Think of a Number (1978) S03E01 - Johnny Ball - video Dailymotion

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00:30Hello and welcome to Think of a Number on Wheels because that's what this week's program is all about. Round wheels, not so round wheels and wheels within wheels. So we'll get on by saying think of a number between one and ten. Five. Five. That'll do. You three there and you two there.

00:56Because we're going to take you back to the days before the wheel was invented when the earth was practically all a wilderness. Oh no, another one.

01:07And to move heavy loads, primitive man used this system. Right, away you go. And pull. Stop. Move that round at the front.

01:17And, mind your fingers. Pull. Oh, stop. Move that round at the front. Pull, slaves. And stop. Oh.

01:31Well, thanks very much, Les. Can you go and sit down? Absolutely. As you can see, this system doesn't work very well. In fact, if you don't have a hard, flat floor, it hardly works at all.

01:45But it was a method that primitive man used for moving heavy loads. Rollers. And the thing about rollers was they were probably the inspiration for the invention of the wheel.

01:57The first wheels rolled out of the inventors' workshops about 3,500 B.C.

02:17Could have been in several places around about the same time. But it definitely happened in Mesopotamia.

02:22This is a Mesopotamian wheel. It's not made of a slice of log. It's two planks, fastened together, and then carved into a round shape.

02:31The Mesopotamians, almost as soon as they'd invented the wheel, started thinking about it in a mathematical way.

02:38They thought that the outer edge of the wheel was about three times the diameter of the wheel.

02:46Have a look at this.

02:47That's a hexagon.

02:52Six-sided figure.

02:54And if you count the sides, one, two, three, four, five, six, you can see that the outer edge of the hexagon is exactly three times the diameter.

03:06However, the wheel is slightly larger, can you see?

03:08Because of the curves.

03:09And in fact, the outer edge of a circle is about three and a seventh times the diameter.

03:18But that just doesn't apply to Mesopotamian wheels.

03:21That applies to all wheels and all circles of any size.

03:26The outer edge, or the perimeter, is always three and a seventh times the diameter.

03:32And we know that proportion by the letter pi.

03:38And we use pi probably because pi was the first letter in the Greek word for periphery, or outer edge.

03:45But to what use is pi?

03:47Well, you can work out lots of things with pi.

03:52Think of a number that's been off the air for a few months now.

03:55But we haven't been idle during that time.

03:57Because as soon as the last program finished, I got this rope, left one end here, and set off around the world.

04:04I did.

04:05With the rope.

04:06And then I brought the other end up here.

04:10See?

04:11And by a fluke, it just reaches.

04:14So there's a rope going all the way around the world.

04:18It's true!

04:19I'll show you.

04:19I'll prove it.

04:20What's this?

04:22You see?

04:24And here you are.

04:28It exactly fits.

04:2940,000 kilometres long, this rope.

04:32Now, there was a chance that it might not reach because we might have to take it over the top of a building or something.

04:36So, when I went around the world, I took this extra rope with me, you see?

04:42It's about 12 metres.

04:44So I could bridge the gap if I needed to.

04:47Here's the thing.

04:49If I join this 12 metres rope onto this rope, which is 40,000 kilometres long, what difference would it make to the circle of rope?

04:58How much bigger would it be?

04:59How far off the earth could you lift the rope all the way around the earth by just adding 12 metres?

05:07Well, you can work it out if you know about pi.

05:09You see, there's your world.

05:1440,000 kilometres round.

05:16Divide it by pi, 3 in the seventh, and you get the distance across, or the diameter.

05:20Now, if you'd increase that distance round by 12 metres, you know the increase in the diameter by dividing that increase, that 12 metres, by pi.

05:31Well, we'll use the Mesopotamian pi and say it's 3.

05:34Divide 12 by 3, and the answer's 4.

05:374.

05:37So your diameter will be 4 metres longer.

05:41So you'll have 2 metres at that end, and 2 metres at that end.

05:43So all the way around the earth, just by adding 12 metres to a 40,000 kilometre rope, you'll be able to lift it 2 metres off the earth, all the way around.

05:58Actually, inaccurate figures, it's 1.9 metres, or 6 foot 2.

