X and the City: Modeling Aspects of Urban Life
Read X and the City: Modeling Aspects of Urban Life Online
Authors: John A. Adam
X
and the
City
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
CONTENTS(See
Chapters 1
and
5
for some estimation questions inspired by these passages.)
Chapter 1
INTRODUCTION:
Cancer, Princess Dido, and the city
Chapter 5
GARDENING IN THE CITY
Chapter 7
NOT
DRIVING IN THE CITY!
Chapter 9
PROBABILITY 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 19
AIR POLLUTION 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 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
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 = t
c
” and “
X
=
N
tot
” 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 would be other city-related applications of mathematics in this book.