by John Gribbin
Imagination and thinking were what the pre-teenage Richard Feynman (like the adult Richard Feynman) was superb at. In one of his favourite anecdotes (or parables, if you prefer), he told how while he was in Cedarhurst he learned how to repair radios. Radio sets were simple in those days, and he had started out, in Far Rockaway, by building his own crystal set, then moved on to fixing some problems for the family. Word spread, and friends and acquaintances used to call him in, rather than go to the expense of calling a regular radio repair man. The highlight of the story comes when a total stranger asks the kid to fix his radio, which makes an awful noise when it is switched on, but then settles down when it has warmed up. The kid paces up and down, trying to work out what is going on, while the owner of the radio gets more and more agitated, muttering about how stupid he has been to ask a little kid to do a man’s job, and asking what Feynman is up to, to which the kid replies, ‘I’m thinking’.
Eventually, having thought things through carefully, the kid realizes that the problem might be solved by reversing the order of two of the tubes (valves) in the radio. He swaps the tubes, switches it on, and it works perfectly. The owner of the set is enchanted, completely converted to the cause of the budding genius, and gets him more work, telling all his friends, ‘He fixes radios by thinking!’20
Now, the point of the story is not that the older Feynman was on some ego trip, boasting about his childhood achievements. It is a story (which happens to be true) about the importance of imaginative thought, and how to solve problems in general. At another level, here is someone who was opposed to what Feynman was trying to do (or, at least, to the way he was trying to do it) who turned around completely to become almost embarrassingly enthusiastic once the technique had been shown to work. So when you know you are right, you should keep your courage in the face of opposition, carrying on the way you know is right. And it also tells us something a little more subtle about Feynman’s character – he did not give up. Faced with a puzzle of any kind, from a neighbour’s broken radio to the fundamental nature of quantum physics, he did not rest until he had solved it (unless, of course, he had promised his sister not to try).
In high school, the pattern continued. Older students would come to him, for example, with tricky geometrical problems they had been assigned in the advanced mathematics class, and he would solve the puzzles – not because he was trying to ingratiate himself with the older boys, but because he couldn’t resist the challenge. As it happens, the reputation he developed for being some kind of whiz at maths did help him socially. He was hopeless at ball games and what were generally regarded as ‘manly’ pursuits, shy with girls, and worried about being thought a ‘sissy’. In What Do You Care What Other People Think? he describes being ‘petrified’ when passing a group of kids playing a ball game in case the ball rolled in his direction and he would be expected to pick it up and throw it back. The ball would always fly out of his hand in totally the wrong direction and everybody would laugh. The fact was, though, that he was simply too useful to the older boys for them to alienate him by making too much fun of these deficiencies.
Richard always tackled those geometry problems (and all other problems) his own way, using techniques that he had developed largely by himself, from first principles. Partly out of a desire to do it himself, partly through Melville’s instruction that you shouldn’t believe anything just because somebody else, no matter how eminent, told it to you, that was the way Feynman would work throughout his scientific life. With his friend Mautner, but largely on his own, he worked out most of the rules of Euclidean geometry for himself. ‘I wanted to find the formula’, he told Jagdish Mehra in 1988. ‘I didn’t care whether it had been worked out by the Greeks or even by the Babylonians; that didn’t interest me at all. It was my problem, and I had fun out of it.’
He was also, as he put it, lucky enough to learn algebra his own way before coming into contact with it at school. His older cousin Robert could never get to grips with algebra, and had a tutor who came to coach him. Feynman was allowed to sit in on these sessions, and quickly learned that in algebra the problem was to find the value of the unknown variable, x, in an equation. While Robert struggled to do this by rote, using rules memorized at school, Feynman appreciated that it didn’t matter how you got the answer, as long as it was the right one. Before he left elementary school, Richard had learned how to solve simultaneous equations – sets of two equations with two unknown quantities, such as
2x + y = 10
and
2y – x = 5
to find the values of both x and y (in this case, x = 3 and y = 4). Then, he made up for himself a problem with four equations and four unknowns.
Hardly surprisingly, by the time Richard came to algebra in high school he was bored to tears by what was on offer. He suffered in silence for a while, then told the teacher that he already knew what she was trying to teach the class. The head of the mathematics department gave him a problem to solve as a test; it was too difficult for him, but he made a good enough stab at it for them to see he really did know something about algebra. So he was put in a special class for the subject, really for students who had failed algebra once and were repeating it, with a teacher, Lillian Moore, flexible enough to cope with Richard’s precocity. It was here that he met a new kind of puzzle. Miss Moore asked the class to solve the equation 2x = 32. Nobody could make head or tail of it. They didn’t have a set of rules for solving that kind of problem. But Richard didn’t need a set of rules; he saw straight away that the solution is x = 5, because 5 twos multiplied together is 32. This kind of thing was self-evident to Richard, and the fact that nobody else in the class felt the same way was one of the first indications he had that he really was different from the other students.
