# Mathematics in Nature: Modeling Patterns in the Natural World

Pages: 416
https://www.jstor.org/stable/j.ctt7rkcn

1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Preface The motivation for the book; Acknowledgments; Credits
(pp. xiii-xix)
4. PROLOGUE Why I Might Never Have Written This Book
(pp. xx-xxiv)

At about eleven years of age, I developed a passion for astronomy. I read everything about it I could get my hands on. My parents were very supportive, but my father, being a farmworker, had a very meager weekly income, about $30 per week at that time, so not a lot of money was available to support my astronomy habit. However, they did have a small insurance policy on my life, which they cashed in for about the equivalent of$50, as I recall. With this I bought a beautiful but somewhat dented old brass-tubed telescope: a 3-inch refractor (with...

5. CHAPTER ONE The Confluence of Nature and Mathematical Modeling
(pp. 1-16)

In recent years, as I have walked daily to and from work, I have started to train myself to observe the sky, the birds, butterflies, trees, and flowers, something I had not done previously in a conscious way (although I did watch out for fast-moving cars and unfriendly dogs). Despite living in suburbia, I find that there are many wonderful things to see: clouds exhibiting wave-like patterns, splotches of colored light some 22 degrees away from the sun (sundogs, or parhelia), wave after wave of Canada geese in “vee” formation, the way waves (and a following region of calm water)...

6. CHAPTER TWO Estimation: The Power of Arithmetic in Solving Fermi Problems
(pp. 17-30)

The second quotation makes much the same point as that by John Tukey over two millennia later (see chapter 1). We may not be as erudite as Aristotle or as brilliant as Enrico Fermi, but we can learn to apply elementary reasoning to obtain “ballpark estimates” to problems (subsequently namedFermi problems) in the manner attributed to that great physicist. The rationale behind such estimates has been well described by Hans Christian von Baeyer, who wrote inThe Fermi Solution,

Fermi’s intent was to show that although, at the outset, even the answer’s order of magnitude is unknown, one can...

7. CHAPTER THREE Shape, Size, and Similarity: The Problem of Scale
(pp. 31-56)

The phrasedimensional analysiswill be used in two senses in this chapter. The first one is somewhat loosely defined, but essentially it concerns the way that physical characteristics of an object (such as surface area, volume, strength, power) vary with its sizeL. “Size” here means any representative linear dimension of the object—height, width, leg length, and so on—provided that dimension is used consistently in all comparisons made for a particular property. This is particularly useful (and valid) when the objects are geometrically similar, and that assumption will be made in much of this section. It is...

8. CHAPTER FOUR Meteorological Optics I: Shadows, Crepuscular Rays, and Related Optical Phenomena
(pp. 57-79)

Before we can sensibly discuss shadows cast by the sun, we need to know how large a source of light (in an angular sense) the sun appears to be in the sky; it is about half a degree of arc, as will be established below.

It is well known that the planets move around the sun in ellipses with the sun at one focus; this, after all, is Kepler’s first law. The perihelion (= closest) distance${r_p}$of the earth from the sun is about$1.47 \times {10^8}$km (or about$9.14 \times {10^7}$mi); the aphelion (= farthest) distance${r_a}$is approximately\$1.52...

9. CHAPTER FIVE Meteorological Optics II: A “Calculus I” Approach to Rainbows, Halos, and Glories
(pp. 80-117)

How much mathematics shall we get into here? Not a great deal, in fact; for those interested in pursuing the mathematics to a much greater level, refer to the author’s review and the many references therein. But first some background material on the history and elementary physics of the rainbow is in order. A good overview of the problem can be found in the book by Banks (1999); for more historical and scientific details, those by Boyer and Lee and Fraser are highly recommended.

The rainbow is at one and the same time one of the most beautiful visual displays...

10. CHAPTER SIX Clouds, Sand Dunes, and Hurricanes
(pp. 118-138)

In this chapter, much of the “scene” is set for the following two chapters on waves and stability, because clouds are wonderful indicators of what kinds of wave motion or atmospheric instabilities are occurring far above us. Although the mathematical descriptions of some of these phenomena (with the exception of the hurricane) are left to chapters 7 and 8, by the time you reach them you will have been exposed to the underlying physics necessary to formulate such descriptions.

Clouds consist of water (in one form or another) and are high altitude fog; or better, fog is cloud at ground...

