A Mathematical Nature Walk

Pages: 288
https://www.jstor.org/stable/j.ctt7sj14

1. Front Matter
(pp. i-viii)
(pp. ix-xiv)
3. Preface
(pp. xv-xviii)
4. Acknowledgments
(pp. xix-xxii)
5. Introduction
(pp. 1-10)

I have long been fascinated by the wonders of nature encountered on a simple walk, even in residential areas, as long as there is a view of the sky, roads lined with trees, and bodies of water within easy walking distance. Such is the case where I live in Norfolk, Virginia, and while most of my nature walks are local in character, I have been fortunate to have had the opportunity to travel farther afield at times, both within the United States and beyond its borders. Many of the topics in this book are consequences of such opportunities. And just...

6. At the beginning . . .
(pp. 11-12)

Q.1:You are looking at a single bright rainbow (the primary bow). Which color is on the top side of the arch?

Q.2: How many colors do you think there are in a typical rainbow?

Q.3: Suppose you see a double rainbow; which color is on the top side of the upper arch (the secondary bow)?

Q.4: Is the region between the two bows typically darker, brighter, or the same as the surrounding sky?

Q.5: How about the region below the primary bow?

Q.6: Have you ever seen anything else closely associated with the primary bow?

Q.7: On a sunny day,...

7. In the “playground”
(pp. 13-23)

In this section, we’ll just attempt some ‘‘applied arithmetic’’ and a little bit of algebra. Even if some of the following estimation problems could be calculated exactly (and most cannot), that is not the point here:mostthings cannot be computed exactly, and frequently all that is necessary is a simple estimate. By breaking down a problem into simpler subproblems, making plausible assumptions and using simple arithmetic, we can genuinely arrive at an order of magnitude ‘‘guesstimate’’ of the answer. Obviously, we may each make somewhat different assumptions, but it is as unlikely that all of mine will beoverestimates...

8. In the garden
(pp. 24-42)

In this section we start to focus more on real-world problems, having ‘‘primed the modeling pump,’’ so to speak. I have learned to notice many things in the garden, while at the same time conveniently missing others, such as the length of the grass . . .

Q.25: Why can I see the ‘‘whole universe’’ in my garden globe? [color plate ]

To see a world in a grain of sand

And a heaven in a wild flower,

Hold infinity in the palm of your hand

And eternity in an hour . . .

—William Blake, ‘‘Auguries of Innocence’’...

9. In the neighborhood
(pp. 43-57)

Q.31: Can you infer fencepost (or bridge*) ‘‘shapes’’ just by walking past them?

Okay, the question is somewhat unusual, but byshapeI mean the cross section of each post. I’ve now left the house behind, and have an unusualinverse problemto consider. When walking around the neighborhood or driving in the countryside, you will see various fences on the side of the road, some of which may be made of posts with rectangular cross sections, and others with circular cross sections. Of course, if you are driving at more than 5 mph it’s probably not easy to tell...

(pp. 58-63)

Q.34: How high is that tree? (An estimate using elliptical light patches)

Using figures 34.1(a) and (b), we may establish a result stated by Minnaert in his delightful bookLight and Colour in the Outdoorsconcerning what I call ‘‘tree pinhole cameras.’’ The essential idea is that the small spaces between the leaves act as ‘‘pinholes,’’ giving rise to the approximately elliptically shaped light patches on the ground. The fact that there is so much shade under a tree, incidentally, is testimony to the effectivness of the foliage; after all, part of its job description is to intercept light and...

11. In the sky
(pp. 64-121)

This is, of course, question 14, ‘‘reloaded,’’ and it leads naturally to question 40.

From figure 39.1, we can approximate the distancexfrom an observer atO, a heighthabove sea level to the horizon atT. This neglects the effect of atmospheric refraction, of course, but since that increases the effective ‘‘distance’’ by about 9% (see the article by French) this result represents a lower bound. Also ignored is the limiting effect of haze. By Pythagoras’ theorem, if the distance$OT = x$, then

\$x\; = \;{[{(R\; + \;h)^2}\; - \;{R^2}]^{{1 \mathord{\left/

{\vphantom {1 2}} \right.