06:02And just think, if you were a 6 foot 2 person, and you went round the earth, by the time you got back, your head would have travelled 12 metres further than your feet.

06:11So you've come back ahead of yourself.

06:12What? Isn't that interesting?

06:15Oh, right.

06:16OK, Australia, pull on the rope.

06:23Pie crops up absolutely everywhere.

06:25Look at this.

06:26We've got panels on the floor.

06:28Can you see?

06:28And you've all got sticks, haven't you?

06:30I want you to throw the sticks.

06:32Just throw them at random, and try to throw them in the middle.

06:35Throw them now!

06:39Hang on, I've got to keep some of them on again.

06:42There's 31 sticks there.

06:47You see, there was a chap called Buffon, lived 200 years ago.

06:51And he used to do things like spinning coins up thousands of times to see how many times they came down heads.

06:58He did it for laughs.

06:59He didn't have much else to do.

07:00Right?

07:00And he did this.

07:02He dropped sticks onto a panel floor to see how many times they landed on the lines.

07:08And he found there was a connection between the sticks and pi.

07:13This is how he worked it out.

07:16He found, you count the number of sticks you've thrown, divide it by pi, double the answer, and that'll be the number on the lines.

07:25Do you think we've got a chance?

07:27I don't know.

07:27Hang on.

07:28I've got a bit of one here, and I'm going to throw that.

07:30Now, we've thrown 31 and a bit sticks.

07:36If you divide that by 3 and a seventh, the answer's 10.

07:39Double it, you get 20.

07:41So, if Buffon was right, 20 sticks should be touching the line.

07:46Let's count them.

07:471, 2, 3, 4, 5, not sure about that one, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, where's the other one?

08:10Oh, it is touching now.

08:1219!

08:13And we're just one out.

08:15Well, there's one there.

08:1620!

08:17We're absolutely right!

08:19You see?

08:19Now, if you threw thousands of sticks, you would have a better chance of getting the right answer.

08:23But there you are.

08:25An incredible thing with sticks dropping, and pi comes into the answer.

08:28The reason for that is this.

08:30A stick can drop in any position, round in a circle.

08:36These sticks are the same width as the panels.

08:39So, the chances of them dropping on a line, or not on a line, depends on pi.

08:45And that's what they found.

08:47And that's what they found.

08:48Okay, pick up your sticks.

08:54Grab a couple.

08:56Grab a couple.

08:56Here, you can have that.

09:07Oh, there you are.

09:08So, what is pi all about?

09:12For instance, what is pi exactly?

09:14Well, pi isn't exactly anything at all.

09:17Pi is 3.14159265358979323846.

09:26But it doesn't stop there.

09:27It goes on forever and ever.

09:29It's an irrational number, which means that it's decimal places go on forever, never repeating, always changing.

09:37There's one fellow memorised pi to 5,050 places.

09:41And in 1973, two Frenchmen, one was a woman, got a computer and worked out pi to 1,000,000 decimal places.

09:52And they published it in a book.

09:55And that's it.

09:57Starting with three at the beginning, and going on, three point, and so on.

10:02I haven't read this book.

10:03I'm waiting for the film.

10:06Catch!

10:08But all this was kicked off by the invention of the wheel.

10:14Which some people say was the most important invention that man has ever perpetrated.

10:20But that's not really true, you see.

10:22The wheel isn't as important as the axle.

10:27Because if the axle isn't in the middle of a wheel, all kinds of strange things can happen.

10:33Oh, anybody ride a bike?

10:43Who wants to go?

10:44You.

10:46The best way to start it is that one low, and that one high.

10:50OK.

10:51On you go.

10:52Try it.

10:52On that one.

10:54You've got to keep peddling.

10:56Have another go.

11:00Go again.

11:01The middle is off centre.

11:03So everything's going up and down.

11:07Oh, thanks a lot.

11:08Give it a round of applause.

11:17The thing is, all centre wheels are used in machines.

11:21Here's another wheel.

11:24And as you see, as it rolls along, there's the wheel.

11:26As it rolls along, the centre takes a path which is a straight line.

11:33But what happens from all these other points where we've made holes?

11:37Well, let's take this one here.

