X and the City

X and the City: Modeling Aspects of Urban Life

John A. Adam
Copyright Date: 2012
Pages: 304
https://www.jstor.org/stable/j.ctt7s3jg
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  • Book Info
    X and the City
    Book Description:

    X and the City, a book of diverse and accessible math-based topics, uses basic modeling to explore a wide range of entertaining questions about urban life. How do you estimate the number of dental or doctor's offices, gas stations, restaurants, or movie theaters in a city of a given size? How can mathematics be used to maximize traffic flow through tunnels? Can you predict whether a traffic light will stay green long enough for you to cross the intersection? And what is the likelihood that your city will be hit by an asteroid?

    Every math problem and equation in this book tells a story and examples are explained throughout in an informal and witty style. The level of mathematics ranges from precalculus through calculus to some differential equations, and any reader with knowledge of elementary calculus will be able to follow the materials with ease. There are also some more challenging problems sprinkled in for the more advanced reader.

    Filled with interesting and unusual observations about how cities work,X and the Cityshows how mathematics undergirds and plays an important part in the metropolitan landscape.

    eISBN: 978-1-4008-4169-1
    Subjects: Mathematics, Statistics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-xii)
  3. PREFACE
    (pp. xiii-xvi)
  4. ACKNOWLEDGMENTS
    (pp. xvii-xx)
  5. Chapter 1 INTRODUCTION: Cancer, Princess Dido, and the city
    (pp. 1-6)

    Although this question was briefly addressed in the Preface, it should be noted that the answer really depends on whom you ask and when you asked the question. Perhaps ten or twelve thousand years ago, when human society changed from a nomadic to a more settled, agriculturally based form, cities started to develop, centered on the Euphrates and Tigris Rivers in ancient Babylon. It can be argued that two hundred years ago, or even less, “planned” cities were constructed with predominately aesthetic reasons—architecture—in mind.

    Perhaps it was believed that form precedes (and determines) function; nevertheless, in the twentieth...

  6. Chapter 2 GETTING TO THE CITY
    (pp. 7-14)

    Before we can explore mathematics in the city, we need to get there. The fastest way is to fly, so let’s hop on a plane. If we assume that the descent flight path occurs in the same vertical plane throughout (i.e., no circling before touchdown—which is very rare these days), then the path is quite well represented by a suitably chosen cubic function [5], as we will see below. Here are some further conditions we impose prior to setting up a mathematical model for the path:

    The horizontal airspeeddx/dt = Uis constant throughout the descent. This is...

  7. Chapter 3 LIVING IN THE CITY
    (pp. 15-34)

    Suppose the city population isNpmillion, and we wish to estimateN, the number of facilities, (dental offices, gas stations, restaurants, movie theaters, places of worship, etc.) in a city of that size. Furthermore, suppose that the average “rate per person” (visits per year, or per week, depending on context) isR, and that the facility is open on averageHhours/day and caters for an average ofCcustomers per hour. We shall also suppose there areDdays per year. This may seem a little surprising at first sight: surelyeveryoneknows thatD= 365 when...

  8. Chapter 4 EATING IN THE CITY
    (pp. 35-40)

    The first time I visited a really big city on more than a day trip, I wondered where people found groceries. I had just left home to start my time as an undergraduate in London, but at home I could either ride my bicycle or use my father’s car to drive to the village store or visit my friends (not always in that order). In London I used the “tube”—the underground transport system (the subway) and sometimes the bus, but I walked pretty much everywhere else. I soon found out where to get groceries, and in retrospect realized that...

  9. Chapter 5 GARDENING IN THE CITY
    (pp. 41-46)

    When I was growing up, I loved to visit my grandfather. Despite living in a city (or at least, a very large town) he was able to cultivate and maintain quite a large garden, containing many beautiful plants and flowers. In fact, whenever I asked what any particular flower was called, his reply was always the same: “Ericaceliapopolifolium!” I never did find out whether he was merely humoring me, or whether he didn’t know! In addition to his garden, he rented a smaller strip of land (an “allotment”) farther up the road where, along with several others, he grew potatoes,...

  10. Chapter 6 SUMMER IN THE CITY
    (pp. 47-62)

    Question: How many squirrels live in Central Park?

