X
and the City
X and the City
MODELING ASPECTS OF URBAN LIFE
John A. Adam
Copyright © 2012 by Princeton University Press
Published by Princeton University Press, 41 William Street, Princeton,
New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock,
Oxfordshire OX20 1TW
press.princeton.edu
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Adam, John A.
X and the city : modeling aspects of urban life / John Adam.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-691-15464-0 1. Mathematical models. 2. City and town life—
Mathematical models. 3. Cities and towns—Mathematical models. I. Title.
HT151 .A288 2012
307.7601'5118—dc23 2012006113
British Library Cataloging-in-Publication Data is available
This book has been composed in Garamond
Book design by Marcella Engel Roberts
Printed on acid-free paper. ∞
Printed in the United States of America
1 3 5 7 9 10 8 6 4 2
For Matthew, who drifted “continentally” to a large city
He found out that the city was as wide as it was long and it was as high as it was wide. It was as long as a man could walk in fifty days . . . In the middle of the street of the city and on either bank of the river grew the tree of life, bearing twelve fruits, a different kind for each month. The leaves of the tree were for the healing of the nations.
—St. John of Patmos
(See Chapters 1 and 5 for some estimation questions inspired by these passages.)
CONTENTS
Preface
Acknowledgments
Chapter 1
INTRODUCTION: Cancer, Princess Dido, and the city
Chapter 2
GETTING TO THE CITY
Chapter 3
LIVING IN THE CITY
Chapter 4
EATING IN THE CITY
Chapter 5
GARDENING IN THE CITY
Chapter 6
SUMMER IN THE CITY
Chapter 7
NOT DRIVING IN THE CITY!
Chapter 8
DRIVING IN THE CITY
Chapter 9
PROBABILITY IN THE CITY
Chapter 10
TRAFFIC IN THE CITY
Chapter 11
CAR FOLLOWING IN THE CITY–I
Chapter 12
CAR FOLLOWING IN THE CITY–II
Chapter 13
CONGESTION IN THE CITY
Chapter 14
ROADS IN THE CITY
Chapter 15
SEX AND THE CITY
Chapter 16
GROWTH AND THE CITY
Chapter 17
THE AXIOMATIC CITY
Chapter 18
SCALING IN THE CITY
Chapter 19
AIR POLLUTION IN THE CITY
Chapter 20
LIGHT IN THE CITY
Chapter 21
NIGHTTIME IN THE CITY–I
Chapter 22
NIGHTTIME IN THE CITY–II
Chapter 23
LIGHTHOUSES IN THE CITY?
Chapter 24
DISASTER IN THE CITY?
Chapter 25
GETTING AWAY FROM THE CITY
Appendix 1
THEOREMS FOR PRINCESS DIDO
Appendix 2
DIDO AND THE SINC FUNCTION
Appendix 3
TAXICAB GEOMETRY
Appendix 4
THE POISSON DISTRIBUTION
Appendix 5
THE METHOD OF LAGRANGE MULTIPLIERS
Appendix 6
A SPIRAL BRAKING PATH
Appendix 7
THE AVERAGE DISTANCE BETWEEN TWO RANDOM POINTS IN A CIRCLE
Appendix 8
INFORMAL “DERIVATION” OF THE LOGISTIC DIFFERENTIAL EQUATION
Appendix 9
A MINISCULE INTRODUCTION TO FRACTALS
Appendix 10
RANDOM WALKS AND THE DIFFUSION EQUATION
Appendix 11
RAINBOW/HALO DETAILS
Appendix 12
THE EARTH AS VACUUM CLEANER?
Annotated references and notes
Index
PREFACE
After the publication of A Mathematical Nature Walk, my editor, Vickie Kearn, suggested I think about writing A Mathematical City Walk. My first reaction was somewhat negative, as I am a “country boy” at heart, and have always been more interested in modeling natural patterns in the world around us than man-made ones. Nevertheless, the idea grew on me, especially since I realized that many of my favorite nature topics, such as rainbows and ice crystal halos, can have (under the right circumstances) very different manifestations in the city. Why would this be? Without wishing to give the game away too early into the book, it has to do with the differences between nearly parallel “rays” of light from the sun, and divergent rays of light from nearby light sources at night, of which more anon. But I didn’t want to describe this and the rest of the material in terms of a city walk; instead I chose to couch things with an “in the city” motif, and this allowed me to touch on a rather wide variety of topics that would have otherwise been excluded. (There are seven chapters having to do with traffic in one way or another!)
As a student, I lived in a large city—London—and enjoyed it well enough, though we should try to identify what is meant by the word “city.” Several related dictionary definitions can be found, but they vary depending on the country in which one lives. For the purposes of this book, a city is a large, permanent settlement of people, with the infrastructure that is necessary to make that possible. Of course, the terms “large” and “permanent” are relative, and therefore we may reasonably include towns as well as cities and add the phrase “or developing” to “permanent” in the above definition. In the Introduction we will endeavor to expand somewhat on this definition from a historical perspective.
