X and the City: Modeling Aspects of Urban Life (2 page)

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.

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

Cancer, Princess Dido, and the city

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 transportation 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.

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,

The term “de-differentiation” means the regression of a specialized cell or tissue to a simpler unspecialized form. There
is
an interesting mathematical link that connects such malignancies with city growth—the
fractal dimension
. This topic will be mentioned in
Chapter 18
, and more details will be found in
Appendix 9
. For now, a few aspects of this analogy will be noted. The degree of border irregularity of a malignant melanoma, for example, is generally much higher than that for a benign lesion (and it carries over to the cellular level also). This is an important clinical feature in the diagnosis of such lesions, and it is perhaps not surprising that city “boundaries” and skylines are also highly irregular (in the sense that their fractal dimension is between one and two). This of course is not to suggest that the city is a “cancer” (though some might disagree), but it does often possess the four characteristics mentioned above
for malignant neoplasms. The question to be answered is whether this analogy is useful, and in what sense. We shall not return to this question, interesting though it is, instead we will end this chapter by examining a decidedly nonfractal city boundary!