\kern-\nulldelimiterspace} 2}}} \approx

since hR. A useful rule of thumb version is easily established if...

12. In the nest
(pp. 122-136)

Q.62: How can you model the shape of birds’ eggs?

My house has a front porch with three pillars, and the inside-facing part of each pillar, at the top, is a popular spot for birds to build nests; so far we’ve been host to doves, sparrows, and finches. However, a strong gust of wind in the right direction often demolishes the nests, sadly. But the parent birds are learning: they have decided that the hanging flower baskets are more desirable locations from which to raise a family, and, at the same time, frustrate the local cat population. Great care has...

13. In (or on) the water
(pp. 137-167)

Q.68: What causes a glitter path? [color plate]

Have you ever noticed golden “cylinders of Sun” (a “glitter path”) on the water as the Sun sets in the west or rises in the east? This beautiful “liquid gold” effect arises, of course, from the reflection of sunlight from the surface of the water, but it is not quitethatsimple. This phenomenon is not reflection from a flat surface, in general—the surface usually is rippled with waves. And since each of these ripples can be a reflecting surface for sunlight (or other light sources), a glitter path is composed...

14. In the forest
(pp. 168-182)

Q.79: How high can trees grow?

The mechanical or elastic properties of trees determine whether or not a tree trunk will buckle under its own weight, and whether or not a branch will bend or break under a load. Thus, stiffness and strength are properties determined by the elasticity of the structure. Consider some facts concerning trees: (i) In relation to its density, wood is stiffer and stronger, both in bending and twisting, than concrete, cast iron, aluminum alloy, or steel; (ii) trees are frugal in their use of resources for growth; (iii) they “use” the “principle of minimum weight”;...

15. In the national park
(pp. 183-193)

Q.84: What shapes are river meanders?

I love looking at river meanders when I’m in a window seat on a plane. I’ve never been very successful at taking good photographs of them for at least three reasons: (i) I’m not the world’s best photographer, (ii) even if I’m seated by a window, I’m often over a wing, and (iii) the amount of air between me and the ground causes much light to be scattered, so the scattered blue light reduces the contrast between the river and its surroundings. And if I’m walking in the vicinity of meanders and oxbow lakes,...

16. In the night sky
(pp. 194-216)

Q.87: How are star magnitudes measured?

When I was about fourteen or fifteen years old, my friend Miffy (a.k.a. Paul Smith) and I would amuse ourselves in our French class by doing calculations using some of the complicated looking formulas in Sir James Jeans’ bookAstronomy and Cosmogony[4]. While outdated (even then) as a textbook of astrophysics, it was fascinating both historically and scientifically, and I still have my copy of it. Since I was pretty good at French, I had done my homework, and as long as we were quietly working on the back row, our teacher probably...

17. At the end . . .
(pp. 217-230)

Since the questions in this book have been prompted by things I have noticed and wondered about on my many walks, both long and short, over the years, it seems to me to be appropriate to conclude with some items that have to do specifically with walking, and endings . . . So without further ado, let’s begin the final section with a question about perambulation!

Q.93: How can you model walking?

No one denies the obvious and numerous health benefits of walking. For a start, it is good for your heart. One recent study indicates that walking at a...

18. Appendix 1: A very short glossary of mathematical terms and functions
(pp. 231-233)
19. Appendix 2: Answers to question 1–15
(pp. 234-237)
20. Appendix 3: Newton’s law of cooling
(pp. 238-239)
21. Appendix 4: More mathematical patterns in nature
(pp. 240-242)
22. References
(pp. 243-246)
23. Index
(pp. 247-248)
24. Back Matter
(pp. 249-249)