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00:30Hello and welcome to Think of a Number.
00:36Anyone fair speech? Anyone fair speech? Anyone fair speech?
00:38Anyone fair speech? Anyone fair speech?
00:41Anyone fair speech?
00:42Anyone?
00:43Have you all got one of these?
00:44Yes.
00:45All of you?
00:46Great.
00:47That didn't take long, because I'm a lightning conductor.
00:50Let's get rid of all this.
00:54Now, you've all got a strip of paper.
00:56Okay. Now, you there. Give me a number between, any number you like, between four and six.
01:02Five.
01:04Five! You don't want to change your mind?
01:07No. Thank goodness for that.
01:09Now, the question is, with a strip of paper, can you make a five-sided figure, or a five-pointed star?
01:17Does anyone know how to do it?
01:19It's one of these things. If you know how to do it, it's easy.
01:22And I'll show you how easy it is. All you do is tie a knot in your strip of paper, cross it over, put your fingers through, and pull the other bit through.
01:31Now, don't pull it too vigorously, and pull it slowly through, and flatten it out as you go.
01:38And then, if you fold the other bit around the back, you've got a five-sided figure.
01:49Anybody done it?
01:51Quite a few of you. Good.
01:52When you've done it, hold it up to the light, and you should be able to see, up to a bright light, you should be able to see a five-pointed star through it.
02:00Can you?
02:02A bit like this one.
02:05See? It's a pentagon, with a pentagram inside it, and another pentagon there.
02:13There you are. A penta... gone.
02:17Right.
02:19Now, there we are.
02:21Now, why have I shown you this?
02:22Well, if you undo your paper again, I'll undo that. Don't bother undoing yours.
02:26Paper, if you discount the thickness, it's really just got a width and a length.
02:31So it's two-dimensional, if you like.
02:33But to tie a knot in it, as we did then, you need the third dimension, because you've got to fold it out there.
02:39You need that third dimension. That's what this program is all about.
02:41Two dimensions, three dimensions, even four dimensions.
02:442D or not 2D?
02:47That's not the question, is it? No.
02:49In 1907, a chap called Hinton wrote a book called An Episode in Flatlands, where he envisaged a flat world.
03:06You see, it's just a disc, and a sea, a few seas, but it's really flat.
03:11And the people live on the edge, a bit like this, like Egyptian paintings.
03:16They can't fall over, because there's no that way.
03:19There's only two dimensions, you see.
03:21But think of the problems.
03:23If you're born facing that way, you can't turn the other way.
03:27Because to do that, your nose has to come out and go in again.
03:30And there's no that way.
03:31So if you're born facing that way, that's the way you stay.
03:34Problems there.
03:36So he thought, I know, I'll have all the fellas facing that way, when they're born,
03:39and all the women facing that way.
03:41That solved a few problems.
03:43One problem it didn't solve, was how did they pass each other?
03:46You see, you can't come out and pass.
03:49So they had to climb over each other.
03:52So he made a law that all the men must lie down and let the women trample all over them.
03:56Which is what happens in the everyday world, really, isn't it?
03:59Isn't it, fellas?
04:01Yeah, the women trample all over us.
04:03Now then, think of where they'd live.
04:06They wouldn't have houses, they'd have flats.
04:09And the doors wouldn't open like that, see?
04:11They'd have to hinge up.
04:13But just think, if somebody's coming in the front door,
04:15at the same time somebody comes in the back door,
04:17your house would fall down.
04:19They'd fall down completely.
04:21So there we are. Think of this.
04:23If you went to work facing this way,
04:25you'd have to come back facing that way.
04:27You'd keep bumping into people.
04:29They'd keep saying, I'm terribly sorry, I wasn't watching where I was coming back from.
04:32So, it would be a world of narrow-minded, flat-footed, thin people,
04:38all bending over backwards saying, you scratch my front and I'll scratch your back.
04:42However, that's not the world we live in, because that's two-dimensional.
04:46We live in a three-dimensional world, aren't we?
04:51And we've got some paper here.
04:54Can I have a piece?
04:56Which we've marked out and cut already.
04:58And I want you to follow this.
05:00And see, you need three dimensions to do this.
05:02You've got a door in your piece of paper.
05:05And I want you to have the door so that it opens that way.
05:08Okay?
05:09And then just fold it back.
05:11And hold it.
05:14Now this right-hand side, fold that back as well.
