Richard Feynman, 1984. Photograph: Tamiko Thiel, CC BY-SA 3.0, Wikimedia Commons.
Recently, I attended a discussion event organized by a Turkish mathematics association called Matematiğin Peşinde, centered around the question, “What good is this maths to us?”
On the bus ride home after the discussion, I recalled a small incident between Richard Feynman and Hans Bethe that I had read about in a book years ago.
Let me briefly explain who Richard Feynman was. Feynman (1918–1988) was one of the most renowned figures in twentieth-century physics. Known for his vibrant personality and keen sense of humour, he not only shared the 1965 Nobel Prize in Physics but also became famous for his ability to explain science.
The other protagonist of this story, Hans Bethe (1906–2005), was one of the leading figures in nuclear physics and received the 1967 Nobel Prize in Physics. The two had worked together before this event. During the Manhattan Project, Feynman served in the Theoretical Division at Los Alamos, which Bethe headed, and Bethe appointed the then-25-year-old Feynman to lead a research group.
Drawing inspiration from a flying plate
The story I’m about to tell is based on “The Dignified Professor”, a chapter in Surely You’re Joking, Mr. Feynman!, which was compiled from recordings of Feynman.
While working at Cornell, Feynman felt scientifically burned out. As he reflected on why he used to love physics, he realised something important: when he was younger, he did not constantly ask himself, “Is this important?” or “What will this contribute to science?” He simply focused on the things that interested him. With that in mind, he gave himself permission to “play” with physics again.
Shortly afterwards, someone in the Cornell dining hall, for whatever reason, tossed a plate into the air. Feynman noticed that as the plate spun, it also wobbled slightly, and that there was a particular ratio between the rotation of the Cornell emblem on the plate and the plate’s wobble.
Adapted from Feynman's Wobbling Plate on the Wolfram Demonstrations Project.
The figures in this interactive tool are not instantaneous speeds. They show the number of revolutions accumulated over a five-second flight. One spin is a 360° rotation of the plate about its own axis; one wobble is a complete revolution of the direction perpendicular to the plate’s surface around the vertical axis. The counters reset at the beginning of each new flight.
There is an interesting detail here. In the book, Feynman gives the ratio the other way round: he says the Cornell emblem rotates twice as fast as the plate wobbles. In Feynman’s Flying Saucer: The Second Serving in SIAM News, Mark Levi suggests that Feynman may later have misremembered the result of his own calculation.
Feynman sat down and began calculating this apparently pointless problem. Then he went to Hans Bethe and told him what he had found.
Feynman, that’s pretty interesting, but what’s the importance of it? Why are you doing it?[1]
Feynman’s answer was very simple:
There’s no importance whatsoever. I’m just doing it for the fun of it.[1]
I think the most interesting part of the story comes next.
Feynman continued to tinker with the equations of the wobbling plate. From there, his thoughts began to flow towards the relativistic motion of electrons, the Dirac equation, and quantum electrodynamics. In the book, he describes the process like this:
It was effortless. It was easy to play with these things. It was like uncorking a bottle: Everything flowed out effortlessly. I almost tried to resist it! There was no importance to what I was doing, but ultimately there was. The diagrams and the whole business that I got the Nobel Prize for came from that piddling around with the wobbling plate.[1]
I should admit something here: rather than directly answering “What good is this maths to us?”, this story answers a broader question: “What is the value of investigating something simply because we are curious about it?”
The details of how Feynman moved from this simple problem to much deeper ideas in physics are missing here, too. The reason is fairly simple: explaining that part properly would probably go well beyond my own knowledge of physics.
But I don’t think you need to know every technical detail of that transition to see the beauty of the story. In Feynman’s own retrospective telling, the moment he stopped trying to produce something important was when he began producing important things again. A problem he initially tinkered with just for fun, one that seemed to serve no practical purpose, became his path back to much bigger ideas.
Perhaps the most beautiful answer the story offers is this: the value of curiosity does not always have to be visible from the very beginning. Sometimes we only discover what something is useful for after we have pursued it.
Getting to know Feynman
Thinking about this story also took me back to how I first got to know Feynman. I read Surely You’re Joking, Mr. Feynman! during my first year at university, and only later realised how much its portrait of him had influenced my perspective on mathematics and science.
