Naming corners in order, and why a rotated square is still a square

0:00Here is a rectangle with its corners labelled A, B, C and D, going round. So the figure is called A B C D. That much looks like pure housekeeping. But try this. Is it also called A B D C? Same four letters, same four corners, and the answer is no. That ordering names nothing at all. Which is strange, if the letters are just labels stuck on corners.

0:28They are not. The order is carrying information, and it is worth finding out what. Watch what happens if you read a name as an instruction to walk. Start at A. Go to B. That is a side, so you travel along the edge of the figure. B to C: another side. C to D: another. D back to A: you are home. The name traced the boundary. Every step went along an edge.

0:57Now try walking A B D C the same way. A to B is fine. But B to D is not a side. To get from B to D you have to cut straight across the middle. That is the whole diagnosis. A name is a walk round the boundary, and a step that leaves the boundary breaks it. There is a small extra thing here that is worth noticing.

1:23Finish that broken walk. After the illegal jump from B to D, you go D to C, which is a proper side. Then C back to A. And that is another cut across the middle. So A B D C did not make one mistake. It made two. And that is not a coincidence. If a walk leaves the boundary, it has to get back on again, and getting back is another jump.

1:51Check every ordering of four letters and you will never find one with exactly one bad step. They always come in pairs. So now we can count, instead of guessing. Four letters can be written in twenty-four different orders. That is four choices, then three, then two, then one. How many of those twenty-four are actually names? A walk round the boundary has two things you get to choose. Where you start, and which way you go.

2:23Four corners to start from, two directions to travel. Four times two. Eight names. Which means sixteen of the twenty-four orderings name nothing whatsoever. And notice A B C D was never the correct one. It was one of eight equally correct ones. Let me give you one to try, because the rule is quick once you trust it. Here is a square, and its corners are labelled S at the top left, P at the top right, Q at the bottom right, and R at the bottom left.

2:58Four names are suggested: P Q S R, S P Q R, R S P Q, and Q R S P. Three of them are genuine names of that square. One is not. Which one? Walk each. S P Q R goes round the boundary, so that is fine. R S P Q starts at a different corner and goes the same way round. Fine. Q R S P likewise. But P Q S R steps from Q to S, and Q and S are opposite corners.

3:33That is a cut across the middle, so P Q S R is the one that is not a name. Now the second half, and it starts with something that feels like a trick question. Here is a square. Take hold of the paper and turn it. Not a lot. Just enough that it is standing on one of its corners. Is it still a square? A great many people say no. They say it has become a diamond.

4:01There is no such category. There is a test, and the honest thing to do is run it. The square test was two lines. All four sides equal, and all four angles ninety degrees. So measure the four sides of the turned figure. Every one of them is exactly what it was before you turned the paper. Nothing was stretched. Now the four angles. Every one is still a right angle.

4:30Both lines pass. It is a square. Not a square-ish thing, not a diamond. It satisfies both conditions, which is the only thing being a square has ever meant. It is worth being clear about why that had to happen, rather than just observing that it did. Look at what a turn actually does to the figure. It changes which side is at the top. It changes which corner is nearest you.

5:00It does not change how long any side is. Turning the paper does not stretch anything. And it does not change any angle. A right angle stays a right angle however you hold it. Now read the square test again. It mentions lengths, and it mentions angles. It never mentions the page, or the top, or which way anything is pointing. So there is nothing in it for a turn to disturb.

5:30Which gives you the next result for free, and I want to be explicit about why it is free. Take a rectangle instead, and turn it onto a slant. Is it still a rectangle? Run the test: opposite sides equal, all angles ninety degrees. The turn did not change a length or an angle, so both lines pass exactly as before. But look back at the argument we just used for the square. Did it use the sides being equal anywhere?

6:03It did not. It only used the fact that turning preserves lengths and preserves angles. So it was never an argument about squares. It was an argument about turning, and it settles any figure whose test is written in lengths and angles. Here is where this becomes properly useful. Put a tilted square on a grid of dots, with every corner sitting exactly on a dot. Now you cannot tell by looking, and you have no ruler. But you can count.

6:36This side goes four dots across and three dots up. So does the next one, and the next, and the last. Four sides that step the same two counts are four sides of the same length. First condition, passed, by counting. And four across and three up, followed by three across and four back, turns you through exactly a right angle each time. Second condition passed. It is a square, and you settled it without measuring anything.

7:10One last thing, tying the two halves together. The naming rule is not tidiness. It is what lets anyone write about a figure without drawing it. If I say the sides A D and B C are equal, you know exactly which two sides I mean, because you know how to read a name. If I say draw the diagonal P R, you know that P and R are opposite corners, because if they were next to each other P R would be a side.

7:41That is the whole payoff. The order tells you which pairs are joined and which are not. And it goes on working after you turn the paper, because turning does not change who is next to whom. So the two halves of this were the same idea twice. A name is not decoration on a picture. It is a walk, and it survives the picture being moved. A square is not a shape you recognise. It is two conditions, and they survive the picture being moved too.

8:15In both cases the thing that matters is written in terms that a turn of the page cannot reach. Which is why you can be handed a figure at any angle, on a grid, with letters at its corners, and still say something certain about it. Next time: constructing one of these from nothing, with the corners in the right order and the check at the end.