Mrs. Perkins's Electric Quilt: And Other Intriguing Stories of Mathematical Physics

PAUL J. NAHIN
Pages: 424
https://www.jstor.org/stable/j.ctt7swmw

1. Front Matter
(pp. i-vi)
(pp. vii-x)
(pp. xi-xii)
4. Preface
(pp. xiii-xxx)
5. 1 Three Examples of the Mutual Embrace
(pp. 1-17)

As the epigraphs illustrate, the role of mathematics as an aid to understanding the world has been evaluated differently by different philosophers. (I side with Nicholas, not Henry, as you have probably guessed.) The central thesis of this book is that physics needs mathematics, but the converse is often true, too. In this opening discussion I’ll show you three examples of what I’m getting at.¹ In the first example you’ll see a physical problem that is not difficult to state, and for which it is only very slightly more difficult to actually calculate a solution. It all comes about so...

6. 2 Measuring Gravity
(pp. 18-23)

In a number of the discussions to follow in this book, we’ll find that we will need to know the numerical value of the gravitational acceleration at the Earth’s surface, usually denoted, as in the first discussion, byg(pronouncedgee). The experimental determination of the value ofgis, in fact, a classic experiment performed every year in thousands of college freshman physics labs worldwide. I remember well when I did it as a freshman in Physics 51 at Stanford (1958). I remember it as a clunky, uninspiring experiment that required watching a highspeed, pulsed spark generator burn holes...

7. 3 Feynman’s Infinite Circuit
(pp. 24-43)

Have you heard of the famous mathematical hotel that has an infinite number of rooms? It has the wonderful feature—unlike any real hotel I’ve ever tried to check into in a strange city at midnight without a reservation—of never being full. Even if you think it’s full, there is always room for more. Here’s why. Let’s suppose itisfull, i.e., there is an infinity of people in the hotel, with each individual person enjoying a private room and with each and every room having a person in it. That night, an infinitymorepeople arrive at the...

8. 4 Air Drag—A Mathematical View
(pp. 44-61)

One of the many anecdotal stories told to illustrate the mathematical genius of the Hungarian-born American mathematician John von Neumann (1903–1957) is the following:

Two bicyclists are 20 miles apart and head toward each other at 10 miles per hour each. At the same time a fly traveling at a steady 15 miles per hour starts from the front wheel of the northbound bicycle. It lands on the front wheel of the southbound bicycle, and then instantly turns around and flies back, and after next landing instantly flies north again. Question: What total distance did the fly cover before...

9. 5 Air Drag—A Physical View
(pp. 62-81)

In the first discussion on air drag I wasn’t terribly concerned about the actual applicability of our results to the real world. That is, while I limited myself to physically plausible air drag force laws, I didn’t take the next step of asking which ones are the force laws that are actually observed. I have already opened the door to more than one law, of course; Stokes’s linear law and the quadratic law. Canbothapply in real life? Yes, but in entirely different situations. So, to be absolutely clear from the start, in this discussion we’ll be interested in...

10. 6 Really Long Falls
(pp. 82-93)

What if, instead of a long fall through Earth’s atmosphere, we had the atmosphere itself, the oceans, the continents—everything, the entire Earth—fall into the Sun? How long would that take? If we allow our imaginations even more freedom, how long would it take to fall from Heaven to Hell? Outrageous as that question no doubt seems, it is raised and answered in both ancient Greek theology and in more recent English poetry. So, as an inverted form of the Earth-into-the-sun question, we might ask how far it is from Earth to Heaven. Mathematical physics can answer both these...

11. 7 The Zeta Function—and Physics
(pp. 94-106)

Following Bellman’s lead, I’m going to take a break from falling objects with this discussion (don’t worry, we’ll get right back to that topic in the next discussion). This will be a mostly mathematical break, but at the end I will make a (speculative) connection with physics. So, consider the almost surely “too simple-looking to be important’’ double integral

$\int\limits_0^1 {\int\limits_0^1 {\frac{1}{{1\; - \;xy}}\,dx\;dy} }$. (7.1)

The integrand of (7.1), integrated over the unit square, is well behaved everywhere except at the single point of the upper right corner of the square, where the denominator vanishes. Looks can be deceiving, of course, and in fact...

12. 8 Ballistics—With No Air Drag (Yet)
(pp. 107-119)

After that “relaxing breather’’ on the pure mathematics of the Riemann zeta function in the previous discussion, let’s get back to tossing stuff. In the earlier discussions on air drag, I limited the physics to strictly up-and-down motions. The more general (and, I think, vastly more intricate) case is when the moving object starts on the journey along its flight trajectory with an initial angleθ0different from the straight-up (or straight-down) case of 90 degrees (measured with respect to the spherical Earth’s “local” surface, which serves as our horizontal reference). Two obvious examples of this are a baseball leaving...

13. 9 Ballistics—With Air Drag
(pp. 120-135)

In this discussion I’ll essentially repeat the previous one, but now with the realistic complication of air drag. You’ll see, when we compare the numerical predictions of the two analyses, that to ignore air drag in ballistic calculations is to accept potentially huge errors. The problem of ballistics with air drag is an old one that can be traced back to a very nasty conflict between two mathematicians in 1718. One, the Scottish-born John Keill (1671–1721), formulated the air drag ballistic problem as a challenge to the other, the Swiss Johann Bernoulli (1667–1748). The two had tangled before...

