Fractal Applications

Fractal Cities

Cities are complex systems that behave in some ways like living organisms. The rules of chaos theory and fractals apply directly to the evolution of cities, and the study of urban patterns allows us to benefit from the experiments of past cultures to shape our own future with as much awareness of the consequences of our actions as possible.

The physical structure of Rome, Italy (above) is completely different from Phoenix, Arizona (below). Rome is much more of a fractal city with its roads going everywhich way and its intermingling of different urban functions, while Phoenix is much more rigid, orderly, and hierarchical. Image courtesy of Google Earth.

The physical structure of a city provides a record of decades or centuries - or millennia - of human activity. Cities that have a historical record of their urban policies - essentially zoning codes - allow us to study the impact of different types of rules on the development of cities.

Small changes in the rules that govern a city can have dramatic impacts on the ways the city forms itself. The impact of the small changes is often amplified through positive feedback loops. Once a city forms a shape, there are often negative feedback loops that serve to stabilize its shape.

Sometimes, with certain rule systems in place, the cities take on fractal characteristics. This is particularly common in medieval cities in Europe and the Middle East. What does a "fractal city" mean? Basically, it is a city that has similar structures at different scales. A large fractal city tends to grow that way by absorbing lots of smaller villages. So the city becomes a collection of villages, and the villages are collections of neighborhoods. The roads in a fractal city tend to go in seemingly haphazard directions, but really they serve as direct connections between various central hubs in a dynamic urban network. A diagonal road can be as much as ~70% (1/sqrt(2)) shorter than two roads in a rectangular grid, so there are efficiencies built in to the geometry of the city.

The processes that form fractal cities are slow and organic and are the results of millions of individual decisions by the citizens about how they want to live. These decisions are made within a framework of principles about what is ok to do in the city. The urban rules from medieval fractal cities are called "proscriptive", which comes from the word "proscribe," which means to forbid, or condemn. This means basically that people and businesses are free to do whatever they want for the most part, as long as they don't do certain unacceptable things.

By contrast, many modern cities have different rules to govern their growth that do not promote organic, fractal patterns. Modern cities tend to have "prescriptive" zoning rules, which prescribe what to do in various areas. For example, a zone may be designated "Industrial" or "Residential" or "Commercial," for example, and those are the only activities permitted in the specified zones. This type of urban rule system results in non-fractal cities whose basic patterns are dictated from above, by the city government.

Phoenix, Arizona is a much younger, more grid-like city than Rome, reflecting in part its different zoning rules. Image courtesy of Google Earth.

Ancient cities evolved over centuries to meet a variety of people's everyday needs. Everyone needs to eat, to sleep somewhere, to shop, to play, to work. Modern cities have been able to evolve the way the have because of the availability of easy, affordable personal transportation: the car. This allows residential zones to exist far from commercial or industrial zones.

Ancient cities did not have this luxury, and they evolved to allow people to have all their needs within immediate walking distance. The fractal structure at many scales that ancient cities display allow cities to be much more pedestrian-friendly and human-scaled.

Why do we care?

In a time of ever-scarcer petroleum and changing global climates, we can learn much from civilizations that have thrived for centuries without relying on individuals each owning cars.

Understanding the impacts of small changes in the zoning rules to the long-term evolution of a city, coupled with the ability to model urban development in computer simulations will help us design optimal cities that will endure and thrive long into the future.

The caption to the diagram above, reproduced from "Reviving the Rule System" by Besim Hakim: "Conceptual representation of the impacts on the local level (three geometric shapes denoting three settlements) by proscriptive meta-principles, and by prescriptive imposed laws. The diagram on the left represents a settlement’s ability to respond freely to local conditions and requirements, but is restrained by an overarching set of meta-principles. This would result in settlements that are diverse in their physical form and exhibit distinct local identity. The diagram on the right represents how prescriptions from a central authority, which is usually far removed from a locality, inhibit creative solutions to local problems. Over time the resulting settlements would tend to be similar to each other"

Question:
In the figure above, which of the two models (answer: left or right) would be more likely to result in a city structure like Phoenix? [ ]

Are Proscriptive or Prescriptive rule system more likely to form cities that are similar to each other? [ ]

Fractivity:
Use the Google Earth program to explore the "Old World" countries of Europe and the Middle East, and the "New World" of the United States. Try to find and identify fractal cities and non-fractal cities in both regions. Sometimes you may find hybrids, which started off with an ancient fractal core that has been surrounded with modern grid-like development, as newer prescriptive rules were imposed.