Math targets cities' essence
New formula relates city size to infrastructure, productivity...
Web edition: September 12, 2013
The notion that cities are all alike borders on blasphemy. Residents
of the world’s great metropolises, from New York to London to Tokyo,
speak of their homes as of a first love or old friend. But decades of
analyses hint that cities, mathematically speaking, might actually all
be the same. Now for the first time, those observations have been tidily
and elegantly drawn together into a formula that describes what a city
is.
That new work is part of a growing field dedicated to the science of
cities. The effort is a timely one: Roughly 75 percent of people in the
developed world now live in urban environments. While much of the
research is in its early days, eventually it may serve as a powerful,
widely used tool for urban planners and policymakers.
The mathematical work is rooted in and reinforces the view “that
cities grow from the bottom up,” says Michael Batty, who trained as an
architect, planner and geographer and went on to found the Centre for
Advanced Spatial Analysis at University College London. “The diversity
of life [in cities] offers greater opportunities for mixing ideas.”
That diversity, which includes dismal poverty, squalid slums and
crime juxtaposed with prosperous businesses, majestic parks and great
art institutions, was much decried in the 19th century. In 1883, for
example, textile designer and artist William Morris lamented England’s
cities as “mere masses of sordidness, filth, and squalor, embroidered
with patches of pompous and vulgar hideousness, no less revolting to the
eye and the mind….”
Discomfort with the notion that cities grew from the bottom up went
along with disdain for disorder and chaos, framing cities as a problem
to be solved. This view prevailed into the 20th century and influenced
postwar urban renewal projects across the United States. The resulting
redevelopment forever changed parts of cities such as Pittsburgh and
Boston, with mixed results.
In the last several decades, however, the view of cities as
disordered systems has begun to change, Batty says. Patterns have
emerged within the chaos. Researchers in economics, physics, complexity
theory and statistical mechanics have observed that certain features of
cities consistently vary with population size.
But the relationships aren’t direct and linear. As a city grows, some
features, such as land area, grow more slowly with respect to
population. This “sublinear” relationship also holds for some aspects of
physical infrastructure, such as the length of pipes and roads: As
population grows, proportionately less infrastructure is required to
support each additional person.
For other characteristics, the reverse is true: Some measures grow
faster with respect to the population. This “superlinear” scaling has
been observed for a number of socioeconomic factors in cities around the
world. Produced wealth, whether measured as income, wages or gross
domestic product, increases at a rate greater than the population. So
does crime. Markers of innovation, including the number of patents
produced and number of jobs in creative fields like the arts and
sciences, also increase superlinearly.
While there’s some quibbling about the exact mathematical values, the
relationships among city characteristics generally hold, says physicist
and complex systems scientist Luís Bettencourt of the Santa Fe
Institute in New Mexico.
Bettencourt, with other researchers at the Santa Fe Institute and
colleagues elsewhere, has been examining these relationships for more
than a decade. Now Bettencourt has created a series of equations,
published in the June 21 Science, that pull the relationships together into a mathematical theory of cities.
Bettencourt’s math stands on four basic assumptions: First, cities
mix varied people together, allowing them to reach each other. Next,
cities are networks that grow gradually and incrementally, connecting
people. Third, human effort isn’t limitless and stays the same
regardless of urban size. And finally, measures of the socioeconomic
output of a city — things like the number of patents awarded or crime
rate — are proportional to the number of social interactions.
Bettencourt’s theory captures the interplay between a city’s
population, its area, the properties of its infrastructure and its
social connectivity. By mathematically describing the tension between a
city’s number of social interactions, their outcome (innovation, for
example, or crime), and the transportation and energy costs of enabling
those interactions, Bettencourt arrives at a parameter that he calls G*.
The closer a city’s value is to G* the more effective it is at
producing positive interactions and all the benefits that flow from
them.
“In a nutshell, the city is the best way of creating a vast,
open-ended social network that minimizes the cost of moving things in
and around an environment,” Bettencourt says. “When people brush up
against each other, that’s when the magic of the city happens — the
social reactor begins to work.”
That conclusion isn’t so surprising, Batty says. Consider how a
concentration of creative genius and technological know-how has made
Silicon Valley into one of the world’s foremost engines of wealth.
Bettencourt’s theory “basically unpacks the equations and then puts it
all together and leads us to what we observe in a clean and elegant
way,” Batty says.
In many respects, the theory formalizes what writer and activist Jane Jacobs articulated in her 1961 book, The Death and Life of Great American Cities.
Earlier scholars’ emphasis on aesthetics and form missed what makes
cities so great, she argued. Cities are a way of sustaining an enormous
number of social interactions through time, she wrote, “a most intricate
and close-grained diversity of uses that give each other constant
mutual support, both economically and socially.”
What urban planners and policymakers will take from Bettencourt’s new
theory remains to be seen. The research suggests that enabling mixing
of people and fostering the creation and spread of ideas is never a bad
idea. It also suggests that city planning should not involve grand,
top-down projects, but perhaps well-considered smaller ones.
“We need to identify the minimal interventions that can lead to the
greatest gains,” Batty says. “Complexity theory teaches us that things
are a good deal more complex than we think, and when we interfere, it
can be at our peril.”
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