Same shape and same size: superimposition as the test

0:00A sign painter has a job. A symbol on one roadside board has to go onto a second board. The second board is blank, and it is standing a long way from the first. He cannot carry the first board with him. So the shape has to travel some other way. That is the whole problem, and it is a better problem than it sounds. Because whatever he writes down has to be enough.

0:27If it is not enough, the second sign comes out different, and nobody finds out until both are standing up. So here is the question. What is the shortest thing you can write down that leaves no room to get it wrong? The obvious answer is tracing paper. Lay it over, follow the outline, carry the sheet. For something small, that works perfectly. But this symbol is a metre across.

0:57Now you need a sheet a metre wide, kept flat, carried without creasing, and lined up exactly on the new board. The method has not become wrong. It has become useless. And there is a second thing wrong with it, which matters more. A tracing tells you nothing. You can copy a shape perfectly and still not know what makes it that shape. So we are going to send the shape as numbers instead.

1:26Here is the symbol on its own, with the board taken away. It is two straight strokes meeting at a point. There is nothing else to it. Give the ends names. A at the top left, B at the bottom where they meet, and C at the top right. Now it can be talked about. The stroke from A to B. The stroke from B to C. And a name for the opening between them: the angle at B, written angle A B C.

1:56The letters are not decoration. They are what lets a shape travel down a phone line. So measure the two strokes. A to B is four centimetres. B to C is eight. Two numbers. Send them, and see what comes back. This is what comes back. Four signs, made by four different painters. Every one of them has a stroke of four centimetres and a stroke of eight. Every one of them followed the instruction exactly.

2:27And no two of them are the same sign. Look at the opening at B. Roughly eighty-six degrees, then seventy, then forty-eight, then ninety-six. Notice that they do not simply open wider as you go along. The third one is the tightest of the four, and the fourth then opens widest of all. Measure straight across, from A to C, and you get four different distances. About eight point seven, seven point six, six point one, and nine point three.

3:01Four different shapes out of one list of two numbers. The list was not enough. So add one more measurement to it. The opening at B. Say that opening is eighty degrees. Now try to draw something different. You cannot. The four centimetre stroke leaves B along one side of that opening, the eight centimetre stroke along the other. Both ends are pinned, and the distance across is forced. Eight point three centimetres.

3:33Here is the thing worth seeing. The opening and that distance are two ways of saying one thing. Widen the angle by any amount and the distance across grows. Close it and the distance shrinks. It never repeats. So the third measurement can be the angle, or the distance across. Either one closes the list, and you need only one. Two figures that agree in shape and agree in size are called congruent.

4:05One is an exact duplicate of the other. Not similar, not close. Exact. And there is a way of settling it with no measuring at all. Lay one figure on top of the other. If they coincide everywhere, they are congruent. That is superimposing, and it is worth being careful about what it is for. Superimposing checks two figures that are already in front of you. It is no help whatever to our sign painter, because his two boards are never in the same place.

4:39The test and the list are answers to two different questions, and this whole topic is really about the list. There is a catch in that test, and it is the thing people get wrong. Here is a shape. And here is the same shape, turned by half a turn. Laid down as they are, they do not coincide anywhere. That does not make them different figures. Turn one of them round first.

5:07Now they coincide exactly, so they were congruent all along. The same goes for a flip. Here is a shape and its mirror image, facing opposite ways. Turn one of them over, lay it down, and it matches. So the test is this. Move it however you like, then see whether it coincides. Turning and flipping are allowed. Stretching is not. Now the mistake this idea invites, and almost everybody makes it once.

5:39Congruent gets heard as same shape. That is half the definition, and the missing half is the half that bites. Here are two triangles. One with sides three, four and five. One with sides six, eight and ten. Work out the angles in each and they agree exactly. Ninety degrees, about thirty-seven, and about fifty-three, in both. Same shape, in every sense of the phrase. But every side of the second one is exactly twice the first.

6:13Lay one on the other and it hangs off the edge. They are not congruent. Shape and size, both. The four signs had matching lengths and differed in shape. These two have matching shape and differ in size. So how short can the list get? That depends entirely on the figure. Take a circle. One number. The radius. That is the entire list. Two circles with the same radius are congruent, and two circles with different radii never are.

6:49And you can show yourself that nothing more is wanted. Suppose I also told you the area, and the distance round the edge. Both follow from the radius. They narrow nothing down. They only repeat what you already had. More measurements are not safer. They hide which measurement is doing the work. Now a rectangle. One number is not enough here, and it is worth seeing why. Suppose I tell you only the area. Say it is six square centimetres.

7:23That could be a half by twelve, a long thin strip. Or one by six. Or two by three. I built three hundred rectangles from sides between half a centimetre and twelve, and counted. Sixty-two of the areas belong to more than one of them. The distance round the outside is no better. Forty-three of those are shared as well. Give the two sides instead, and every one of the three hundred is pinned down exactly.

7:53A circle takes one number. A rectangle takes two. The list is as long as the figure makes it. Back to strokes meeting at a point, and add a third stroke. Three arms leaving one junction. And look at the way this one sits. Two of the arms run almost straight through the junction, so most of the figure reads as one long stroke. The third branches off it at close to a right angle.

8:24It is lopsided, and that is on purpose. Nothing says three arms have to be evenly spread. Evenly spread would put a hundred and twenty degrees between each pair. This one has about a hundred and seventy on one side. So how long is the list now? Three lengths, one for each arm. And then the angles between them. Two of those will do, because the third is whatever is left over out of three hundred and sixty.

8:55Five measurements. Two arms took three, three arms take five, and each new arm after that costs you two more. One last thing, and it is a puzzle rather than a lesson. Here is a five by five board. Twenty-five squares. Shade the middle one and leave it out. That leaves twenty-four. Cut those twenty-four into six pieces, and every piece congruent to every other. Six pieces out of twenty-four squares, so four squares to a piece.

9:29Four squares can be joined in five different ways, and I tried all five. Only one of them will tile this board. The L. Three squares in a line with a fourth turned off the end. And here is the cutting. Six L pieces, each an exact duplicate of the others, turned and flipped into place, which is exactly what congruent has meant all along.