05:18And then fold it again.
05:22Okay?
05:23Now stick your fingers through the gap.
05:25Like that.
05:26Bring the other tab over.
05:28And there's a bit of sticky tape on it.
05:30Stick it down.
05:32And you should have six ones.
05:36And on the other side, six twos.
05:38So you could fold it like this.
05:40And you've got a book of ones.
05:42And inside, you've got twos.
05:45Or you could have a book of twos.
05:47And inside, you've got...
05:50...ones.
05:53Threes!
05:54Alright, let's have a book of threes.
05:55And inside, you've got twos, haven't you?
05:57No, you've got fours.
05:59So it's a four-sided book.
06:01And it's easy to make, isn't it?
06:03Have one of those.
06:05Now those are called...
06:07...tetra-tetra-flexigons.
06:09But this is called a hexa-hexa-flexigon.
06:12Because it's got six sides and it's got six faces, if I can find them.
06:16Now, if I crimp that, push that in.
06:20And open it up.
06:22Oh!
06:24A shock.
06:26It's an horrible face.
06:28Now, if we crimp that...
06:30Oh, push that in.
06:32And open it up.
06:33What have we got?
06:35Number five.
06:37I see.
06:39So that's one and a five and a face.
06:41I don't know what that is.
06:44We'll be back at the face again.
06:46Right, let's...
06:47...see if we can get something else out of this face.
06:50Number three!
06:52Right on this side.
06:55Number four!
06:57Our designer, when he made this, he got his tie caught in the middle.
07:02And as he flexed it, he dragged him in and we haven't seen him since.
07:07Number one!
07:09Now, where's this one take us?
07:12Oh, look!
07:14What's it say?
07:15Help!
07:16I'm a prisoner inside this hexa-hexa-flexigon.
07:18That's our designer.
07:19Now, that's got six sides.
07:21But, of course, you can jumble them up.
07:22And to make it, you need a strip of paper.
07:24Not as big as this.
07:25But this sort of shape.
07:26In triangles.
07:27And it's got to be numbered like this.
07:28Z-1-2-3-1-2-3.
07:29And a blank one at the end.
07:31And then, turn that over.
07:33And renumber it on the other side.
07:35Like that.
07:36Then you've got to know how to fold it.
07:38And that's the problem.
07:39Put you right into us and we'll tell you how to do it.
07:41These are flat flexigons.
07:43But you can have three-dimensional flexicons.
07:46Flexi-cubes!
07:47Who's got flexi-cubes?
07:50Give them a flex.
07:56A little bit like worry beads, aren't they?
07:59They go round and round and round.
08:00Drive your crackers.
08:02You can't really talk about flexing paper or folding paper
08:07without talking about origami, can you?
08:09And we've got some origami birds that flap.
08:12Come on, Paul.
08:13And they all flap.
08:16And there's thousands of origami designs.
08:18And everybody who's ever folded paper
08:20have always made paper airplanes.
08:22Who've got paper airplanes?
08:23Give them a fly, then.
08:25Here they come.
08:26Whoa!
08:29Now, it's plain to see.
08:31No, not plain to see.
08:32It's obvious that these are three-dimensional models
08:34and they fly in three dimensions.
08:37But artists don't need to fold paper.
08:42Just from a two-dimensional piece of paper,
08:44they can give the impression of three dimensions
08:47using their artistic art.
08:49This picture was designed by an artist called Escher.
09:09And as you can see, it's a staircase
09:11that seems to go up and away from us.
09:15And it looks like a real staircase.
09:17That's because he's got his perspective right.
09:19And he's obeyed the laws of watching
09:22where his vanishing points are.
09:25When you look at railway lines,
09:27the lines aren't parallel as you look down the track.
09:29They meet.
09:31The nearer they get to the horizon,
09:32they seem to come closer together
09:33until eventually they meet on the horizon.
09:35Similarly, when you look up at a tall building,
09:37the walls of the building seem to come in
09:39and get nearer and nearer together.
09:41Now, that's the effect you get by looking up.
09:43So to get that effect on paper,
09:45you have to have lines that converge at one point.
09:53And as you can see, all his vertical lines converge at that point right up there
10:03with the strings attached.
10:04It's a vanishing point or a convergence point.
10:07Now, there are three planes to this picture.
10:10If you look at this wall,
10:12everything on that plane converges on a point over there.
10:16And you can see
10:18how the landing and all the steps converge on that point.
