Maths and physics provide the legal sporting advantage
Sport is a great way of enthusing young minds about maths. There is an abundance of maths and physics in sport. John Barrow has collected 100 bite-size reflections on all sorts of sport and shows how a basic understanding of maths and physics can provide the sporting advantage.
Robyn Williams: You may have noticed that all our stories so far have had something to do with space. So how, if we now turn to sport, do you get the cosmos onto the track? Well, the answer is with John Barrow who is a cosmologist and a very famous one, winner of the Templeton Prize, and someone who wrote about the anthropic universe, how it all seems just right for humans. Professor Barrow is basically a maths a man, which is why he has written Mathletics: 100 Amazing Things about Sport.
John Barrow: I've always been interested in sport, and long ago I used to do track athletics very seriously and some other sports when I was younger. So I've always had an interest in the way that mathematics and physics interfaces with sport and how you can use sport as a very good example to try to motivate some ideas in mathematics and physics, particularly to school kids, some of whom think they are not interested in maths and can't see how it would be relevant to them. And there's a type of student who is very interested in sport and had never imagined that there was something to be gained by thinking mathematically about it. I think on the other hand sport is a really nice example…lots of new types of problems that people in applied maths hadn't really thought about.
Robyn Williams: And your book has got two pages per topic, so it's bite-size and it's very accessible.
John Barrow: Yes, so it's two or three pages on 100 topics. So I just picked 100 things where maths tells you something interesting. So it might be how could Usain Bolt run faster, what's his weak point, and it's his start. You look at how slow his reaction time is, and then what would happen if he exploited the strongest wind assistance that he could gather and he went to the highest altitude that was allowed.
Robyn Williams: He'd be like lightning.
John Barrow: He can go down to 9.46 or so for 100 metres without becoming a better runner, so it's just by tuning certain things that are within his control. Another interesting one is right-handedness and left-handedness. So about 10% of the population is left-handed, and in combat sports like boxing and fencing, who's got the advantage? It's a simple little game theory problem. The left-handers have an advantage because they are far more used to competing against right-handers because so many of their opponents are right-handed, whereas the right-handed fighters very rarely encounter a left-hander. So in these combat sports, the left-handers have an advantage. And I was interested to see that when I checked this at the London Olympics, you looked at the fencing competitors, there's a huge fraction, it was over 40% of the competitors were left-handed. So you could see the effect of this gradually leading to an equalisation of the right and left-handedness.
But then I realised that there were other cases that people perhaps hadn't noticed. I mean, tennis and athletics are examples of sports where a choice is made in the setup of a sport which can either bias it towards the right-hander or to the left-hander. So in tennis the choice is that the opening serve is in the right-hand court, and that means when you follow through all the successive points, that the left-hander serves on their forehand a much more often than the right-hander as you move from side to side.
And in athletics we run anticlockwise around the track. If you have an event like the 400 metre hurdles, this really favours the right-handed competitor who will lead to the hurdle with their left leg, and that means that they can run closer to the inside of the lane, whereas the left-hander who leads with their right leg needs to move towards the outside of the lane and is very susceptible to disqualification. That competitor dragging their trail leg inside the hurdle is a common form of disqualification.
In baseball you see it. If you are a right-hander, you slog the ball off to the leg side, you are then moving in the opposite direction to the one in which you want to run to first base, whereas the left-hander is already then moving in that direction that they are going to run off. And the ultimate prejudice I'm afraid is polo, so I'm sure you're a keen polo player…
Robyn Williams: Oh you could tell, can you?
John Barrow: Yes. Polo is defined to be right-handed, so you are only allowed to swing that mallet on the right-hand side of the horse, so with your right hand. So that's for safety reasons for the courses, because you don't want people from right and left swinging at each other's horses. So if you are a left-hander, polo is not going to be for you.
Robyn Williams: And were these things generally known? Did you just put them together, or did you do a bit of calculation and work most of this stuff out yourself?
John Barrow: It was half and half. So some are known and then some were things like rowing configurations and so forth that I'd thought up or I had thought about for a long time.
Robyn Williams: When you took your book and these ideas to the public, especially the young public, did they respond?
John Barrow: Yes, they were very enthusiastic. There are lots of things that they can do, there are little experiments. The reaction time one is interesting. So if you analyse sports where you need to react, whether it's cricket, the batsmen, tennis player, someone to the starting pistol, baseball, you work out how long do people have to sight the ball and hit it. So these sports, they look rather different, the 22 yards from one crease to the other, it's a bit different from in baseball or the diagonal distance across the tennis court, but they've obviously evolved that the time that the receiver has to respond is very similar between 0.1, 0.2 of a second.
