The shortest and longest crossing inside a rectangle

0:00Here is a rectangle. Its long sides are seven centimetres, and its short ones are four. Label the corners A, B, C and D, going round. Now put a point on each of the two short sides. X sits somewhere on the left-hand side, and Y sits somewhere on the right. Neither one is pinned down. X can slide anywhere from A down to D, and Y anywhere from B down to C.

0:28Draw the crossing between them. That is a length which changes as the two points move, and the question is how much, and between what and what. Before measuring anything, guess. Where should X and Y go to make that crossing as short as it can possibly be? And where to make it as long as it can possibly be? Take a moment with both. A very common first guess is that the closest position is the one at the top, with both points tucked up near A and B.

1:04And a common guess for the farthest is that it must be at a corner, and that there is one such place. It is worth having your instinct written down before the measuring starts, because measuring has a way of confirming whatever you already believed. And the two answers turn out to be different in kind, which is the part worth waiting for. So measure. Put X half a centimetre below A, and Y three centimetres below B.

1:36The crossing comes out seven point four centimetres. Move them. X one centimetre down, Y one centimetre down. Seven centimetres exactly. Once more. X two centimetres down, and Y four centimetres down, which puts Y right at the bottom corner. Seven point three. Three positions, three numbers, and so far no pattern at all. They are not even going in a direction. The drops on the left were a half, then one, then two, and the lengths went down and then back up.

2:14Three sentences like those are already hard to hold on to. So write them differently. One column for how far X is below A. One for how far Y is below B. One for the length of the crossing. The same three facts, but now they sit underneath each other, and you can run your eye down a column. That is the entire reason for a table. Nothing new has been recorded. It has been recorded somewhere the eye can actually use it.

2:49And now something shows up. The crossing was shortest in the row where the two drops were the same number. Here is why that happens. However you place X and Y, the sideways part of the journey is always identical: seven centimetres, straight across. The only thing that ever changes is the up-and-down part. And that is not either drop on its own. It is the difference between them. Slide both points down by the same amount and the crossing does not change at all.

3:24It is only ever answering to the gap between the two. And because that gap is usually small, most positions are crowded near the bottom of the range. Out of every position the two points can take, more than half give a crossing within half a centimetre of the shortest one. Which tells you exactly where to look for the shortest one. Make that difference zero. Put X and Y level with each other, at the same height on their two sides.

3:56Now the crossing has no up-and-down part left in it at all. It runs straight across. And straight across is seven centimetres, because that is how wide the rectangle is. So the shortest crossing is seven. Notice that nothing about the four centimetre sides came into that. Only the seven. But it is still a thing you measured, and there is a better reason available. Look at what sits above the crossing when X and Y are level.

4:29A four-sided figure, with corners A, B, Y and X. What kind of figure is it? Its top and its bottom are both horizontal. Its two sides are both vertical. So all four of its angles are right angles, and it is a rectangle. And in a rectangle, opposite sides are equal. XY and AB are opposite sides of that rectangle. So XY is AB. Not approximately, and not as measured — the same length, for the same reason any rectangle's opposite sides are.

5:06And look at what that argument never did. It never measured the crossing. It read the answer off a side that had already been drawn before either point was put on the figure. Now the part that is easy to walk straight past. There is not one closest position. There are as many as there are heights on a four centimetre side. Level at the top, level at the bottom, level anywhere in between.

5:36Every single one of them gives seven centimetres. On a millimetre ruler you could count forty-one of them. In truth there is no counting them at all. Pick any height you like, put both points at it, and you get seven. The shortest crossing is not a place. It is a whole family of places, and the answer is identical at every one of them. The other end of the range behaves completely differently.

6:06To make the crossing long, you want the difference between the two drops to be as big as you can get it. The biggest difference available is the full four centimetres. One point at the very top of its side, the other at the very bottom of its side. And that can happen in exactly two ways. X at A with Y at C. Or X at D with Y at B.

6:32Two positions. Not a family this time — two. And everything in between is strictly longer than level and strictly shorter than those two. Nothing anywhere ties. And look at what those two crossings actually are. X at A and Y at C is the line from one corner of the rectangle to the corner opposite it. X at D and Y at B is the other one. So the longest crossing is not some new length that appeared. It is a corner-to-corner line, and a rectangle has two of them.

7:09Measure them both and they come out the same: a shade over eight centimetres. About eight point one. That those two lines are equal to each other is worth noticing on its own. It is not an accident of these numbers. It is true of every rectangle there is, and it is the next thing worth proving. One test, to find out what the argument actually used. Keep the seven centimetre width, but make the short sides ten times longer. Forty centimetres instead of four.

7:45What happens to the shortest crossing? Nothing at all. Still seven centimetres, still reached whenever the two points are level, still for the same reason. The longest one does change. It stretches out with the sides, and now it is over forty centimetres. So the floor was never about the sliding sides. It was the distance across, and the sliding sides had nothing to do with it. So what was this exploration really about?

8:18A length that is not a number but a range. It has a floor of seven and a ceiling of about eight point one, and it never once leaves that span, however the two points are placed. And the two ends of it are not alike. The bottom is reached everywhere the points are level. Endlessly many positions, all giving the same answer. The top is reached at two positions only, and nowhere else.

8:46And the whole span is barely a centimetre wide. Four centimetres of sliding buys you about ten millimetres of length. One length, two questions, and two completely different kinds of answer.