A point fixes a location; a segment is the shortest route between two points

0:00Take a pencil and put a dot on a page. Now sharpen the pencil, and do it again. Smaller dot. Sharpen it more. Smaller still. Keep going. Sharper pencil, finer needle, whatever you like. The dot gets smaller and smaller. And here's the thing — it never gets to nothing. However fine your tip is, the mark you make still has some width. It's still a tiny blob of graphite.

0:29So the dot is never actually the thing we want. It's always an approximation of it. That sounds like a problem. It's about to turn into the first idea of the chapter. Here's how your book handles it. It doesn't try to draw the thing. It defines it. A point determines a location. That's what it's for. It tells you where. And then the crucial part. A point has no length. No breadth. No height.

1:00None. Not a little bit. Not too small to see. None at all. Which means a point is not a very small dot. Smallness belongs to the dot, not to the point. The dot on the page is a model. It's a stand-in, on paper, for something that has a position and nothing else. And once you accept that, all the confusion goes away, because you stop trying to picture it and start using it.

1:30Your book gives you three things to hold in your hand as models for a point. The sharp tip of a compass. The tip of a sharpened pencil. The pointed end of a needle. And notice what all three have in common. They're the sharpest things in your geometry box. But every one of them, under a microscope, is rounded and rough and definitely has width. None of them is a point. Each of them is the best a physical object can do at being one.

2:04That's what a model is. Not the thing. Something you use to think about the thing. Now, a practical problem, and it's the reason your book does something that looks trivial. Here are three dots on a page. Tell me about the middle one. You can't. Not without pointing at the screen. There's nothing to say. So we give them names. Capital letters. This is Z. This is P. This is T.

2:33And now you can talk. Point Z is above point P. Point T is furthest right. That's not decoration. Every single statement in the rest of this chapter is about named objects. Without labels there's no geometry — there's just a picture you can gesture at. Right. Take a rectangular sheet of paper. Fold it once, anywhere you like, and press the fold down. Now open it out and look at the crease.

3:05That crease is straight, and it runs from one point to another. Call the ends A and B. And your book uses it as the first picture of the next idea: a line segment. But hold on, because we have to be careful here, and this is where it gets interesting. Here's A, and here's B. Now let me draw you a route from A to B. There. That's a route.

3:31And here's another one, going a different way round. And another. And another. And I could keep drawing routes from A to B all day, and never run out. So if I say to you the path from A to B — which one do I mean? That's the problem. There are infinitely many, so the phrase doesn't pick out anything. And here's the fix, and it's one word. Out of all those routes, one of them is the shortest. Exactly one. And it's the straight one.

4:02That is the line segment from A to B. Not a path. The shortest path. And now the phrase means something, because there's only one shortest. That's why the word shortest is in the definition. Take it out, and segment A B stops naming anything in particular. Two more things about that segment, and they're both small but worth being sure about. The points A and B are called the end points. That's the book's own word for them.

4:35And here's the question people get wrong. Do the end points belong to the segment, or are they just the boundary? They belong. The segment is the shortest path together with its two end points. Think about the crease again. Does the crease stop just short of the corner of the fold? No. It runs all the way. The ends are part of it. Now the second small thing, and this one is a set-up for later.

5:05I can write this segment as A B. Or I can write it as B A. Are those different? No. They're the same segment. The segment is a set of points — the shortest route, and the two ends. Writing the letters the other way round doesn't change which points are in it. So A B equals B A. That's a statement about the object, not about the spelling. I'm flagging this now because a bit later in the chapter you'll meet rays. And for rays, the order absolutely does matter.

5:40So don't file this away as a general rule about letters. It's a specific fact about segments. Let's use it. Here's a figure from later in the chapter. Five points, marked and labelled. L, M, P, Q, R. They're joined up in a zig-zag, in that order. Question: how many line segments are in this figure? Have a look before I say. Count them off. Here they are. L to M. M to P. P to Q. Q to R.

6:13Four segments. Not five. And it's worth being clear about why it's four. There are five points, but the path is open — it doesn't join back up. Each segment needs two points, and consecutive segments share one. Five points, joined in a chain, gives you four links. Now a better question, and this is the first one in the chapter where you need the definition rather than the picture.

6:43For each of those five points: how many segments does it lie on? Start with L. L is at the end of the chain. It's on L M, and that's all. One segment. Same at the other end. R is only on Q R. One segment. Now M. M is in the middle of the chain. It's the end of L M, and it's also the start of M P. Two segments.

7:09And the same for P, and for Q. Two each. So the pattern is: the two points at the ends lie on one segment, and the three in the middle lie on two. And notice why, because the why is more useful than the count. A point inside the path is shared by the two segments that meet there. A point at the end has nothing on one side of it.

7:35If I added a sixth point on the end, you could tell me the answer without drawing anything. So where does that leave us? A point fixes a location, and nothing more. A segment is the shortest route between two points, and it carries its end points with it. But notice what a segment has that a point doesn't. It has ends. It stops. And that turns out to be the interesting bit, because you can ask: what if it didn't?

8:06What if you took a segment and let it run on for ever at one end? Or at both? That's a ray, and that's a line. And they're what your book does next. Before that, here's one for you. Draw four points, and join them in a closed loop — so the last one joins back to the first. How many segments now? And how many does each point lie on? It's not the same answer as our zig-zag, and I'd like to know why.

8:35Put it in the comments. See you in the next one.