11:41This is on the outer edge of the wheel.

11:43And see what happens, what path it takes as we roll the wheel.

11:47Oh, hang on.

11:49I'm not drawing.

11:52That's it.

12:01They're humps.

12:03They're called cycloids.

12:05It's quite a neat sort of pattern.

12:06Let's try another one.

12:13What about this hole here?

12:14It's halfway between the edge of the wheel and the centre.

12:17What kind of path will that take?

12:19Off we go.

12:20I've missed in the middle.

12:26Never mind.

12:28There you are.

12:30And it's sort of a wavy line.

12:33Rather like the path that the axle on the clown's bike was taking.

12:37It's the same path my bum was taking.

12:39Oop, open that.

12:40OK.

12:41Now, this is like a train wheel.

12:44A train wheel is really two wheels.

12:46A smaller one and a bigger one.

12:47The small one runs on the line.

12:50And the big one is a flange which keeps it on the tracks.

12:55Now, what about a point on the edge of the flange of a train wheel?

13:04This point here.

13:06Let's see what path it takes.

13:07That is.

13:19Sort of a cycloid, but it's got loops in it.

13:22And if you think about it, to make a loop, the pen must be going that way at the bottom here.

13:27Which means that if a train is coming in this direction, 125 miles an hour,

13:35at all times, certain points on the bottom of the wheels are travelling backwards.

13:42Now, cycloids of all kinds are used in machines.

13:47And there are other kinds of cycloids.

13:50Hypercycloids.

13:51Wheels within wheels.

13:54See what you can draw with those.

13:55Now, this is an interesting one.

14:11Can I have a go?

14:13Here you go.

14:15This wheel here is half the diameter of the outer wheel.

14:19You see?

14:19And if you take a point on the edge of this wheel, the strangest thing happens.

14:24Watch this.

14:25You follow its path.

14:27Instead of a curve, it follows a straight line.

14:32And as the wheel revolves round and round, so that straight line is repeated backwards and forwards.

14:38So, what you've done is converted rotary action into straight line action, or linear motion.

14:46Now, here's something that does the reverse.

14:49You've all got nuts with strings on.

14:51Can somebody lend me one for a second?

14:53And then, what I want you to try and do, you can swing it round.

14:58That's quite easy.

14:59But what I want you to try and do is this.

15:02Get it swinging, and then pull, and get it to revolve, just by pulling the other nut backwards and forwards.

15:09Great, yes.

15:17One, two.

15:19That's it.

15:20And you don't have to swing it.

15:22What you've done is produced rotary motion using linear motion.

15:27And all these principles, cycloids, off-centre wheels, linear motion converted to rotary motion, and rotary motion converted back to linear motion,

15:36they're all principles used in machinery.

15:38So, let's go back thousands of years and look at a very primitive machine.

15:56Two thousand years ago in China, Chinese use water wheel and invent the cam.

16:08Water wheel turns shaft, and then cam turn also.

16:17But cam have advantage of turning rotary motion into linear motion, lifting load, like so.

16:25As cam turn, load is lifted until suddenly...

16:30It drop on head.

16:33Anyway, that's what it does.

16:35And that was the invention of the trip hammer.

16:38The Chinese invented that, and it was a very, very early mechanical device.

16:44Machines didn't come to this country for quite some time.

16:48First, the water wheel, and then, about 1191, Britain saw its very first windmill.

17:00The rotary action of the sails is converted into rotary action in a different direction.

17:18And for that, gears were needed.

17:21The same gears can turn the fan tail at the back.

17:26As the fan tail turns, it revolves the whole top of the windmill so that the sails face the wind.

17:32And that is where your linear motion, circular motion, comes in.

17:37Because the wind comes in a linear direction and turns the sails.

17:49Now, that windmill is a model of an actual windmill in Dis, in Norfolk, which was built about 1860.

17:58There, it was built to replace one that blew down.

18:02However, 1860 was a very good year, because it was the middle of the Industrial Revolution.

18:09Look at this.

18:09You could say this was a pair of water wheels.

18:27But the water isn't driving them.

18:30They're being driven by steam power.

18:33Here are your two steam pistons.