    Central Park in New York City runs from 59th Street to 110th Street [6]. At 20 blocks per mile, this is 2.5 miles. Central Park is long and narrow, so we will estimate its width at about 0.5 mile. This gives an area of about 1 square mile or about 2 square kilometers.

    It’s difficult to estimate the number of squirrels in that large an area, so let’s break it down and think of the area of an (American) football field (about 50 yards X 100 yards). There will be more than...

  11. Chapter 7 NOT DRIVING IN THE CITY!
    (pp. 63-72)

    As we have remarked already, cities come in many shapes and sizes. In many large cities such as London and New York, the public transportation system is so good that one can get easily from almost anywhere to anywhere else in the city without using a car. Indeed, under these circumstances a car can be something of an encumbrance, especially if one lives in a restricted parking zone. So for this chapter we’ll travel by bus, subway, train, or quite possibly, rickshaw. Whichever we use, the discussion will be kept quite general. But first we examine a situation that can...

  12. Chapter 8 DRIVING IN THE CITY
    (pp. 73-88)

    As noted in the previous chapter, in many very large cities a car is not needed at all. But there also are many cities and towns where it is essential to have a vehicle. One advantage of driving over public transportation can be that one does not have to keep stopping at intermediate locations on the way to one’s destination (traffic permitting). Of course, the daily commute can be extremely frustrating when it is of the stop-and-start variety, and the gas consumption becomes prohibitive. At the time of writing the price of petrol in the UK is far higher (by...

  13. Chapter 9 PROBABILITY IN THE CITY
    (pp. 89-96)

    There are many topics under the umbrella “road traffic” that are amenable to mathematical investigation. Examples include traffic flow on the open road and at intersections, parking problems, accident rates, design of road systems for new towns and expanding cities, traffic lights and other control systems, transportation and scheduling problems, to name but a few. And as in so many aspects of mathematical modeling, there are two basic choices: to define the problem in adeterministicor aprobabilisticcontext. The former can subdivide farther into continuum or discrete (or equivalently, macroscopic or microscopic) approaches, each with its own advantages...

  14. Chapter 10 TRAFFIC IN THE CITY
    (pp. 97-106)

    There is a fundamental relationship between the flow of trafficqin vehicles per unit time, the concentrationkin vehicles per unit distance, and the speeduof the traffic. It isq = ku. In general each of these quantities is a function of distance (x) and time (t), but the formq(k)=ku(k) may also be valuable. Another useful quantity is the spacing per vehicle,s=k-1. Ifs0is the minimum possible spacing, that is, when the vehicles are stationary (or almost so), then$k_j = s_0^{ - 1} $is referred to as thejam concentration; but it has nothing to do...

  15. Chapter 11 CAR FOLLOWING IN THE CITY—I
    (pp. 107-112)

    Don’t some cars inevitably follow others, and not just in the city? They certainly do, but the phrase as used here means that we model traffic by identifying each car as a separate object, not just part of the flow of a fluid called “traffic.” We’ll start by setting up a particular type of differential equation for this (now) discrete system.

    Suppose that the position of thenth car on the road isxn(t). If we disallow passing in this model we can assume that the motion of any car depends only on that of the car ahead. A simple...

  16. Chapter 12 CAR FOLLOWING IN THE CITY—II
    (pp. 113-120)

    I love to watch clouds; their changing forms are indicative of the different kinds of hydrodynamical process that are present in the upper atmosphere, such as convection, shear flow, and turbulence. Unfortunately, I am rather prone to do this while driving. Probably the worst example of this occurred many years ago when my wife and I were on our way to the local hospital (she was in labor with our third child). I won’t elaborate here, except to say that she rightly urged me to concentrate on the road. Distractions such as cloud-watching while driving increase the reaction time for...

  17. Chapter 13 CONGESTION IN THE CITY
    (pp. 121-128)

    What percentage of our (waking) time do we spend driving? In the United States, a typical drive to and from work may be at least half an hour each way, frequently more in high density metropolitan areas. So for an 8-hr working day, the drive adds at least 12–13% to that time, during much of which drivers may become extremely frustrated. (Confession: I am not one of those people; I am fortunate enough to walk to work!)

