What "parallel" actually claims, and why it is hard to check

0:00Two lines can be related to each other in more ways than you might expect. And before comparing any of them, it helps to have words for what happens where they touch. Two segments can cross, and where they cross there is a point. That point can sit part way along both of them, with the segments carrying on past it. Or one segment can stop exactly there, at the point where it meets the other.

0:28The place a segment stops is called an endpoint. The place exactly halfway along it is called the midpoint. So a meeting is described by saying where it happens on each segment, and at what angle. Here are five pairs of segments, scattered about, and they do five different things. This pair crosses, and the crossing point is part way along both. So does this pair, at a point over here.

0:58These two share an endpoint. One point, and it is where both of them stop. The opening between them measures a hundred and fifteen point three degrees. Then two pairs that do not touch at all. This pair runs along with the gap between them looking much the same at both ends. And this pair also runs along together, but the gap is clearly narrower on the right. Which raises the obvious question about the two pairs that miss.

1:31Would they meet, if you carried them on? Take the narrowing pair first. Extend both, keeping each one straight. The gap shrinks. Keep going and it looks like they are heading for each other. But the paper runs out, and the answer has not arrived. Now the other pair, the one whose gap looked steady. Extend those too, and watch carefully, because a very slow closing looks like no closing at all.

2:00The paper runs out again. I am not going to tell you the answers, and it is worth being clear why. The trouble is that a drawing is about twenty centimetres wide, and the claim is about for ever. Here is what that costs, in numbers. Take two lines whose gap closes by one millimetre across the width of a page. One millimetre. On a page. You would be doing well to notice that at all.

2:30Those two lines are a bit over a quarter of a degree apart. And if they start eight millimetres apart, they meet eight pages further on. Make it worse. Let the gap close by less than the width of the pencil line you drew it with. They still meet, eleven pages away, and there is nothing on your page that could have told you. You have been surrounded by these since long before anybody named them.

2:59The white keys of a piano. The slats of a bench. A run of upright railings. In each case the same shape repeats, and the edges keep pace with one another. In a photograph of a keyboard the edges seem to lean towards each other, which is worth noticing and setting aside. That is a fact about photographs, and we will come back to it another time. Take a rectangular sheet of paper. Its opposite edges keep pace; the edges that meet at a corner make a square corner.

3:34Fold it in half and the crease keeps pace with both of those edges. Three lines now, where there were two. Fold it again and there are five, then nine, then seventeen, each fold dropping a new one midway between every neighbouring pair. So here is what the word actually claims, and it has two halves. Two lines are parallel if they lie on one flat surface, and if they never meet, however far either one is extended in either direction.

4:08Read the second half again, because it is stronger than it looks. Not that they do not meet on your page. That they never meet at all. Which is precisely the claim your page cannot check, as we just saw. Now the first half, which almost everybody drops when they repeat this. One flat surface. A tabletop, a sheet of paper, a windowpane. Why does that first half matter? Here is the case that shows it.

4:39Rule a straight line across the top of a desk. Now rule another one across a board standing up behind the desk. One line lies flat. The other stands upright, running left to right along the board. Extend both as far as you like, in both directions. They never meet. They cannot. One of them is always above the other, by the height of the board. And they are not parallel, because they do not point the same way at all.

5:10Never meeting is only half. Without the flat surface, that half proves nothing. Back to a flat page, then, and to a question worth being careful with. Here is a field of dots with nine segments drawn on it, at all sorts of slopes. Which pairs of them appear to be parallel? Appear. That word is doing a lot of work, and it is there on purpose. Look for a while and you will make some pairs. Two verticals, obviously.

5:42Two that lean the same way by what looks like the same amount. And then one or two you keep going back and forth on. That hesitation is the honest part. Your eye has given you what it can, and it is not certain. But look again at what these segments are drawn on. The dots are in a grid, evenly spaced, which changes the question completely. Take this segment. From one end to the other it goes seven dots across and three dots up.

6:16Seven across, three up. That is not a judgement. That is counting. And two segments point the same way exactly when their counts are multiples of each other. Seven and three, against fourteen and six. Same direction, and I never looked at the slope. It also explains why some are harder to draw than others. A step of one across and one up fits the grid sixty four ways; seven across and three up fits only twelve.

6:49Once you have decided, there is a way of writing the decision onto the drawing. A small arrowhead is drawn on each of the lines you are claiming keep pace. Same mark, same set. Both lines carry it, because the claim is about the pair. If a second set on the same drawing is also a matched pair, they get double arrowheads. That way two claims can sit on one figure without anybody having to guess which is which.

7:20And a small square in the corner still says two lines are at right angles. None of these marks make anything true. They record a decision somebody made, and they say what that decision was, which is not the same thing. Now a small test, and it is designed to be unfair. One segment on the left. Two more on the right, fanning out from a shared point. One of those two keeps pace with the segment on the left. The other does not.

7:52Which one? Take your time. Here is what you are up against. Those two slopes differ by about half a degree. Over a segment six centimetres long, that puts their far ends six tenths of a millimetre apart. That is less than the thickness of the line they are drawn with. I am not going to tell you which. Not because it is a trick, but because looking genuinely cannot settle it.

8:22So where does that leave you? With a claim about for ever, and a drawing about the width of your hand. With an eye that cannot separate half a degree, and a ruler that is not much better. You might reach for distance. Measure the gap here, measure it there, see if it changed. That instinct is a good one, and staying the same distance apart really does follow from being parallel.

8:49But it is not what the word claims, and two measurements on a short drawing carry the same problem as before. What actually settles it is neither looking nor measuring gaps. It is a third line, drawn straight across both. Because of where that line cuts them, the angles it makes will answer the question outright, and that is where this goes next.