What stayed with me was not simply that Feynman was a physicist, but the way he approached problems: tinkering with something, trying to solve it in his own way, asking questions that might seem silly, and sometimes working on a problem simply because it was fun. It was an approach to science that resembled what I already loved about mathematics.
Study hard what interests you the most in the most undisciplined, irreverent and original manner possible.[2]
The book’s account of his time at Los Alamos contains a good example of this mindset. In “Safecracker Meets Safecracker”, Feynman describes his fascination with opening safes. Rather than forcing locks open, he explored the tolerances in their mechanisms and how predictable people could be when choosing combinations. Numberphile’s Safe Cracking with Feynman video recounts these stories and some of the methods he used.
Some readers may instead remember that Feynman was briefly portrayed in the film Oppenheimer. But beyond the anecdotes and that brief portrayal, I think one of the best ways to get a sense of him is to listen to recordings of his talks.
One of my favourites is The Relation of Mathematics and Physics, in which he explains the relationship between mathematics and physics.
If you don’t want to set aside an hour, the roughly ten minutes beginning at 44:18 give a particularly good summary of Feynman’s general perspective on the differences between mathematics and physics.
Playing with numbers
Feynman’s idea of “playing” did not appear only in grand problems of physics. Throughout the book, he brings the same attitude to everyday, apparently trivial problems. One of my favourite examples is a calculation contest he entered against someone in Brazil who used an abacus.
His opponent asked him for . Now it is your turn:
Try it firstI recommend spending at least 2 minutes on this problem before looking at the solution.Show the solutionHide the solution
At first glance, this looks like a rather daunting number to calculate mentally. But Feynman immediately noticed something:
So the answer must be only slightly greater than 12.
Now, 1729.03 is only 1.03 greater than 1728. Proportionally, this increase is
of the original number.
A small approximation Feynman remembered from calculus comes into play here: when a number changes by a very small amount, the proportional change in its cube root is approximately one third of the proportional change in the number itself.
If you are curious where this comes from, the terms to look up are linear approximation and first-order Taylor approximation.
The increase above 12 is therefore approximately
which gives
and hence
There’s no need to assume that Feynman calculated in his head all at once. He could approach the division step by step: for example, , and then, by gradually reducing the remaining difference, he could first arrive at and then at approximately . The fact that he first wrote down and then added two more digits in the story is entirely consistent with this kind of calculation.
That was Feynman’s trick: instead of calculating the intimidating-looking cube root directly, he measured how far the new number lay from the familiar point .
Feynman also knew that this method would not work so neatly for an arbitrary number. At the end of the chapter, he explains how lucky he was that his opponent happened to choose 1729.03:
So I never could teach him how I did cube roots or explain how lucky I was that he happened to choose 1729.03.[3]
Between inspiration and criticism
Before finishing, it is worth taking a brief step back. Feynman’s scientific curiosity, problem-solving style, and perspective on science are inspiring. But that does not mean his personality or every action in his life should be idealised.
I should also say that some passages in Surely You’re Joking, Mr. Feynman!, particularly stories involving women, made me uncomfortable. They made me look more critically at the entertaining and unconventional portrait of Feynman that the book had built up for me.
I don’t see a contradiction here: we can learn a great deal from some of a person’s ideas and work while criticising other things they did.
Still, while writing this post, I realised again what has stayed with me most about Feynman: the pleasure of tinkering with a problem without knowing where it will lead. Perhaps that was why, years later, the story of the wobbling plate came back to me on that bus.
Notes
- Richard P. Feynman, “The Dignified Professor”, Surely You’re Joking, Mr. Feynman!, Edward Hutchings (ed.), W. W. Norton, 1985.
- Richard P. Feynman, letter to J. M. Szabados, 30 November 1965, in Michelle Feynman (ed.), Perfectly Reasonable Deviations from the Beaten Track: The Letters of Richard P. Feynman, Basic Books, 2005, p. 206.
- Richard P. Feynman, “Lucky Numbers”, Surely You’re Joking, Mr. Feynman!, Edward Hutchings (ed.), W. W. Norton, 1985.