18:34But they're unusual in that they oscillate, or rock backwards and forwards.

18:40So that their linear motion can be converted into rotary motion by the use of cranks.

18:47But now, with the Industrial Revolution, you've got all the principles of rotary action and wheels being used together in machines.

18:56You see the eccentric wheels, just like the clown's bicycle.

18:58And you've got the gears, that drives another piston, which provides the engine with water.

19:07And the whole thing drives the water wheels, and pulls along a paddle steamer through the water.

19:13Victorian engines were made in wands.

19:25But nowadays, engines are made sometimes by the thousand.

19:29And whereas that engine was powerful enough to pull a boat across a lake, engines now have the power to carry well over a hundred people up into the air and take them halfway round the world.

19:44This is an aircraft engine, and it's brand new.

20:14It's called the RJ-500, and it's going to be built by Rolls-Royce and a Japanese company combined.

20:20And when I say it's brand new, it's quite ahead of its time, because it won't go into production until 1986.

20:28So we're seeing it for the very first time.

20:31Have a look.

20:35You used to have propellers on aircraft that went out to the sides.

20:38The propellers have been taken into the aircraft now, and they suck air in and compress it.

20:45And then there are some more vanes behind there, which once again channel the air and choke it down to a narrower entrance where it's really compressed.

20:55Then it goes through some more fans who compress it even more.

21:00And when it's compressed just as much as it can be, it comes through here, the combustion chamber where the fuel is added, and then it passes through turbines, which are once again great whirling fans going at incredible speeds, which thrust the whole lot out of the back.

21:16And the back end gets very hot.

21:22And this engine is going to be revolutionary in that it's going to be quieter and more economic.

21:30And once again, it's converting linear motion, the wind coming this way.

21:36Rotary motion is pushing it through.

21:38And so all this rotary motion produces linear motion thrusting the whole thing that way.

21:42And as the aircraft flies that way, if it carries on around the world, once again, you're back to rotary motion.

21:49Let's have a trick.

21:51Have you got some cards?

21:52Thanks.

21:53I want you to grab a card each.

21:55Here you are.

21:57Now, you remember we were talking about pi and remembering it to 5,000 decimal places?

22:01Well, I can't remember pi to 5,000 decimal places, but one thing I can do, can you come out?

22:09I think, is remember the ten-figure numbers on the back of those cards.

22:12I'm going to turn my back.

22:13You look at them.

22:15Hold your cards up so that the two-figure numbers are facing this way.

22:18I want you to choose any two-figure number.

22:21Yeah.

22:21Tell me.

22:2225.

22:2325.

22:23Everybody else, put their cards down, and I'm going to try and tell you the ten-figure number on the back of card number 25.

22:327, 5, 2, 7, 9.

22:386, 5, 1, 6, 7.

22:44Is that right?

22:47Amazing.

22:48Thanks very much.

22:48Actually, you could do that, because it's a trick.

22:54It's got something to do with memory, but it's really a trick.

22:56If you'd like to know how to do that trick, or anything else about the programme, drop me a line.

22:59Johnny Ball.

23:00Think of a number.

23:01BBC Television, London, W1A, 1AA.

23:05But remember this.

23:07Next time you see a machine, it's the wheels that you don't see inside,

23:11which keep the wheels that you do see going round.

23:13And when you think about the mathematics of circles, there's more to pi than meets the eye.

23:26Goodbye.

23:27Goodbye.

23:28Goodbye.

23:29Goodbye.

23:30Goodbye.

23:31Goodbye.

23:32Goodbye.

23:33Goodbye.

23:34Goodbye.

23:35Goodbye.

23:36Goodbye.

23:37Goodbye.

23:38Goodbye.

23:39Goodbye.

23:40Goodbye.

23:41Goodbye.

23:42Goodbye.

23:43Goodbye.

23:44Goodbye.

23:45Goodbye.

23:46Goodbye.

23:47Goodbye.

23:48Goodbye.

23:49Goodbye.

23:50Goodbye.

23:51Goodbye.

23:52Goodbye.

23:53Goodbye.

23:54Goodbye.

23:55Goodbye.

Think of a Number (1978) S03E01 - Johnny Ball

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