14. 10 Gravity and Newton
(pp. 136-169)

With two exceptions, all of the previous discussions have used Newton’s analytical formulation of gravity, and in this discussion I want to continue that thread, but now in a much broader context than that of simply working with falling objects: here I’ll add a bit more historical commentary, along with some additional interesting calculations. Before we are through, we will blow up planets and shrink stars! In the crudest sense, it has never been hard to “discover’’ gravity. After all, everybody who has ever lived did that the first time they fell down. But why does everything on Earth, without...

15. 11 Gravity Far Above the Earth
(pp. 170-185)

In the opening quotation, taken from a letter Newton wrote to his friend Edmond Halley (the man who got thePrincipiainto print), Newton’s anger was directed at Robert Hooke (1635–1703), the Curator of Experiments at the Royal Society of London. Halley had just written to Newton that Hooke wanted to be mentioned in thePrincipiaas at least a co–discoverer of the inverse square law of gravity simply because he had in the past proposed it. Hooke could do nothing beyond proposing, however, and all the wonderful mathematical discoveries in thePrincipiaare due to Newton’s analyses...

16. 12 Gravity Inside the Earth
(pp. 186-214)

Most people, when faced with the above opening question, almost immediately think of the Earth’s rotation, probably because of the common familiarity just about everybody has (by age five) with the tendency of a merry-go-round to toss a rider off when it spins. For the Earth, this “tendency” appears as an acceleration component opposite to that of Earth’s gravity. My wife, who has never had a physics course in her life—she was an art history major in college—came up with this explanation in less than five seconds after I ran the opening test question by her. A nonrotating...

17. 13 Quilts & Electricity
(pp. 215-232)

Newtonian gravity is a fascinating subject, but I’ve been discussing it for a long while now, and I (and perhaps you, as well) would enjoy a break from it. So, let’s do something completely different here for just a bit. Would you be surprised if I told you that there is an intimate connection between the artistic world of quilting and the mathematical world of electric circuit theory?There is. And I don’t mean either the light bulb above or the electronic circuitry inside a modern quilt sewing machine.¹

I’m talking about the fundamental conservation laws of energy and of...

18. 14 Random Walks
(pp. 233-260)

In September 1904, at the International Congress of Arts and Science in St. Louis, Sir Ronald Ross (1857–1932) gave an interesting talk (reprinted a year later inScience)¹ on how to mathematically model the spread of mosquitoes from a breeding pool. As the vectors of such devastating diseases as yellow fever and malaria, understanding how these insects migrate was an important practical problem of the day. Yellow fever in particular had some years before laid waste to the French effort to build a Panama Canal. Ross was not a mathematician by training—he was a British medical doctor who...

19. 15 Two More Random Walks
(pp. 261-284)

Perhaps the best known drunkard’s walk is calledBrownian motion, named after the Scottish botanist Robert Brown (1773–1858) who, in 1827, observed (under a microscope) the chaotic motion of tiny grains of pollen suspended in water drops. (This motion had been previously noted by others, but Brown took the next step of publishing what he saw in an 1828 paper in thePhilosophical Magazine.) This motion exhibits a number of interesting physical characteristics: the higher the temperature and the smaller the suspended particle the more rapid the motion, while the more viscous the medium the slower the motion; the...

20. 16 Nearest Neighbors
(pp. 285-298)

Well, with an opening like the one above you may be wondering if your author has himself been driven around the bend with all the random walks and MATLAB coding we’ve been doing. Actually, there is one more random walk I want to show you, with a totally unexpected application to physics, but I’ll defer that until the next discussion. Here we’ll take a break from random walks but still keep our feet in probability (and just a bit of MATLAB, too). Discussing the nearest neighbor problem in terms of cannibals is simply for fun, of course, but similar mathematical...

21. 17 One Last Random Walk
(pp. 299-320)

In this discussion I’ll show you a remarkable connection between random walks and electrical resistor networks. It is, at first glance, all too easy to dismiss the mathematics of resistors as trivial, but that would be a very big mistake. To set the stage for the rest of this discussion, then, let me immediately give you an example of the not so obvious ability of resistors to help us understand some very nontrivial mathematics. Consider the inequality

$\frac{{(a\; + \;b)\,(c\; + \;d)}}{{a\; + \;b\; + \;c\; + \;d}}\; \geqslant \;\frac{{ac}}{{a\; + \;c}}\; + \;\frac{{bd}}{{b\; + \;d}},\;a,\;b,\;c,\;d\; > \;0$.

Can you prove that this is so? Also, assuming the inequality is valid, under what conditions does equality hold? Here’s how to...

22. 18 The Big Noise
(pp. 321-372)

During the autumn quarter of my sophomore year (1959–1960) at Stanford I took Math 130, my first course in differential equations. The assigned textbook was Ralph Palmer Agnew’s classicDifferential Equations. I have kept my copy of Agnew all these past fifty years since, and as I write it is open on my desk in front of me to page 40. On that page is a problem I’ve always remembered because it described such a novel idea (to me) that I was immediately and forever fascinated by it. This will be our final discussion of the book (final serious...

23. 19 Electricity in the Fourth Dimension
(pp. 373-384)

Back in Section 17.3 I showed you how to analyze the three-dimensional resistor cube, made from one-ohm resistors, with the result that an applied voltage difference of one volt across two opposite vertices would result in a current into and out of the cube of 6/5 amperes. That is, the so-calledbody diagonalresistance of the cube is 5/6 ohm. You may not have even thought of it then—we take “seeing” in 3-D so much for granted—but one reason we could solve the problem so easily (once we had the symmetry and superposition trick in hand) is because...

24. Acknowledgements
(pp. 385-386)
Paul J. Nahin
25. Index
(pp. 387-392)
26. Back Matter
(pp. 393-393)