10:22Similarly, everything on this plane here,
10:24where this chap's walking here,
10:26all those lines converge at that point over there.
10:28Now, we've covered some bits up,
10:31because if we'd shown you them before,
10:32it would confuse you.
10:34Have a look.
10:36People living on the wall.
10:38Unless you turn it this way around.
10:41And then you see it's another real world.
10:44We're looking down on it this time.
10:46And because we're looking down on it,
10:47all the verticals converge not at a point above,
10:49but at a point below.
10:51And here is the string.
10:54And you can see how they all converge.
10:57There.
10:58And there, you see.
10:59And it looks like another real picture.
11:01If we take this one,
11:03there's another world.
11:05You have to turn it right round.
11:08And there you've got this other world here.
11:12But all three pictures,
11:14it's almost like three dimensions,
11:16but they all use the same convergence points.
11:18And as a result,
11:19they all seem to live together.
11:20These two fellas,
11:21they're using the same staircase,
11:23although they're living in different dimensions.
11:25It's very confusing.
11:26Three different worlds,
11:27but everything seems to work.
11:29So, he really didn't design this picture
11:32around a square canvas,
11:34or a square piece of paper,
11:36but around, or within,
11:39an equilateral triangle
11:41in two dimensions.
11:43Now, all triangles are two-dimensional.
11:47Aren't they?
11:49You can put triangles on flat surfaces.
11:51Here's a triangle.
11:52I'll put my hand through it like that.
11:54But if you look at it,
11:55it looks a bit twisted, doesn't it?
11:56If you look down here,
11:57along here, and up there.
11:58Why is that?
12:00I mean, I put my hand through it like that.
12:03I can't put my hand through it like this, can I?
12:06I did it!
12:09It can't be possible.
12:11Well, it can,
12:12because this isn't a triangle at all.
12:15Three pieces of wood,
12:17stuck together at odd angles.
12:19But from one set position,
12:24it's a triangle.
12:28You see, your eye is kidding,
12:29believing that it really is a triangle.
12:31Your eye can be fooled.
12:32Sometimes it can't sort out
12:34the information that's given to it.
12:36Have a look at this.
12:38Now, what could this be?
12:44An oil rig, perhaps?
12:46A confetti maker?
12:50Could be anything.
12:52But as you look from different angles,
12:54it's very difficult to understand what it might be.
12:56Anybody have any idea?
13:00You could be close.
13:08It's a flag.
13:09The Union Jack.
13:10But you can only see it
13:11from one particular point.
13:13And the camera is trying to get to that point now.
13:16And there it is.
13:18A little bit of adjustment.
13:20And there's the Union Jack.
13:23And yet, from any other angle,
13:26it's chaos.
13:28You see, your eye can't even detect the distances
13:30away from the camera that these things are.
13:33Your eye isn't very good at judging distance
13:35if it doesn't have enough information.
13:38Over here, I've got a cube,
13:41a pyramid,
13:42a sphere,
13:43and a cylinder.
13:45And I'm going to let you look at them.
13:46You in the audience,
13:47and you at home.
13:48And I want you to say
13:49which one is nearest to the camera.
13:51Are you ready?
13:53Look.
13:56That's enough.
14:00Triangle.
14:01The pyramid.
14:03The...
14:04The square.
14:05The cube.
14:06The ball.
14:07The circle.
14:08The ball.
14:09The circle.
14:10Oh.
14:11All right.
14:12Bit confused, aren't they?
14:13Right.
14:14No, they're not all the same.
14:15They are different distances.
14:16Which one is furthest away?
14:18Have a look.
14:19The cylinder.
14:20The cylinder.
14:21The square.
14:22The cylinder.
14:23The square.
14:24The cylinder.
14:25Okay, that's enough.
14:26The square.
14:27The square and the cylinder.
14:28So people have said the square is nearest and furthest away.
14:31Which is it?
14:32Have a look again.
14:33And I'll move the square.
14:34And I'll move the square.
14:43Furthest away.
14:44Which one was nearest?
14:46The cylinder.
14:48The cylinder's nearest.
14:50Wrong.
14:52The sphere.
14:53The sphere.
14:54Wrong.
14:56The triangle's nearest.
14:57Right.
14:58If you don't believe me, have a look at all of them.
15:02And we confuse you, you see, because they're all different sizes.
15:05So your eye can be fooled.
