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What I wanna do in this video[br]is explore the types of 2D shapes

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we can construct,[br]by taking planar slices of cubes.

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So what am I talking about?

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Let's say we wanna deconstruct a square:[br]

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How can we slice a cube with a plane[br]to get the intersection of this cube,

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and that plane to be a square?

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We'll imagine that that plane[br]we are to cut, just like this,

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the square is maybe the most obvious one, [br]

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so it cuts the top right over there,

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it cuts the side, right over here,

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it cuts the side, I guess on the back[br]of the glass cube, where you will see it,

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right over there, dotted line,

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and then it cuts this, right over here. [br]So you can imagine a plane that did this,

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and if I wanna draw the broader plane,[br]I can draw it like this.

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Let me see if I can do a decent, [br]an adequate job

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at drawing the actual, [br]as you see you'd say a part of the plane,

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that is cutting this cube.[br]It will look - it could look something...

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it could look something like this. [br]And I can even color in

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the part of the plane that you could [br]actually see it the cube were opaque,

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if you couldn't see through it. [br]But if you could see through it

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you would see this dotted line,[br]and the plane would look like that.

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So a square is [br]a pretty straightforward thing to get,

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if you're doing a planar slice of a cube.

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But what about a rectangle? [br]How can you get that?

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And at any point, I encourage you:[br]

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Pause the video and try to think about it[br]on your own:

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How can you get these shapes[br]that I'm talking about?

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Well for a rectangle[br]you can actually cut like this:

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So, if you cut this side like this,

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and then cut that side like that,

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and then you cut this side like that[br]- I think you'd see where this is going,

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this side like that,

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and then you cut the bottom,[br]right over there,

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then the intersection of the plane[br]that you are cutting with;

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so, the intersection, let's see:[br]

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This could be the plane [br]that I'm actually cutting with,

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so the intersection of the plane[br]that I'm cutting with,

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and my cube is going to be[br]a rectangle.

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So it might look like this, and once again[br]let me shade in the stuff,

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if you kind of view this, [br]if you imagine the plane

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just like one of those huge blades[br]that magicians use to saw people in half,

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or pretend like they are - they give us [br]

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the illusion of sawing people in half,

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it might look something like this.

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Ok! So you'll go like: [br]"Ok, that's not so hard to digest,

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that if I intersect a plane with a cube,[br]I can get a square, I can get a rectangle.

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But what about triangles?"

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Well, once again pause the video[br]if you think you can figure it out,

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triangles are not so bad. [br]You can cut this side over here,

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this side right over here,

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and this side right over here,

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and then, this is it - of course [br]I can keep drawing the plane,

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but I think you get the idea - [br]

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this would be a triangle. [br]

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There's different types of triangles[br]that you can construct.

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You could construct[br]an equilateral triangle,

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so as long as this cut is the same length[br]as this cut right over here,

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is the same length as that upper length,

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or the length that intersects [br]on this space of the cube,

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that's gonna be an equilateral triangle.

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If you pushed this point up more,[br]actually I'd do that in a different color,

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you were going to have [br]an isosceles triangle.

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If you were to bring this point[br]really, really close, like here,

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you would approach having a right angle, [br]but it wouldn't be quite a right angle:

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you'd still have these angles[br]who'd still be less than 90º,

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you can approach 90º.[br]So you can't quite have

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an exactly a right angle, [br]and so since you can't get to 90º,

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sure enough you can't get near to 91º,[br]so actually you're not gonna be able to do

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an obtuse triangle either.

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But you can do an equilateral, [br]you can do an isosceles,

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you can do scalene triangles.

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I guess you could say you could do[br]the different types of acute triangles.

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But now let's do [br]some really interesting things:

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Can you get a pentagon[br]by slicing a cube with a plane?

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And I really want you to pause the video[br]and think about it here,

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because that's such a fun thing. [br]Think about it:

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How can you get a pentagon[br]by slicing a cube with a plane?

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All right, so here I go,

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this is how you can get a pentagon[br]by slicing a cube with a plane:

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Imagine slicing the top[br]- we'll do it a little bit different -

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so imagine slicing the top, [br]right over there, like this,

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Imagine slicing this backside, [br]like that,

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this back side that you can see,[br]quite like that,

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now you slice this side,[br]right over here, like this,

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and then you slice this side[br]right over here, like this.

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This could be,[br]and alike I'm gonna draw the plane

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- yet maybe it won't be so obvious[br]if I try to draw the plane -

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but you get the actual idea:[br]if I slice this, the right angle

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(not any right angle - 90º - [br]but 'the right angle' - the proper angle.

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Actually I shouldn't use 'right angle',[br]that would confuse everything.)

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If I slice it in the proper angle,[br]

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that I'm doing [br]with the intersection of my plane

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then my cube is going to be[br]this pentagon, right over here.

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Now let's up the stakes something,[br]let's up the stakes even more!

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What about a hexagon?

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Can I slice a cube in a way,[br]with a 2D plane,

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to get the intersection of the plane[br]on the cube being a hexagon?

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As you could imagine, [br]I wouldn't have asked you that question

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unless I could. [br]So let's see if we can do it.

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So if we slice this, right over there,[br]if we slice this bottom piece,

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right over there,[br]and then you slice this back side,

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like that, and then you slice[br]the side that we can see right over there,

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(This side, I could have made it[br]much straighter)

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So hopefully you get the idea![br]

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I can slice this cube[br]so that I can actually get a hexagon.

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So, hopefully, this gives you [br]a better appreciation

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for what you could actually do [br]with a cube,

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especially if you're busy slicing it[br]with large planar planes

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- or large planar blades, in some way -

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There's actually more to a cube[br]that you might have imagined in the past.

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When we think about it,[br]there's six sides to a cube,

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and so it's six surfaces to a cube,[br]

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so you can cut [br]as many as six of the surfaces

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when you intersect it with a plane,[br]

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and every time you cut [br]into one of those surfaces

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it forms a side. [br]So here we're cutting into four sides,

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here we're cutting into four surfaces[br]of four sides,

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here we're cutting into three,[br]here we're cutting into five

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- we're not cutting into the bottom[br]of the cube, here -

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and here we're cutting into all six[br]of the sides, all six of the surfaces,

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of the faces of this cube.