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Math Fun in the Urban Jungle – A Tale of More than Two Cities


Glen Whitney stands at a point on the surface of the Earth, north latitude 40.742087, west longitude 73.988242, which is near the center of Madison Square Park, in New York City. Behind him is the city’s newest museum, the Museum of Mathematics, which Whitney, a former Wall Street trader, founded and now runs as executive director. He is facing one of New York’s landmarks, the Flatiron Building, which got its name because its wedge- like shape reminded people of a clothes iron. Whitney observes that from this perspective you can’t tell that the building, following the shape of its block, is actually a right triangle—a shape that would be useless for pressing clothes—although the models sold in souvenir shops represent it in idealized form as an isosceles, with equal angles at the base. People want to see things as symmetrical, he muses. He points to the building’s narrow prow, whose outline corresponds to the acute angle at which Broadway crosses Fifth Avenue…

The cross street here is 23rd Street,” Glen Whitney says, “and if you measure the angle at the building’s point, it is close to 23 degrees, which also happens to be approximately the angle of inclination of the Earth’s axis of rotation.”

That’s remarkable,” he is told.

Not really. It’s coincidence.” He adds that, twice each year, a few weeks on either side of the summer solstice, the setting sun shines directly down the rows of Manhattan’s numbered streets, a phenomenon sometimes called “Manhattanhenge.” Those particular dates don’t have any special significance, either, except as one more example of how the very bricks and stones of the city illustrate the principles of the highest product of the human intellect, which is math.

Cities are particular: They are shaped by their histories and accidents of geography and climate. Thus the “east-west” streets of Midtown Manhattan actually run northwest-southeast, to meet the Hudson and East rivers at roughly 90 degrees, whereas in Chicago the street grid aligns closely with true north, while medieval cities such as London don’t have right-angled grids. But cities are also, at a deep level, universal: the products of social, economic and physical principles that transcend space and time. A new science—so new it doesn’t have its own journal, or even an agreed-upon name—is exploring these laws. We will call it “quantitative urbanism.” It’s an effort to reduce to mathematical formulas the chaotic, exuberant, extravagant nature of one of humanity’s oldest and most important inventions, the city.

The systematic study of cities dates back at least to the Greek historian Herodotus. In the early 20th century, scientific disciplines emerged around specific aspects of urban development: zoning theory, public health and sanitation, transit and traffic engineering. By the 1960s, the urban-planning writers Jane Jacobs and William H. Whyte used New York as their laboratory to study the street life of neighborhoods, the walking patterns of Midtown pedestrians, the way people gathered and sat in open spaces. But their judgments were generally aesthetic and intuitive … “They had fascinating ideas,” says Luís Bettencourt, a researcher at the Santa Fe Institute, a think tank better known for its contributions to theoretical physics, “but where is the science?” Bettencourt, a physicist, practices a discipline that shares a deep affinity with quantitative urbanism. Both require understanding complex interactions among large numbers of entities: the 20 million people in the New York metropolitan area, or the countless subatomic particles in a nuclear reaction.

The birth of this new field can be dated to 2003, when researchers at SFI convened a workshop on ways to “model”—in the scientific sense of reducing to equations—aspects of human society. One of the leaders was Geoffrey West, who was also a theoretical physicist, but had strayed into biology, exploring how the properties of organisms relate to their mass.
An elephant is not just a bigger version of a mouse, but many of its measurable characteristics, such as metabolism and life span, are governed by mathematical laws that apply all up and down the scale of sizes. The bigger the animal, the longer but the slower it lives: A mouse heart rate is around 500 beats per minute; an elephant’s pulse is 28. If you plotted those points on a logarithmic graph, comparing size with pulse, every mammal would fall on or near the same line. West suggested that the same principles might be at work in human institutions.

Out of that meeting emerged a collaboration that produced the seminal paper in the field: “Growth, Innovation, Scaling, and the Pace of Life in Cities.” In six pages dense with equations and graphs, West, Lobo and Bettencourt, along with two researchers from the Dresden University of Technology, laid out a theory about how cities vary according to size.

The relationship is captured by an equation in which a given parameter—employment, say—varies exponentially with population. In some cases, the exponent (power) is 1, meaning whatever is being measured increases linearly, at the same rate as population. Household water or electrical use, for example, shows this pattern; as a city grows bigger its residents don’t use their appliances more. Some exponents are greater than 1, a relationship described as “superlinear scaling.” Most measures of economic activity fall into this category; among the highest exponents the scholars found were for “private [research and development] employment,” 1.34; “new patents,” 1.27; and gross domestic product, in a range of 1.13 to 1.26. If the population of a city doubles over time, or comparing one big city with two cities each half the size, gross domestic product more than doubles. Each individual becomes, on average, 15 percent more productive. Bettencourt describes the effect as “slightly magical,” although he and his colleagues are beginning to understand the synergies that make it possible. Physical proximity promotes collaboration and innovation, which is one reason the new CEO of Yahoo recently reversed the company’s policy of letting almost anyone work from home.

Lastly, some measures show an exponent of less than 1, meaning they increase more slowly than population. These are typically measures of infrastructure, characterized by economies of scale that result from increasing size and density. New York doesn’t need four times as many gas stations as Houston, for instance; gas stations scale at 0.77; total surface area of roads, 0.83; and total length of wiring in the electrical grid, 0.87.

Remarkably, this phenomenon applies to cities all over the world, of different sizes, regardless of their particular history, culture or geography. Mumbai is different from Shanghai is different from Houston, obviously, but in relation to their own pasts, and to other cities in India, China or the U.S., they follow these laws. “Give me the size of a city in the United States and I can tell you how many police it has, how many patents, how many AIDS cases,” says West, “just as you can calculate the life span of a mammal from its body mass.

Whitney walks swiftly through Madison Square Park to the Shake Shack, a hamburger stand famous for its food and its lines. He points out the two service windows, one for customers who can be served quickly, the other for more complicated orders. This distinction is supported by a branch of mathematics called queuing theory, whose fundamental principle can be stated as “the shortest aggregate waiting time for all customers is achieved when the person with the shortest expected wait time is served first, provided the guy who wants four hamburgers with different toppings doesn’t go berserk when he keeps getting sent to the back of the line.

At the Times Square subway station, Whitney buys a fare card, in an amount he has calculated to take advantage of the bonus for paying in advance and come out with an even number of rides, with no money left unspent. On the platform, as passengers rush back and forth between trains, he talks about the mathematics of running a transit system … The calculation, simplified, is this:

Multiply the number of people on the express train by the number of seconds they will be kept waiting while it idles in the station. Now estimate how many people on the arriving local will transfer, and multiply that by the average time they will save by taking the express to their destination rather than the local.

This can lead to the potential savings, in person-seconds, for comparison. The principle is the same at any scale, but it is only above a certain size of population that the investment in dual-track subway lines or two-window hamburger stands makes sense. Whitney boards the local, heading downtown toward the (Math) museum.

(from the Smithsonian Magazine, ) Read more here

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NCEA Level 2 Algebra Problem. Using the information given, the shaded area = 9, that is:
y(y-8) = 9 –> y.y – 8y – 9 =0
–> (y-9)(y+1) = 0, therefore y = 9 (can’t have a distance of – 1 for the other solution for y)
Using the top and bottom of the rectangle,
x = (y-8)(y+2) = (9-8)(9+2) = 11
but, the left side = (x-4) = 11-4 = 7, but rhs = y+? = 9+?, which is greater than the value of the opp. side??
[I think that the left had side was a mistake and should have read (x+4)?]

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