15:07But sometimes it can sort out two dimensions and three dimensions.
15:09It's lucky we only have three dimensions, isn't it?
15:12Or have we only got three dimensions?
15:28Centuries ago, man believed the Earth was flat.
15:32And then he discovered the world was round.
15:34But that didn't change the shape of the world.
15:36It only changed his thinking.
15:38He thought it was flat.
15:39If you walk far enough, you'd fall off the edge.
15:40And then he suddenly thought it was round.
15:42And if you walk and walk and walk, you'll come back to the same point.
15:46So, we know about the Earth now, but what do we know about space?
15:50How far up can you go?
15:53Forever and ever?
15:54Will space carry on forever?
15:55Will you fall off the end of space?
15:59Perhaps there's a wall round it.
16:01If there is, what's on the other side of the wall?
16:05Difficult to know, isn't it?
16:07Perhaps it's bent.
16:09And if you go on long enough, you'll come back to the place you started.
16:13Nobody can be sure.
16:15Have a look at it from another point of view.
16:18If you thought the Earth was flat,
16:21and you stood at the North Pole,
16:22and you started to coil some rope round,
16:24the coil would get bigger and bigger, wouldn't it?
16:27And you'd think, yes, the Earth's flat.
16:28It's bound to get bigger because the Earth's flat,
16:30so the coil would get bigger.
16:31Now, as you kept coiling, still believing the Earth was flat,
16:34when you got to the equator,
16:37you may suddenly think, hang on, the circle's not getting any bigger.
16:41It's getting smaller.
16:43And you'd be very confused because you believe the Earth is flat.
16:47And when you got to the South Pole,
16:49you'd realise that you weren't outside this circle,
16:52you were inside it, inside this coil,
16:55and it was coming around you.
16:56And your mind would go,
16:59wouldn't it?
17:01Right, let's look at space from the same point of view.
17:03If you bandaged the Earth,
17:05lots and lots of bandages still got bigger and bigger.
17:06Bigger, I mean, further enveloping the sun,
17:09enveloping the stars, and getting bigger and bigger.
17:11More and more bandages.
17:12How long could you go on?
17:13Wouldn't there come a time, perhaps,
17:15when the bandages didn't seem to be getting any bigger?
17:19When suddenly, you weren't bandaging the outside of a sphere,
17:22but the inside of a sphere.
17:25And as you bandaged and bandaged,
17:28the sphere got smaller,
17:29and you may find that you're trapped inside a sphere of bandages.
17:33Well, we can't know whether this would happen,
17:36but mathematicians, you see,
17:38have to consider all the possibilities,
17:40and perhaps then they can come up with the answers
17:43and find out what space is all about.
17:46Whether there is another dimension to space.
17:49Now, while we're talking like this,
17:51you're watching television at home
17:53on two-dimensional television sets, aren't you?
17:56A flat screen.
17:58You can't really be sure
18:00whether you're seeing me in three dimensions.
18:01You know somehow that I'm further down from the backing
18:03because you can see a bit of floor, perhaps,
18:05and you can feel I'm walking in front of it.
18:08So you get a feeling of three dimensions,
18:09but your screen's really only two-dimensional.
18:12Why can't we have three-dimensional television?
18:15We've got three-dimensional cameras,
18:16and we've had them for a long time.
18:21These cameras are very old.
18:24They're called stereoscopic cameras,
18:26and the first ones were made round about 1850.
18:28This one was made in 1897.
18:31Here's another one.
18:32They take two pictures at once.
18:36This one is unique.
18:37It's got three lenses.
18:39So you can set it for stereoscopic pictures
18:42or panoramic pictures.
18:45It's set for stereoscopic pictures,
18:47and these two shutters will open and close now.
18:51There.
18:52There.
18:53There's no film in it now.
18:55Here's some more.
18:56There's a viewfinder there.
18:58And this is the only stereoscopic camera
19:02that you can buy these days.
19:04It's still being produced.
19:05Now, these cameras all take two pictures.
19:09Here are some of them.
19:16And to see, or to get the three-dimensional effect,
19:20you need a viewer.
19:22Here's a viewer here.
19:24There's a picture in it.
19:26You look through here,
19:28and the lenses superimpose the two pictures.
19:31And what you see is three-dimensional.
19:35It's fabulous.
19:36I wish you could see this.
19:37But you can't.
19:39You could, you know,
19:40if you're all sitting at home with one of these.