Human reaction time is believed can't be faster than 0.1 of the second, and so that's why in athletics if you respond to the starter's gun faster than that you are defined as false starting and you are out. The very best starters, people who will respond in 0.12 or so of a second, and they can do that consistently, often they are hurdlers. So Colin Jackson was probably the best. The hurdlers are very disciplined because they're going to have to do something very quickly after they've started, it's not just go hell for leather but you've got to be very controlled. So Sally Pearson in Australia and Colin Jackson.
So this response time, interestingly people have known for a good 50 or 60 years that it's very slightly slower in women on the average than men. And long ago when this was first discovered it was thought, well, it must be something to do with the fact that such a small fraction of women drive cars, you know, in the 1930s. But clearly that's not the case today. But the gap of the average in statistical distribution stayed the same. What I noticed is when you looked at the false starts statistics at the Beijing Olympics, there were fewer false starts in the women's events by exactly the fraction you would expect given their different reaction times. People don't know why it is. It must be linked, presumably, just to strength and power, which leads to the men's sprint times being faster than the women. So it's not just the psychological response.
So there are lots of interesting statistical problems. League tables, okay, how many points do you give for a win, what do you take into account? I mean, soccer in this country, there is an elaborate system of three points for a win, one for a draw, and goal difference. If you just use goals scored, everything would come out to be exactly the same at the same league table. If you had goal difference, you know, not surprisingly it pretty much also would. So the points are all a bit of a smokescreen.
Cricket of course is complicated, and swimming. People often ask me what was the most interesting thing that I learned overall, and it was that swimming is far and away the most technical sport and the sport where mathematics and mathematical physics has made the biggest impact.
If you go back to 1968, the men's 100 metre freestyle, if you recorded about 52.7 seconds I think it was, you would have won the gold medal in the men's 100 metres. Now this would hardly get you in the women's team in most countries. So the world record has come down to 46.8, so nearly 6 seconds for 50 seconds of action. But what's happened in the running over that same period of time? 400-metre running, there was a world record set in Mexico in 1968, 43.8 seconds. The world record now, 43.2, it's come down 0.6 of a second. So there's not much you could do in the running, you are running through air, but in the case of swimming, Counsilman and other mathematicians really got to grips with analysing what happens when strokes are pulled through the water.
So many of the things that you learned in swimming lessons at school turn out to be wrong. You don't pull in a straight line next to your body, you move your hand in a question-mark shape so you are pulling on water that's not already moving. And you don't keep your fingers completely close together because the optimal operation is for them to be slightly separated because then the little jets of water between the fingers produce more drag pull. And underwater filming, people analysing strokes, and the sudden appearance of enormously tall swimmers. You didn't used to see that in the '60s and early '70s. And Mark Spitz, all those gold medals, you see a picture of him, long hair, no cap, big moustache…what's going on? You would have thought that anybody would have guessed that the extra drag would play a role.
Robyn Williams: From your moustache?
John Barrow: Yes. So people wised up rather quickly. The very, very tall swimmers…an interesting case…so the drag on a swimmer, there is a sort of bubbly turbulence around the edge of your body, there's waves, but as you go faster you tend to create a sort of long wave trough, a sort of bow trough because you create a pressure difference between ahead of you and behind you. And at an appropriate speed you can get caught in that wave trough, and when you are caught in it you can't go any faster. For you and me, we are never going to worry about this, but the taller you are, the less likely you will ever be caught in the wave trough because at the maximum speed you will go you will be bigger than that wave length for the trough.
So it's a complex business, and lots of interesting things to study; the efficiency of different strokes, diving, and the famous old problem that used to be asked of maths students, you know, when you drop a cat it always lands on its feet, but it's not spinning when it starts, the spin overall is conserved yet it sort of rotates and does all sorts of complicated things on the way down but lands not spinning on its feet. It's the same with high board divers. So you can see they are not spinning when they start, and I think you are not allowed to use your feet to generate torque, but they spin, and yet angular momentum spin is conserved, it starts at zero. So what's going on?
And you notice when they start to spin in one direction they will throw one arm across their chest and another one in the opposite direction across the head. So they are producing equal and opposite rotations which cancel out. So despite appearances they keep the overall rotation zero by setting up rotations which are in counter directions. So the cat doesn't need to read about all this, it just does it. And falling cats is an emotive subject, but I've seen a film of cats falling 30 storeys from the ground, falling off window sills in New York skyscrapers, but just scamper off on the sidewalk when they land.
Robyn Williams: Mathletics he called it, John Barrow's book on 100 amazing things about sport. He is Professor of Mathematical Sciences at Cambridge.
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