11. CHAPTER SEVEN (Linear) Waves of All Kinds
(pp. 139-172)

Wave motion can occur in a wide variety of situations in the natural world, and it is something with which we are all familiar. We can observe waves on the surfaces of oceans and lakes or when a pebble is dropped into a pond; waves are generated and propagated when a musical instrument is played (correctly or otherwise), when a radio station transmits programs, or when a solar flare occurs. In the previous chapter we noted that clouds can often be indicators of wave motion, especially in the presence of wind. Violent disturbances on or below the earth’s surface, such...

12. [Illustrations]
(pp. None)
13. CHAPTER EIGHT Stability
(pp. 173-193)

The second quotation above is from the very first professional presentation I made in fear and trepidation as a first-year graduate student. I remember the occasion as if it were yesterday; I was talking about a particular class of waves that may exist in the atmosphere of the sun and other stars (or perhaps just in my mind), and the above line brought a laugh. I needed it, and so did the audience. Now, about thirty years later, I had intended to use it in the next chapter, which introduces the topic of nonlinear waves, but since this present chapter...

14. CHAPTER NINE Bores and Nonlinear Waves
(pp. 194-212)

What is a bore? The answers will vary depending on whether one is at a cocktail party, the banks of the River Severn in England, or the Bay of Fundy in Nova Scotia (to name but two of many geographical locations). We will focus our attention on tidal bores, which, as David Lynch points out in an article of the same title, are remarkable hydrodynamic phenomena. A tidal bore is the incoming tide in the form of a wave (technically, it is called asolitarywave) moving upstream in a river that empties into the sea. Even after the bore...

15. CHAPTER TEN The Fibonacci Sequence and the Golden Ratio (τ)
(pp. 213-230)

Although originally considered by Fibonacci of Pisa in 1202 in connection with (idealized) rabbit population growth, the infinite set of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . has a wonderful variety of properties and applications (but not as many as some have claimed—the article by Markowsky, discussed below, deals with various misconceptions about the golden ratio τ in connection with many human “constructions”). Before discussing those properties, I should clarify what I mean by the word “idealized” above. The rabbits were supposed to reproduce according to strict rules concerning their...

16. CHAPTER ELEVEN Bees, Honeycombs, Bubbles, and Mud Cracks
(pp. 231-253)

A polygon, as the old joke goes, is not a dead parrot (but then neither do geometers move in the best circles). There is something fascinating about the symmetries of a regular polygon, and when polygons are approximated in nature, such configurations capture the attention of even the most casual observer. When outside, look up, and on occasion you may see hexagonal convection-cell clouds; look around, and you may see mud cracks or salt flats exhibiting the same type of pattern; the Giant’s Causeway on the Antrim coast in Northern Ireland is composed of some 40,000 basalt column formations, most...

17. CHAPTER TWELVE River Meanders, Branching Patterns, and Trees
(pp. 254-294)

As a caveat to some of this chapter (river meanders), and to a lesser extent part of the previous one (mud cracks), it should be reemphasized that, while the mechanisms and principles inherent in these mathematical models may represent part of the “truth” behind the observed patterns, they certainly do not qualify as “the whole truth and nothing but the truth.” Indeed, as discussed at some length in chapter 1, this is the case to a greater or lesser degree for all mathematical models. In any field of scientific endeavor, pragmatically speaking, no model can fully encapsulate all the data...

18. CHAPTER THIRTEEN Bird Flight
(pp. 295-308)

In this chapter we discuss some principles relating to flight in general and bird flight in particular. Some of these principles pertain to the type of dimensional arguments discussed in chapter 3. The first topic to reencounter is that ofwing loading. The power necessary for sustained flight for birds and airplanes is proportional to the wing loading, which is the weight of the bird or airplane divided by the area of the wings. For geometrically similar objects, weight increases as the cube of the length of the bird (or plane), and wing area as the square of the length,...

19. CHAPTER FOURTEEN How Did the Leopard Get Its Spots?
(pp. 309-335)

(. . . or legs, or tail, or toes; or the feathers, or wings in a bird. . . .) Before we can even attempt an answer to this question (or these questions), it is necessary to try to understand something about the phenomenon of diffusion. Denny and Gaines have done an excellent job explaining the physical and mathematical principles behind diffusion, and in considerable detail, but since you are not reading their book at this moment, for completeness I will attempt to explain the ideas more succinctly but somewhat heuristically here.

Diffusion arises because of random molecular motion; generally...

20. APPENDIX Fractals: An Appetite Whetter . . .
(pp. 336-340)
21. BIBLIOGRAPHY
(pp. 341-355)
22. INDEX
(pp. 356-359)