    On 25 January 2011 my local newspaper carried an article entitled “Slow Motion.” The article took data from the American Community Survey...

  18. Chapter 14 ROADS IN THE CITY
    (pp. 129-134)

    This can be quite a complicated quantity to calculate, depending as it does on the type of road network and distribution of starting points and destinations, among other factors. Smeed (1968) assumed a uniform distribution of origins and destinations for both idealized and real UK road networks, and with some simplifying assumptions, concluded that the range of values lay between 0.70A1/2and 1.07A1/2, with a mean of 0.87A1/2, A being the area of the town center (assuming this can be suitably defined). Obviously the factorA1/2renders the result dimensionally correct. With this behind us, the next stage is to...

  19. Chapter 15 SEX AND THE CITY
    (pp. 135-148)

    We can think about the growth of cities in several ways, none of them prurient, despite the title of this chapter. Perhaps the most obvious one is how the population changes over time; another is how the civic area changes; yet another might be the number of businesses or companies in the city; and so on. These statistical properties are generally referred to asdemographics, and they can include gender, race, age, population density, homeownership, and employment status, to name but a few. In this chapter we shall focus attention on some the simplest possible models of population growth in...

  20. Chapter 16 GROWTH AND THE CITY
    (pp. 149-158)

    In light of the above comments, it is somewhat surprising that fairly general quantitative patterns of urban population densityρ(population per unit area) can nevertheless be formulated for single-center cities. As far back as 1951 Colin Clark, a statistician, compiled such data for 20 cities, and found thatρdeclined approximately exponentially with distance from city centers (though “city center” is not always easily defined in practice;central business districtis perhaps a better term). If we restrict ourselves to the special but important case of circular symmetry, whereinρ=ρ(r),rbeing the radial coordinate, then naturally we expect...

  21. Chapter 17 THE AXIOMATIC CITY
    (pp. 159-166)

    In this chapter we try to be a little more formal by defining axioms for an equilibrium model of acircularcity. By using the word “equilibrium” here, we mean that the “forces” attracting people to live in the city are balanced at every point by the “forces” that “repel” them. Of course, there are no forces in the physical sense of that word, but by analogy with the balance between gravity and pressure gradients in stars it is possible to suggest certain forms of “coercion” that persuade individuals to live exactly where they do. Let’s get started.

    We make...

  22. Chapter 18 SCALING IN THE CITY
    (pp. 167-178)

    No, by “scaling in the city” we don’t mean what Spiderman does in his various movie adventures . . . There is, according to Brakman et al. (2009, available in Oxford’s online resource center),

    a remarkable regularity in the distribution of city sizes all over the world, also known as the “Rank-Size Distribution.” Take, for example, Amsterdam, the largest city in the Netherlands and give it rank number 1. Then take the second largest city, Rotterdam, and give it rank number 2. Keep on doing this for those cities for which you have data available, possibly selecting only cities exceeding...

  23. Chapter 19 AIR POLLUTION IN THE CITY
    (pp. 179-190)

    I spent several years as a student living in London, but fortunately I never had to experience something that plagued the city in the first half of the twentieth century: smog ( = smoke + fog). The last major occurrence of London smog was in 1952, and while estimates vary, it is thought that as many as 12,000 people died in the weeks and months following the outbreak. Basically, smog is caused by the chemical reaction of sunlight with chemicals in the atmosphere.

    But more generally, what is air pollution? Essentially, it is the presence of substances in the atmosphere...

  24. Chapter 20 LIGHT IN THE CITY
    (pp. 191-208)

    With such particles suspended in the atmosphere for sometimes days or weeks at a time, smog presents a danger to health, but in London it was also known as a “pea souper” because one could not see one’s hand in front of one’s face! In fact, as a result of the Great London Smog of 1952 (caused by the smoke from millions of chimneys combined with the mists and fogs of the Thames valley), the Clean Air Act of 1956 was enacted. With this in mind we now turn to the topic of how air pollution may affectvisibility.

    We...