That difference came to the fore when Richard became the star of the school maths team, competing with other New York high schools in the ‘Interscholastic Algebra League’. The algebra team would travel to different schools to compete with their maths whizzes. There were five members of each team, and they would be given problems that required what would nowadays be called lateral thinking to solve, with a strictly limited time in which to solve them – typically 45 seconds. Each member of the team worked independently, and could write anything he wanted on the paper in front of him. All that mattered was that before the time was up each competitor had to draw a circle around the one number on the paper that was his answer to the problem. The problems were deliberately chosen so that although they could, of course, be solved ‘by the rulebook’, it would be just about impossible to do so in the time available; but they were easy once you saw the short cut (or invented your own short cut). Feynman always won these competitions, writing down his number and ostentatiously drawing a circle around it, often on an otherwise blank piece of paper, usually before the other competitors had really got to grips with it at all. The practice served him well in later life, when he retained the ability to solve algebraic problems quickly and neatly, without ploughing through the textbook methods.
So Richard learned a little maths in high school, although he always claimed that he didn’t learn any science at all there, because he was always ahead of what was being taught in class. The kind of biology, physics and chemistry taught in Far Rockaway High School in the 1930s was already familiar to him from the Encyclopaedia Britannica, his own tinkering (for example with electricity), and informal conversations with his teachers and others. Even the maths he learned while at high school was largely self-taught – the big new thing for him in those years was calculus, which he learned from two books, Calculus Made Easy, by S. P. Thompson (St Martin’s Press, New York, 1910) and Calculus for the Practical Man by J. E. Thompson (Van Nostrand, New York, 1931), one of a series of ‘practical man’ guides to mathematics that Richard devoured around the time he left elementary school and went to high school.
But two mathematical experiences that Richard had while in high school did stick with him for the rest of his life. One gave him an insight into what it
was like for ordinary students; the other shaped his entire subsequent career.
The glimpse of mathematical mortality came when Richard was introduced to solid geometry, the study of shapes in three dimensions, in high school. He was completely thrown, and couldn’t understand what the teacher was getting at at all, although he could use the rules the teacher gave in order to carry through calculations properly. For once, he was in the same position as students who used the rules of algebra to solve equations without understanding what was going on. Then, the penny dropped. After a couple of weeks, he realized that the mess of lines being drawn on the blackboard was indeed meant to represent three-dimensional objects, not some crazy pattern in two dimensions. Everything came into focus, and he never had any trouble with the subject again. As far as science was concerned, ‘it was my only experience of how it must feel to the ordinary human being’, he later said.21
In 1933, the Feynman family visited the World’s Fair in Chicago; a year later, Richard began his final year in high school, and made the mathematical encounter that was to shape his career.
He owed the encounter to the Depression. That year, a new physics teacher, Abram Bader, joined the school. He had been working for a PhD at Columbia University, under the Austrian-born physicist I. I. Rabi, whose work on the magnetic properties of fundamental particles would bring him the Nobel Prize in 1944. But Bader ran out of money, and had to drop out of research to become a teacher. He quickly appreciated Feynman’s unusual abilities, lending him a book on advanced calculus, and often talking to him, out of class, about scientific matters. Once he explained something called the Principle of Least Action. They discussed the topic only once, but the whole scene stuck in Feynman’s mind for the rest of his life. He was so excited by the idea that he remembered everything about the occasion – exactly where the blackboard was, where he was standing, where Mr Bader was standing, and the room they were in. ‘He just explained, he didn’t prove anything. There was nothing complicated; he just explained that such a principle exists. I reacted to it then and there, that this was a miraculous and marvelous thing to be able to express the laws in such an unusual fashion.’22
The ‘miraculous and marvelous thing’ can be understood in terms of the flight of a ball tossed from the ground through an upper-storey window. In this context, the term ‘action’ has a precise meaning. At any point in its flight, you can calculate the difference between the kinetic energy of the ball (the energy of the ball’s motion, related to its speed) and its potential energy (the gravitational energy the ball possesses because of its height above the ground). The action is the sum of all these differences, all along the path of the ball through the air (action can be calculated in a similar way for charged particles moving in electric or magnetic fields, including electrons moving in atoms). There are many different curves the ball could follow to get through the window, ranging from low, flat trajectories to highly curved flight paths in which it goes far above the window before dropping through it. Each curve is a parabola, one of the family of trajectories possible for a ball moving under the influence of the Earth’s gravity. All this Feynman knew already. But Bader reminded him that if you know how long the flight of the ball takes, from the moment it leaves the thrower’s hand to the moment it reaches the window, that rules out all but one of the trajectories, specifying a unique path for the ball. And then he told him about the Principle of Least Action.
