form natural barriers instead of covering them over or ignoring them. When barriers are placed in the right places, they should have holes in them to allow pedestrians to go through, otherwise connections will be severed.

The theory of the urban web is organized around three principles: nodes, connections, and hierarchy. These principles, coupled with mathematical theories, give guidelines for good urban design. The goal of these guidelines is to create organized complexity. One way to measure this organized complexity is to use the pattern measure. The pattern measure is a formula applied to a two-dimensional symbol array that results in a number. The number that results gives an idea of the liveliness of the pattern as perceived by people.

The pattern measure is based upon two calculations. The first is the Temperature calculation. This simply describes the variety of symbols in the pattern. To calculate it, one takes the number of element-symbols and subtracts one. The other requirement is the Harmony calculation. It represents the level of symmetry in the pattern. One point is added for every internal sub-symmetry in the pattern. Examples of sub-symmetries are reflectional symmetry across the axis, reflectional symmetry about the diagonal, and rotational symmetries. The maximum number of possible symmetries for a particular size grid is a constant value.

One calculate a life measure L of a pattern by multiplying the temperature T and the harmony H. A complexity factor C can be obtained by multiplying the temperature T by the maximum H value minus the desired pattern's H value. This way of calculating complexity rewards patterns with fewer symmetries because the smaller the number of symmetries or harmony H, the higher the complexity C will be. The example below shows how this works.

I II III IV 0 0 + 0 + + + 0 0 0 0 0 0 0 0 + L (life) 0 1 1 3 C (complexity) 0 5 5 3

The patterns above are based on a two-symbol system in a two by two grid. The numbers tabulated correspond to our subjective notion of what is interesting. This can be seen informally because the simple pattern in I receives a score of 0 and 0, while the symmetric and varied pattern in IV gets a score of 3 and 3.

The pattern measure can be extended to apply to systems with more symbols and larger grids. The complexity of the calculations increases with the increasing size of the grid, because more symmetries are possible. The pattern measure can also be applied to existing architectural forms. It captures the information inherent in a pattern and expresses it in a way that corresponds to our intuitive notion of life or complexity in a pattern. The L value gives a rough estimate of how strongly we connect with a design. These measures could be used in the design or evaluation process to get an idea of how interesting a design is.

In summary, the mathematical theory of the urban web shows some similarities between the mathematical theories and good urban design principles, with good being defined as pleasing to humans. The connections, nodes, and hierarchy of the urban web should have organized complexity in order to achieve greatness. A pattern measure, based on simple calculations, can show the strongly humans connect to a design. These techniques show the mathematical underpinnings of urban design principles.

**Bibliography**

Alexander C., S. Ishikawa, M. Silverstein, M. Jacobson, I. Fiksdahl-King and S. Angel, A Pattern Language. (New York: Oxford University Press, 1964).

Salingaros, Nikos A., Theory of the Urban Web, Journal of Urban Design. Vol. 3 (1998), pp53-71.

Salingaros, Nikos A., A Pattern Measure, Environment and Planning B. June 1999.

This research paper was written December 16, 1999 for 11.328 Urban Design Skills.

*Created 2004-12-16, Last Modified 2011-07-24, © Shailesh N. Humbad*

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