  25. Chapter 21 NIGHTTIME IN THE CITY—I
    (pp. 209-220)

    What is the difference between night and day in the city? That may appear to be a silly question, but humor me here. During the day, the sun provides the light we need, outdoors at least. Artificial lighting may be necessary inside a building, depending on the number and location (or even existence) of windows. In the country at night, the only sources of light are the moon, planets, and stars, when they are visible, barring the occasional encounter with a UFO. By contrast, most of the heavenly bodies are not seen in the city (with the possible exception of...

  26. Chapter 22 NIGHTTIME IN THE CITY—II
    (pp. 221-232)

    In a city at night, owing to artificial light sources, we may witness substantial modifications to light pillars, ice crystal halos, and rainbows when the weather conditions are right for producing them. The sources of light are much nearer than the sun (or moon) and thus the light from them is divergent, not parallel. This can give rise to some quite amazing three dimensional “surfaces” that differ significantly from their daytime (parallel light) counterparts. In order to appreciate this, some geometry associated with surfaces of revolution will be required. But before getting into that, let’s ask a much more basic...

  27. Chapter 23 LIGHTHOUSES IN THE CITY?
    (pp. 233-246)

    Lighthouses in a town or city, you ask? Certainly; there are over a thousand in the United States (though many of them are no longer in use), and Michigan has the most lights of any state, with more than 150 past and present lights. A state-by-state listing of all U.S. lighthouses may readily be found online, as well as listings for lighthouses in Europe and elsewhere. Many of these are close to or within city boundaries. London’s only lighthouse (no longer functioning as such) is located at Trinity Quay Wharf in London’s docklands. By contrast, New York City has several....

  28. Chapter 24 DISASTER IN THE CITY?
    (pp. 247-254)

    Hollywood is very fond of making disaster movies about volcanic eruptions, earthquakes, meteoric impacts, and alien invasions, among other threats to us Earthlings. In July 1994 there was an alien invasion of sorts—on the planet Jupiter. The following news flash [41] can be found on California Institute of Technology’s Jet Propulsion Laboratory ( JPL) website:

    From July 16 through July 22, 1994, pieces of an object designated as Comet P/Shoemaker-Levy 9 collided with Jupiter. This is the first collision of two solar system bodies ever to be observed, and the effects of the comet impacts on Jupiter’s atmosphere have...

  29. Chapter 25 GETTING AWAY FROM THE CITY
    (pp. 255-260)

    Why would one wish to get out of the city, at least for a time? To escape the possible impending doom discussed in the previous chapter? There are many other reasons why we might wish to take a break from city life, such as dissatisfaction with continued traffic congestion and the hectic pace of life, or just a desire to experience nature in all its variety. My favorite seasons are spring and fall (autumn). The season of autumn can be one of great beauty, especially where the foliage changes to a bright variety of reds, oranges, and yellows.

    It’s a...

  30. Appendix 1 THEOREMS FOR PRINCESS DIDO
    (pp. 261-262)
  31. Appendix 2 PRINCESS DIDO AND THE SINC FUNCTION
    (pp. 263-268)
  32. Appendix 3 TAXICAB GEOMETRY
    (pp. 269-272)
  33. Appendix 4 THE POISSON DISTRIBUTION
    (pp. 273-276)
  34. Appendix 5 THE METHOD OF LAGRANGE MULTIPLIERS
    (pp. 277-278)
  35. Appendix 6 A SPIRAL BRAKING PATH
    (pp. 279-280)
  36. Appendix 7 THE AVERAGE DISTANCE BETWEEN TWO RANDOM POINTS IN A CIRCLE
    (pp. 281-282)
  37. Appendix 8 INFORMAL “DERIVATION” OF THE LOGISTIC DIFFERENTIAL EQUATION
    (pp. 283-286)
  38. Appendix 9 A MINISCULE INTRODUCTION TO FRACTALS
    (pp. 287-290)
  39. Appendix 10 RANDOM WALKS AND THE DIFFUSION EQUATION
    (pp. 291-296)
  40. Appendix 11 RAINBOW/HALO DETAILS
    (pp. 297-302)
  41. Appendix 12 THE EARTH AS VACUUM CLEANER?
    (pp. 303-308)
  42. ANNOTATED REFERENCES AND NOTES
    (pp. 309-316)
  43. INDEX
    (pp. 317-319)