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Ways to cross-section a cube

Learn about all the different ways we can obtain cross-sections of a cube.

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Video transcript

What I wanna do in this video is explore the types of 2D shapes we can construct, by taking planar slices of cubes. So what am I talking about? Let's say we wanna deconstruct a square: How can we slice a cube with a plane to get the intersection of this cube, and that plane to be a square? We'll imagine that that plane we are to cut, just like this, the square is maybe the most obvious one, so it cuts the top right over there, it cuts the side, right over here, it cuts the side, I guess on the back of the glass cube, where you will see it, right over there, dotted line, and then it cuts this, right over here. So you can imagine a plane that did this, and if I wanna draw the broader plane, I can draw it like this. Let me see if I can do a decent, an adequate job at drawing the actual, as you see you'd say a part of the plane, that is cutting this cube. It will look - it could look something... it could look something like this. And I can even color in the part of the plane that you could actually see it the cube were opaque, if you couldn't see through it. But if you could see through it you would see this dotted line, and the plane would look like that. So a square is a pretty straightforward thing to get, if you're doing a planar slice of a cube. But what about a rectangle? How can you get that? And at any point, I encourage you: Pause the video and try to think about it on your own: How can you get these shapes that I'm talking about? Well for a rectangle you can actually cut like this: So, if you cut this side like this, and then cut that side like that, and then you cut this side like that - I think you'd see where this is going, this side like that, and then you cut the bottom, right over there, then the intersection of the plane that you are cutting with; so, the intersection, let's see: This could be the plane that I'm actually cutting with, so the intersection of the plane that I'm cutting with, and my cube is going to be a rectangle. So it might look like this, and once again let me shade in the stuff, if you kind of view this, if you imagine the plane just like one of those huge blades that magicians use to saw people in half, or pretend like they are - they give us the illusion of sawing people in half, it might look something like this. Ok! So you'll go like: "Ok, that's not so hard to digest, that if I intersect a plane with a cube, I can get a square, I can get a rectangle. But what about triangles?" Well, once again pause the video if you think you can figure it out, triangles are not so bad. You can cut this side over here, this side right over here, and this side right over here, and then, this is it - of course I can keep drawing the plane, but I think you get the idea - this would be a triangle. There's different types of triangles that you can construct. You could construct an equilateral triangle, so as long as this cut is the same length as this cut right over here, is the same length as that upper length, or the length that intersects on this space of the cube, that's gonna be an equilateral triangle. If you pushed this point up more, actually I'd do that in a different color, you were going to have an isosceles triangle. If you were to bring this point really, really close, like here, you would approach having a right angle, but it wouldn't be quite a right angle: you'd still have these angles who'd still be less than 90º, you can approach 90º. So you can't quite have an exactly a right angle, and so since you can't get to 90º, sure enough you can't get near to 91º, so actually you're not gonna be able to do an obtuse triangle either. But you can do an equilateral, you can do an isosceles, you can do scalene triangles. I guess you could say you could do the different types of acute triangles. But now let's do some really interesting things: Can you get a pentagon by slicing a cube with a plane? And I really want you to pause the video and think about it here, because that's such a fun thing. Think about it: How can you get a pentagon by slicing a cube with a plane? All right, so here I go, this is how you can get a pentagon by slicing a cube with a plane: Imagine slicing the top - we'll do it a little bit different - so imagine slicing the top, right over there, like this, Imagine slicing this backside, like that, this back side that you can see, quite like that, now you slice this side, right over here, like this, and then you slice this side right over here, like this. This could be, and alike I'm gonna draw the plane - yet maybe it won't be so obvious if I try to draw the plane - but you get the actual idea: if I slice this, the right angle (not any right angle - 90º - but 'the right angle' - the proper angle. Actually I shouldn't use 'right angle', that would confuse everything.) If I slice it in the proper angle, that I'm doing with the intersection of my plane then my cube is going to be this pentagon, right over here. Now let's up the stakes something, let's up the stakes even more! What about a hexagon? Can I slice a cube in a way, with a 2D plane, to get the intersection of the plane on the cube being a hexagon? As you could imagine, I wouldn't have asked you that question unless I could. So let's see if we can do it. So if we slice this, right over there, if we slice this bottom piece, right over there, and then you slice this back side, like that, and then you slice the side that we can see right over there, (This side, I could have made it much straighter) So hopefully you get the idea! I can slice this cube so that I can actually get a hexagon. So, hopefully, this gives you a better appreciation for what you could actually do with a cube, especially if you're busy slicing it with large planar planes - or large planar blades, in some way - There's actually more to a cube that you might have imagined in the past. When we think about it, there's six sides to a cube, and so it's six surfaces to a cube, so you can cut as many as six of the surfaces when you intersect it with a plane, and every time you cut into one of those surfaces it forms a side. So here we're cutting into four sides, here we're cutting into four surfaces of four sides, here we're cutting into three, here we're cutting into five - we're not cutting into the bottom of the cube, here - and here we're cutting into all six of the sides, all six of the surfaces, of the faces of this cube.