One of the most important principles in physics is the conservation of energy – the total amount of energy associated with the ball (in this example) stays the same. Some of this energy is in the form of gravitational potential energy, which depends on its height above the surface of the Earth (strictly speaking, on its distance from the centre of the Earth). When the ball rises, it gains gravitational potential energy; when it falls, it loses some of this energy. The only other relevant form of energy possessed by the ball is its energy of motion, or kinetic energy. Higher speeds correspond to greater kinetic energy. At the moment the ball leaves the thrower’s hand, it has a lot of kinetic energy because it is moving fast. As it rises, some of this kinetic energy is lost, traded for gravitational potential energy, and it slows down. At the top of its trajectory, it has minimum kinetic energy and maximum potential energy, then as it falls down the other side of the curve it gains kinetic energy and loses potential energy. But the total, the sum of (kinetic + potential) energy is always the same.
All this Feynman knew. But what he didn’t know was that given the time taken for the journey, the trajectory followed by the ball is always the one for which the difference, kinetic energy minus potential energy, added up all along the trajectory, is the least. This is the Principle of Least Action, a property involving the whole path.
Looking at the curved line on a blackboard representing the flight of the ball, you might think, for example, that you could make it take the same time for the journey by throwing it slightly more slowly, in a flatter arc, more nearly a straight line; or by throwing it faster along a longer trajectory, looping higher above the ground. But nature doesn’t work that way. There is only one possible path between two points for a given amount of time taken for the flight. Nature ‘chooses’ the path with the least action – and this applies not just to the flight of a ball, but to any kind of trajectory, at any scale. Mr Bader didn’t work out the numbers involved, or ask Feynman to work them out. He just told him about the principle, a deep truth which impressed the high school student in his final year before going on to college.
It’s worth a slight detour to give another example of the principle at work, this time in the guise of the Principle of Least Time, because it is so important both to science and to Feynman’s career. This version of the story involves light. It happens that light travels slightly faster through air than it does through glass.‡ Either in air or glass, light travels in straight lines – an example of the Principle of Least Time, because, since a straight line is the shortest distance between two points, that is the quickest way to get from A to B. But what if the journey from A to B starts out in air, and ends up inside a glass block? If the light still travelled in a single straight line, it would spend a relatively small amount of time moving swiftly through air, then a relatively long time moving slowly through glass. It turns out (see Figure 1) that there is a unique path which enables the light to take the least time on its journey, which involves travelling in a certain straight line up to the edge of the glass, then turning and travelling in a different straight line to its destination. The light seems to ‘know’ where it is going, apply the Principle of Least Action, and ‘choose’ the optimum path for its journey.
The connection between mathematics and physics highlighted by the Principle of Least Action reinforced a growing fascination that Richard had had with this area of science right through high school. While working with radio receivers, building his own circuits and working out how to tune them, he had come across equations describing the behaviour of these practical objects that involved the Greek pi, the ratio of the circumference of a circle to its diameter. Although there were circular (or cylindrical) coils in these circuits, it is also possible to work with square coils, and pi came into the equations whatever the shape of the coils. There was some deep link between physics and mathematics, which Feynman did not understand, but which intrigued him. Although still known as a whiz at maths, his fascination was really with physics.
Figure 1. Light travels faster through air than through glass. So the quickest journey from A to B that is partly through air and partly through glass is not the (dotted) straight line from A to B, but there is a unique ‘path of least time’ made up of two straight lines. This is a special case of the Principle of Least Action at work. The dotted lines to the right show an example of a path that takes longer than the path of least time (solid lines).
We have emphasized the role of science in young Richard’s life because it was, indeed, the main thing in his life. He went through the educational system in what
seemed, superficially, a conventional way, but actually learned his science for himself, outside the system (including teaching himself about relativity theory from books while still in high school). He found school boring, but sailed through examinations with ease, appearing, in that respect, to have been a model student.
How clever did he have to be, to do all that? Joan Feynman once sneaked a look at the results of the standard IQ tests that both she and her brother had taken in high school.23 Her score was 124, his was 123, so she could always claim to be smarter than he was. It is notoriously true that IQ tests are only any good at measuring the ability of people to do IQ tests, but much later in life Feynman took great delight in being able to quote his IQ score when invited to join the organization Mensa, which is exactly the kind of ‘club’ for the self-important that he despised. Unfortunately, he replied, he could not join Mensa because his IQ was not high enough for them.
But that didn’t stop his being a genius, because some kinds of genius cannot be measured in IQ tests. The mathematician Mark Kac, who was born in Poland but spent most of his career in the United States, once explained that there are two kinds of genius. One is the kind of person that you or we would be just as good as, if only we were a lot more clever. There is no mystery about how their minds work, and once what they have done is explained to us we think we could have done it, if only we had been bright enough. But the other kind of genius is really a kind of magician. Even after what they have done is explained to us, we cannot understand how they did it. ‘Richard Feynman’, said Kac in 1985, ‘is a magician of the highest caliber.’24
But in spite of being clearly different from his peers in this way, even as a child, and in spite of his fears of being thought a sissy, Richard wasn’t what would now be called a ‘nerd’. He had a handful of close friends, some interested in science, others on the humanities side; and his feet were kept firmly on the ground in those Depression days by the need to work at odd jobs to earn spending money.