John A. Adam - X and the City: Modeling Aspects of Urban Life

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 townsMathematical models. I. Title.
HT151 .A288 2012
307.7601'5118dc23 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 for some estimation questions inspired by these passages.)

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 CITYI

Chapter 12
CAR FOLLOWING IN THE CITYII

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 CITYI

Chapter 22
NIGHTTIME IN THE CITYII

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?

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 didnt 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 cityLondonand 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 cityafter 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 would be other city-related applications of mathematics in this book.