This book is an eclectic collection of topics ranging across city-related material, from day-to-day living in a city, traveling in a city by rail, bus, and car (the latter two with their concomitant traffic flow problems), population growth in cities, pollution and its consequences, to unusual night time optical effects in the presence of artificial sources of light, among many other topics. Our cities may be on the coast or in the heartland of the country, or on another continent, but presumably always located on planet Earth. Inevitably, some of the topics are multivalued; not everything discussed here is unique to the city—after all, people eat, garden, and travel in the country as well!
Why X and the City? In the popular culture, the letter X (or x) is an archetype of mathematical problem solving: “Find x.” The X in the book title is used to introduce the topic in each subsection; thus “X = tc” and “X = Ntot” refer, respectively, to a specific length of time and a total population, thereby succinctly introducing the mathematical topics that follow. One of the joys of studying and applying mathematics (and finding x), regardless of level, is the fact that the deeper one goes into a topic, the more avenues one finds to go down. I have found this to be no less the case in researching and writing this book. There were many twists and turns along the way, and naturally I made choices of topics to include and exclude. Another author would in all certainty have made different choices. Ten years ago (or ten years from now), the same would probably be true for me, and there wou
ld be other city-related applications of mathematics in this book.
Mathematics is a language, and an exceedingly beautiful one, and the applications of that language are vast and extensive. However, pure mathematics and applied mathematics are very different in both structure and purpose, and this is even more true when it comes to that subset of applied mathematics known as mathematical modeling (of which more below). I love the beauty and elegance in mathematics, but it is not always possible to find it outside the “pure” realm. It should be emphasized that the subjects are complementary and certainly not in opposition, despite some who might hold that opinion. I heard of one mathematician who referred to applied mathematics as “mere engineering”; this should be contrasted with the view of the late Sir James Lighthill, one of the foremost British applied mathematicians of the twentieth century. He wrote, somewhat tongue-in-cheek, that pure mathematics was a very important part of applied mathematics!
Applied mathematics is often elegant, to be sure, and when done well it is invariably useful. I hope that the types of problem considered in this book can be both fun and “applied.” And while some of the chapters in the middle of the book might be described as “traffic engineering,” it is the case that mathematics is the basis for all types of quantitative thought, whether theoretical or applied. For those who prefer a more rigorous approach, I have also included Chapter 17, entitled “The axiomatic city.” In that chapter, some of the exercises require proofs of certain statements, though I have intentionally avoided referring to the latter as “theorems.”
The subtitle of this book is Modeling Aspects of Urban Life. It is therefore reasonable to ask: what is (mathematical) modeling? Fundamentally, mathematical modeling is the formulation in mathematical terms of the assumptions (and their logical consequences) believed to underlie a particular “real world” problem. The aim is the practical application of mathematics to help unravel the underlying mechanisms involved in, for example, industrial, economic, physical, and biological or other systems and processes. The fundamental steps necessary in developing a mathematical model are threefold: (i) to formulate the problem in mathematical terms (using whatever appropriate simplifying assumptions may be necessary); (ii) to solve the problem thus posed, or at least extract sufficient information from it; and finally (iii) to interpret the solution in the context of the original problem. This may include validation of the model by testing both its consistency with known data and its predictive capability.
At its heart, then, this book is about just that: mathematical modeling, from “applied” arithmetic to linear (and occasionally nonlinear) ordinary differential equations. As a little more of a challenge, there are a few partial differential equations thrown in for good measure. Nevertheless, the vast majority of the material is accessible to anyone with a background up to and including basic calculus. I hope that the reader will enjoy the interplay between estimation, discrete and continuum modeling, probability, Newtonian mechanics, mathematical physics (diffusion, scattering of light), geometric optics, projective and three-dimensional geometry, and quite a bit more.
Many of the topics in the book are posed in the form of questions. I have tried to make it as self-contained as possible, and this is the reason there are several Appendices. They comprise a compendium of unusual results perhaps (in some cases) difficult to find elsewhere. Some amplify or extend material discussed in the main body of the book; others are indirectly related, but nevertheless connected to the underlying theme. There are also exercises scattered throughout; they are for the interested reader to flex his or her calculus muscles by verifying or extending results stated in the text. The combination of so many topics provides many opportunities for mathematical modeling at different levels of complexity and sophistication. Sometimes several complementary levels of description are possible when developing a mathematical model; in particular this is readily illustrated by the different types of traffic flow model presented in Chapters 8 through 13.