19:42And you could focus it on the television.
19:43Then we could show pairs of pictures.
19:46A bit like these.
19:47They were taken when I was much older.
19:49All right.
19:50Let's try it.
19:51Let's show you two pictures of me.
19:53There we are.
19:54If I hold my hand out here,
19:55you can see that the two cameras
19:58are looking for slightly different angles.
19:59Now, with a viewer to focus your eyes
20:02and superimpose the two pictures,
20:03you could possibly see me in three dimensions.
20:05I could probably pop out
20:06and have a cup of tea.
20:08Milk and one sugar.
20:10Right?
20:11The thing is,
20:12we might one day get three-dimensional television
20:14because there is now a camera
20:16that takes three-dimensional pictures
20:18but doesn't produce two prints.
20:21Produce one print.
20:22And look at this.
20:24Here's the camera.
20:25Developed in America,
20:26but it's going to be made,
20:27mass-produced in Scotland.
20:28And it's got four lenses.
20:30It takes four pictures.
20:32But where the new technology comes in
20:35is the way those pictures are printed.
20:37They're all printed onto one piece of paper.
20:39The four pictures.
20:41What happens is this.
20:42The four pictures are shredded
20:43into very thin strips.
20:4564,000 strips to an inch.
20:47And they take the first strip from the first picture,
20:49the second strip from the second picture,
20:50the third strip from the third picture,
20:51the fourth strip from the fourth picture,
20:53then the fifth strip from the first picture again.
20:56And so on.
20:57And they build up a picture,
20:58put them all on one piece of paper,
21:00and then put a plastic lens on the top.
21:04And they get three-dimensional pictures.
21:07These pictures aren't really any thicker than normal pictures.
21:15But when you look at them, you'll see they're three-dimensional.
21:19If I move them slightly,
21:22you'll see the tree
21:24and the window in the background
21:27moving.
21:29And you get a feeling that you can see behind the boy.
21:34And you see three dimensions.
21:37Have a look at this one.
21:39This one
21:40is a girl
21:41by a waterfall.
21:42And if you look carefully,
21:46the water beneath her
21:49seems to be nearer to the camera
21:51and the trees behind her
21:54much further away.
21:59And here's another one
22:01of a girl coming up some steps.
22:04And you can see
22:06that the railing
22:08at the top of the steps
22:09seems to move slightly.
22:12Gives the impression
22:14that she actually is
22:15a few steps down.
22:18So this technology
22:20can produce three-dimensional pictures.
22:22Perhaps one day
22:24we'll have cameras producing
22:25three-dimensional television.
22:27Let's have a three-dimensional trick.
22:29Who got flexigans?
22:30Pick them up.
22:31You've got a question mark on the front.
22:32But if you flex them,
22:33you'll find numbers inside.
22:34Several numbers.
22:35And I want you to select one of them.
22:37And who's our calculator girl?
22:38Come on.
22:40All right.
22:41Switch your calculator on.
22:43Now, when they're showing numbers,
22:45I want you to punch them in there
22:46with a plus in between each
22:47so we can add them up.
22:48And when you've got eight numbers in there,
22:50don't press equal sign
22:51until I tell you.
22:52Okay?
22:53I'll face the other way.
22:54I'll even close my eyes.
23:01Finished!
23:03I'll look round.
23:04Let me have a look at the numbers very quickly.
23:05Oh, yes, I'll tell you.
23:06It's...
23:08Could have added that up easily, couldn't I?
23:10216.
23:12Correct!
23:19It's a trick, isn't it?
23:20Of course it's a trick.
23:21And if you'd like to know how to do it,
23:22drop me a line.
23:23Johnny Ball.
23:24Think of a number.
23:25BBC Television.
23:26W-1-A-1-A-A.
23:28That's about the trick
23:29or anything else in the programme.
23:31Especially the paper folding.
23:33You see, this is the last programme in this series.
23:35But if we send you all our bonf on paper folding,
23:37you'll have enough to keep you going
23:39until we're on television again.
23:41Talking about television,
23:42it's two-dimensional, you know.
23:44You keep your eyes open.
23:45It's very difficult to know whether I'm thick or not.
23:47What I mean is whether I'm three-dimensional
23:50or whether I'm two-dimensional.
23:52Goodbye till next time.
23:53APPLAUSE
24:17we need to have to talk.
24:23We need this to talk.
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