2D vs. 3D objects
So I have a three-dimensional solid right over here. And I want to imagine what type of a shape I would get if I were to make a vertical cut. And just to refresh ourselves, or give us a sense of what a vertical cut is, imagine if this was made out of jello or something kind of fairly soft. But it's still a three-dimensional solid. And I were to make a cut-- let's say I were to make a cut right over here. So let's say I had this big, sharp metal thing-- let me draw it like that. So you have this big, sharp metal thing. Let me draw it a little bit neater. So you have this big, sharp metal thing that I'm going to cut right over here. So this is the thing that I'm going to make the cut. And I'm going to go straight down. This is a vertical cut that we're talking about. So this is the thing that I'm going to cut with. Let me make it big enough so that it can capture the shape that will result. So this thing right over here, it's right in front. And I'm going to cut-- I'm going to make it go straight down and cut through this jello, or whatever you want to call it, this rectangular pyramid of jello. And what would be the resulting shape of the intersection between the jello and this thing that I'm using to cut it? And now I encourage you to pause your video and think about what the resulting shape would be. And the shape would be in two dimensions. Because this purple surface is a two-dimensional-- you could view it as part of a plane. And so where this intersects when you cut down this rectangular pyramid is the shape we're looking for. So I encourage you to pause the video and think about it or try to come up with it on your own. So let's think about it. And let me draw the rectangular pyramid again. So that's the same one. And now let me see what it would look like once I've done my cut, once I've brought this thing down. So then this is where I cut. So I cut it right over here. And then it'll exit the bottom. It'll cut along this side like that, cut along that side like that. And then it'll exit the bottom right over there. And so let me draw my whole thing. And so once I slice it down, it will look like this. My best shot at drawing it-- it will look like this. This is a vertical cut. So I've brought this thing down. And now the intersection between the thing that I'm cutting with and this pyramid is going to be this shape right over here. It cut into the top right over there. It would get all the way to the bottom right over there. And along this side, it would cut right there. And along that side, it would cut right over there. So what's the resulting shape in two dimensions of essentially the intersection between the slicer and the jello? Well, it would be this thing. It would look like a trapezoid. Let me do that in a new color. I'm overusing that one color. It would look like a trapezoid. So this would be the resulting shape. So the resulting shape would look like this. Just like that, if you made the cut right over there.