# Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction

PAUL J. NAHIN
Pages: 400
https://www.jstor.org/stable/j.ctt7rk7v

1. Front Matter
(pp. i-vi)
(pp. vii-ix)
3. INTRODUCTION
(pp. x-xxviii)

These two views, from two of the most famous physicists of the last century, on the relationship between physics and mathematics, are quite different. Both won the Nobel Prize (Einstein in 1921 and Feynman in 1965), and their words, however different, deserve some thought. Einstein’s are ones without sting, carefully crafted to perhaps even bring a surge of pride to the practitioners of the rejected mathematics. Feynman’s, on the other hand, are just what we have come to expect from Feynman—brash, outrageous, almost over-the-top. Feynman’s comment really goes too far, in fact, and I think it was uttered as...

4. 1 FEYNMAN MEETS FERMAT
(pp. 1-15)

In the Introduction I mentioned that we will often use a computer in this book. Indeed, modern computer software is a powerful tool for mathematical physicists, and in this first chapter I want to give you a simple illustration of this. (I have strong personal views on the usefulness of computers in society in general, and not just as a tool for mathematical physicists. In Chapter 8 of this book are illustrations of some of those views using the device of short fictional stories published more than thirty years ago.) As the epigraph to this chapter suggests, my example involves...

5. 2 JUST FOR FUN: TWO QUICK NUMBER-CRUNCHING PROBLEMS
(pp. 16-26)

Heroic calculation of certain particular numbers (π, e, and$\sqrt 2$, for example) has long had a special attraction for persons blessed with genius-level intellects—that is, for people who, you would think, should have had far better things to do! In myAn Imaginary Tale: The Story of$\sqrt {-1}$(Princeton, N.J.: Princeton University Press, 1998, 2007, 2010), for example, I mention Newton’s calculation of π to sixteen decimal places, an achievement the great man felt obliged to explain with the words, “I had no other business at the time.”

I used the Newton story as a lead-in to the...

6. 3 COMPUTERS AND MATHEMATICAL PHYSICS
(pp. 27-81)

The opening quotation gets right to the heart of mathematical physics, with its passion for calculation. If you can’t calculate the value of whatever it is you are talking aboutand get an answer in agreement with experiment, then the conclusion is that there is at leastsomething(maybe even a lot) you don’t understand. Being able to calculate something that nobody else has been able to is the fantasy dream of all young mathematical physicists. (Being able to simply understand what some young person has just calculated for the first time is the dream of alloldmathematical physicists!)...

7. 4 THE ASTONISHING PROBLEM OF THE HANGING MASSES
(pp. 82-130)

In the FPU experiment we had a sophisticated system of coupled harmonic oscillators, a system capable of vibrating at a number of different frequencies―most generally, in fact, at more than one of the frequencies and even, perhaps, at all of them simultaneously. The geometry of the FPU experiment was simple in one sense: it did not involve any gravitational potential energy considerations. That was because all the masses and springs in the system were at the same elevation (which we implicitly took as our zero reference level). The only potential energy in the FPU system was that stored in the...

8. 5 THE THREE-BODY PROBLEM AND COMPUTERS
(pp. 131-217)

Isaac Newton (1642–1727) is a man who almost certainly needs no introduction to readers of this book. Newton’sPrincipiatransported gravity from a topic about which just about everybody had an opinion, based on how they personally imagined God had constructed the world, to one unambiguously described in dispassionate mathematics.¹ This momentous intellectual achievement allowed some people (those possessing sufficient analytical ability) to actuallyderive without speculationthe observable effects of gravity. This is not to say, however, that that is necessarily an easy thing to do. As long as one talks about the Sun and the Earth alone,...

9. 6 ELECTRICAL CIRCUIT ANALYSIS AND COMPUTERS
(pp. 218-287)

In this chapter I want to show you a high-intensity number-crunching computer software application that is very different from MATLAB. It is one used every day by professional electrical engineers worldwide, but, like MATLAB (and personal computers and video game consoles, for that matter), it would have been considered a fantasy when I was in high school. (Just to get the terminology straight—afantasyis even more “far out” than isscience fiction!) In addition, this chapter will also discuss how electronic circuits were first used in the 1940s to solve mathematical problems on what are calledanalogcomputers....

10. 7 THE LEAPFROG PROBLEM
(pp. 288-296)

As the final number-crunching analysis of this book, consider the following pretty little problem that I came across when reading a recent “Ask Marilyn” column, a problem posed to her by a reader (identified only as M. Schwartz, in Ventura, California):

A friend and I once went from his house to mine with one bike. I started walking and he rode the bike. When he got a couple of blocks ahead, he left the bike on the sidewalk and started walking. When I got to the bike, I started riding, passing him, and then left the bike a couple of...

11. 8 SCIENCE FICTION: WHEN COMPUTERS BECOME LIKE US
(pp. 297-327)

I think it is pretty close to the mark to say that the large majority of physicists, mathematicians, and engineers, in their youth, read at least a few science fiction (SF) stories or novels. Certainly the “scientific romances” of H.G. Wells, such asThe Time MachineandWar of the Worlds, or the “advanced engineering” works of Jules Verne, such as hisJourney to the Center of the Earth, From the Earth to the Moon, and20,000 Leagues Beneath the Sea, would be on the reading list of any youngster with the sort of imaginative mind that would find appeal...

12. 9 A CAUTIONARY EPILOGUE
(pp. 328-334)

The speed of any computer is fundamentally limited by how fast its various component parts can send and receive signals among themselves―that is, by the speed of light and by how far those signals have to travel. We can’t do anything about the speed of light, but one way to increase the speed of a computer is simply to make it smaller. That means the computer’s volume decreases. Suppose, just to be specific, a computer has the shape of a sphere with radiusr. Asr→ 0, the volume decreases asr³. If the energy required to power our...

13. APPENDIX (FPU COMPUTER EXPERIMENT MATLAB CODE)
(pp. 335-336)
14. SOLUTIONS TO THE CHALLENGE PROBLEMS
(pp. 337-370)

The Buckled Railroad Track Problem. This problem has been around for a long time, generally posed as I have stated it. That is, one only has to provide aroughanswer (inchesversusfeetversusyards). A significantly more sophisticated version asks for the value of the maximum buckle accurate, say, to five decimal places. You can find an analysis to that more difficult question in the terrific book by Forman S. Acton (see note 7 in Chapter 3 again),Real Computing Made Real: Preventing Errors in Scientific and Engineering Calculations(N.J.: Princeton University Press, Princeton, 1996, pp. 222–223)....

15. ACKNOWLEDGMENTS
(pp. 371-372)
Paul J. Nahin
16. INDEX
(pp. 373-376)
17. ALSO BY PAUL J. NAHIN
(pp. 377-377)