In writing this book I have studied many articles both online and in the literature. Notes identifying the authors of these articles, denoted by numbers in square brackets in the text, can be found in the references. A more general set of useful citations is also provided.
ACKNOWLEDGMENTS
Thanks to the following for permissions:
Achim Christopher (Figures 23.1 and 23.10)
Christian Fenn (Figure 22.5)
Skip Moen (Figures 3.4 and 21.1)
Martin Lowson and Jan Mattsson, for email conversations about their work cited here.
Larry Weinstein for valuable feedback on parts of the manuscript.
Alexander Haußmann for very helpful comments on Chapter 22.
Bonita Williams-Chambers for help with Figure 10.2 and Table 15.1.
My thanks go to Kathleen Cioffi, who oversaw the whole process in an efficient and timely manner. I am most appreciative of the excellent work done by the artist, Shane Kelley, who took the less-than-clear figure files I submitted and made silk purses out of sows’ ears! Many thanks also to the book designer, Marcella Engels Roberts, for finding the illustrations for the chapter openers and designing the book. Any remaining errors of labeling (or of any other type, for that matter) are of course my own.
I thank my department Chair, J. Mark Dorrepaal, for arranging my teaching schedule so that this book could be written in a timely fashion (and my graduate students could still be advised!).
As always, I would like to express my gratitude to my editor, Vickie Kearn. Her unhurried yet efficient style of “author management” s(m)oothes ruffled feathers and encourages the temporarily crestfallen writer. She has great insight into what I try to write, and how to do it better, and her advice is always invaluable. And I hope she enjoys the story about my grandfather!
Finally, I want to thank my family for their constant support and encouragement, and without whom this book might have been finished a lot earlier. But it wouldn’t have been nearly as much fun to write!
X
and the City
Chapter 1
INTRODUCTION
Cancer, Princess Dido, and the city
To look at the cross-section of any plan of a big city is to look at something like the section of a fibrous tumor.
—Frank Lloyd Wright
X = ?: WHAT ARE CITIES?
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 century more and more emphasis was placed on economic structure and organizational efficiency. A precursor to these ideas was published in 1889 as a book entitled City Planning, According to Artistic Principles, written by Camillo Sitte (it has since been reprinted). A further example of this approach from a historical and geographical perspective, much nearer our own time, is Helen Rosenau’s The Ideal City: Its Architectural Evolution in Europe (1983). But there is a distinction to be made between those which grow “naturally” (or organically) and those which are “artificial” (or planned). These are not mutually exclusive categories in practice, of course, and many cities and towns have features of both. Nevertheless there are significant differences in the way such cities grow and develop: differences in rates of growth and scale. Naturally growing cities have a slower rate of development than planned cities, and tend to be composed of smaller-scale units as opposed to the larger scale envisioned by city planners.
“Organic” towns, in plan form, resemble cell growth, spider webs, and tree-like forms, depending on the landscape, main transportatio
n routes, and centers of activity. Their geometry tends to be irregular, in contrast to the straight “Roman road” and Cartesian block structure and circular arcs incorporated in so many planned cities, from Babylonian times to the present [1]. Some of the material in this book utilizes these simple geometric ideas, and as such, represents only the simplest of city models, by way of analogies and even metaphors.
ANALOGIES AND METAPHORS
Was Frank Lloyd Wright correct—do city plans often look like tumor cross sections writ large? Perhaps so, but the purpose of that quote was to inform the reader of a common feature in modeling. Mathematical models usually (if not always) approach the topic of interest using idealizations, but also sometimes using analogies and metaphors. The models discussed in this book are no exception. Although cities and the transportation networks within them (e.g., rail lines, roads, bus routes) are rarely laid out in a precise geometric grid-like fashion, such models can be valuable. The directions in which a city expands are determined to a great extent by the surrounding topography—rivers, mountains, cliffs, and coastlines are typically hindrances to urban growth. Cities are not circular, with radially symmetric population distributions, but even such gross idealizations have merit. The use of analogies in the mathematical sciences is well established [2], though by definition they have their limitations. Examples include Rutherford’s analogy between the hydrogen atom and the solar system, blood flow in an artery being likened to the flow of water in a pipe, and the related (and often criticized) hydraulic analogy to illustrate Ohm’s law in an electric circuit.
Analogy is often used to help provide insight by comparing an unknown subject to one that is more familiar. It can also show a relationship between pairs of things, and can help us to think intuitively about a problem. The opening quote by Frank Lloyd Wright is such an example (though it could be argued that it is more of a simile than an analogy). One possibly disturbing analogy is that put forward by W.M. Hern in the anthropological literature [3], suggesting that urban growth resembles that of malignant neoplasms. A neoplasm is an abnormal mass of tissue, and in particular can be identified with a malignant tumor (though this need not be the case). To quote from the abstract of the article,
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