 Hello, and welcome to screencast 121 part B, which is a part two of working with definitions. Working with definitions and instantiating them in particular is so important that the more practice we get the better. And so we're going to practice a little bit with this, with an example that's not in your textbook. It's not even based on numbers, but on geometry. And that's the definition of a polygon. So a polygon is something you probably know what it is. If it'd be a good idea before you read this definition, I'm going to show you to just take a second, pause the video and draw some examples of what you think a polygon is, maybe some examples of things that aren't polygons. Just to kind of get straight in your mind what you think it is, because we're going to have to go on this definition, I'm going to show you here. So after you have done that, let's read the definition here. This is taken from Wikipedia. It says that a polygon is a flat shape that consists of straight lines that are joined to form a closed chain or circuit. So you notice in this definition, there's sort of three major pieces to this that seem to be emphasized, the notion that a polygon has to be flat or something that isn't sticking up or curving in 3D, that it consists of straight lines and not semicircles or arcs or curvy things. And then this last one, closed chain, now flat shape and straight lines are pretty straightforward. We don't really necessarily need definitions for that, but closed chain, you might wonder quite legitimately, what is a closed chain? It's kind of hard to move forward with this definition until we understand what that is. So this happens all the time in mathematical definitions. Mathematical definitions build upon themselves and they're typically defined in terms of things that have definitions of their own, like closed chain. So what is a closed chain? Let's make sure we know what that is. Well on the Wikipedia page, actually under closed chain is a hyperlink and it will take you to this page here that contains this text. The actual term closed chain is a closed chain. Well what's a chain first of all? Well here's the definition of a chain. So notice we have to recurse ourselves a few steps backward in this definition. We have the definition of polygon that depended on knowing what a closed chain was and we can't really understand what a closed chain is until we know what a straight up chain is. So what is a chain? Well it's a connected series of line segments and if you weren't clear on what a line segment is, that's got a definition too. So it's a connected series of line segments period. So let's instantiate that definition up here at the top of the slide. So what does a chain look like? Well it's a connected series of line segments. So something like this would be considered a chain or one line segment would be considered a chain or a couple that kind of do like this would be considered a chain. But something like this would not be like a two line segments that are sort of separated from each other. It's not connected. That's the word connected there is really important in that definition. So these are chains, this isn't. That's instantiating the definition. Now since now that I think we're, if you're clear on what a chain is, let's talk about what a closed chain is. Well it's one a chain in which the first vertex coincides with the last one. Okay, so actually none of these guys we drew here would be considered closed chains because the first vertex is not connected with the last one. Okay, so here's what a closed chain might look like. And here's the line segment, drawing my chain, draw it down here, and then maybe that's supposed to connect up right there. Okay, so that's what a closed chain would look like. Okay, so now we understand the three major portions of the definition of polygon, flat, straight, and closed. So let's move on to a little concept check here. And this is a something I'm going to give you, although in practice you would be doing this yourself. And there are some some figures here, six figures I'm going to draw. And I want you to determine whether these are polygons or not. Okay, so the first one here is just a straight triangle. And by the way, you've probably noticed my handwriting is not the greatest on these things. Forgive me. And just kind of assume that unless I say so, all the lines you see here are perfectly straight. So triangle, this sort of a shape here, it's got a dent in it. This shape here, which is going to be like the previous one, except that point is going to actually connect with the lower line segment there. This shape, which is going to be kind of in an X form. And there's no intersection vertex right here, the lines are simply overlap and there's no vertex. And then lastly, this sort of a U shaped, there's vertices at all these corners here, in case you can see them. And then finally, this one, which is these are truly not straight. I'll even draw a little loop like this, because it's kind of fun. There we go. And that's f. Okay, so question is, which according to the definition, which of these six figures are polygons? Now go with the definition. Don't go with what you think a polygon is, but what does the definition specify a polygon is positive definition, pause the video, and come back when you're ready. Okay, so you might be surprised that all of the first four are polygons, but neither of the last two are polygons. Some of these are surprising and some aren't, but there are some surprises in here for sure. Let's look at the first four. Well, I think it's pretty clear that the triangle one is a polygon. But the question is why? How do we know it's a polygon? It's not because of our intuition. It's because it's hits all the important specifications of the definition. It's a flat shape. Yes, it's consists of straight lines. Yes. And those lines are joined to form a closed circuit. So if I kind of go around counterclockwise one, two, three, one, the last vertex is connected to the first one, that makes it a closed chain, according to our other definition. So that's a polygon. So as this one, even though it's got kind of a dent, you might have been wondering, does the dent disqualify from being a polygon? Well, not this time. It's not what we call a convex or regular polygon. Is it a flat shape? Yes. Is it consist of straight lines? Yeah. Is it a closed chain or circuit? Yeah, I can go kind of go around in order like this. And I end up the last vertex is connected to the first one. Even this one is considered a polygon. Is it flat? Yes. Is it straight? Yes. Does it form a closed chain? Yes. 1234531. You might consider, well, wait a minute, that was the last vertex here, but I can still get around and call this and call this maybe like I hit this vertex twice. There's nothing in the definition of closed chain that rules out duplicate vertices that I saw. Okay, so this would be considered a polygon under that definition. This one would also be considered a polygon, even with the intersection of the line segments, because this is actually even easier, I think, to understand. If I kind of go around in a little tour like this 12341, that's definitely a closed chain. And this is flat and made up a straight line. So those are all considered polygons. These last two are a little problematic. I will say that one of the things you need to realize about mathematics is that many times the same term will have different definitions. And so you might look at this and say, I don't want to consider those polygons. This definition stinks. I'd like a different definition or at least a modified definition that maybe says a flat shape consisting of straight lines that don't intersect. You know, and maybe if I modify that definition, I would have something that I really like. But without modifications, these two guys have to be considered polygons. There are other definitions of polygons out there, particularly in some of the applications that I personally work with that specifically rule out intersections. So this guy and this guy would not be considered polygons. However, this definition, which is the only one that matters right now, does specify those are polygons. Like it or not, okay? So what that means about our definition is another question. But under the definition, those would be polygons. So those are good instances of definition, even though I don't necessarily like them, because they push the boundaries of what the definition means, is particularly what does a closed chain mean? These last two are pretty clearly not polygons. This one's not a closed chain. You'd have to connect those first and last vertices together. And so that's not a closed chain, hence not a polygon. And this one's pretty clearly not, because the lines aren't straight. So let's wrap up what we've learned here. So instantiating a definition is pretty interesting. You have to construct examples and non-examples that really push the boundaries of the definition and the terminology. Many times you're reading a definition, you will come into a term that you don't understand. And so in that case, what do you do? You go look up its definition. And a well-constructed definition will always let you do this. And what you want to look out for are extreme examples and non-examples that really do push the boundaries and help you understand the object that you are working with. That's it for now. Thanks for watching.