[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.66,0:00:02.66,Default,,0000,0000,0000,,What we're going to\Nexplore in this video
Dialogue: 0,0:00:02.66,0:00:09.80,Default,,0000,0000,0000,,are polyhedra, which is just\Nthe plural of a polyhedron.
Dialogue: 0,0:00:09.80,0:00:13.20,Default,,0000,0000,0000,,And a polyhedron is a\Nthree-dimensional shape
Dialogue: 0,0:00:13.20,0:00:15.71,Default,,0000,0000,0000,,that has flat surfaces\Nand straight edges.
Dialogue: 0,0:00:15.71,0:00:19.13,Default,,0000,0000,0000,,So, for example, a\Ncube is a polyhedron.
Dialogue: 0,0:00:22.84,0:00:31.05,Default,,0000,0000,0000,,All the surfaces are flat, and\Nall of the edges are straight.
Dialogue: 0,0:00:31.05,0:00:34.11,Default,,0000,0000,0000,,So this right over\Nhere is a polyhedron.
Dialogue: 0,0:00:34.11,0:00:35.59,Default,,0000,0000,0000,,Once again, polyhedra is plural.
Dialogue: 0,0:00:35.59,0:00:37.82,Default,,0000,0000,0000,,Polyhedron is when\Nyou have one of them.
Dialogue: 0,0:00:37.82,0:00:39.32,Default,,0000,0000,0000,,This is a polyhedron.
Dialogue: 0,0:00:39.32,0:00:41.79,Default,,0000,0000,0000,,A rectangular pyramid\Nis a polyhedron.
Dialogue: 0,0:00:41.79,0:00:42.97,Default,,0000,0000,0000,,So let me draw that.
Dialogue: 0,0:00:42.97,0:00:45.07,Default,,0000,0000,0000,,I'll make this one a little\Nbit more transparent.
Dialogue: 0,0:00:45.07,0:00:47.74,Default,,0000,0000,0000,,Let me do this in a\Ndifferent color just for fun.
Dialogue: 0,0:00:47.74,0:00:51.60,Default,,0000,0000,0000,,I'll make it a magenta\Nrectangular pyramid.
Dialogue: 0,0:00:51.60,0:00:55.49,Default,,0000,0000,0000,,So once again, here I\Nhave one flat surface.
Dialogue: 0,0:00:55.49,0:01:01.25,Default,,0000,0000,0000,,And then I'm going to have\Nfour triangular flat surfaces.
Dialogue: 0,0:01:01.25,0:01:04.63,Default,,0000,0000,0000,,So this right over here, this\Nis a rectangular pyramid.
Dialogue: 0,0:01:04.63,0:01:06.48,Default,,0000,0000,0000,,Now, it clearly\Nlooks like a pyramid.
Dialogue: 0,0:01:06.48,0:01:08.17,Default,,0000,0000,0000,,Why is it called a\Nrectangular pyramid?
Dialogue: 0,0:01:08.17,0:01:13.38,Default,,0000,0000,0000,,Because the base right\Nover here is a rectangle.
Dialogue: 0,0:01:13.38,0:01:18.15,Default,,0000,0000,0000,,So these are just a few\Nexamples of polyhedra.
Dialogue: 0,0:01:18.15,0:01:20.73,Default,,0000,0000,0000,,Now, what I want to think\Nabout are nets of polyhedra.
Dialogue: 0,0:01:20.73,0:01:22.90,Default,,0000,0000,0000,,And actually, let me draw\Nand make this transparent,
Dialogue: 0,0:01:22.90,0:01:26.63,Default,,0000,0000,0000,,too, so we get full appreciation\Nof the entire polyhedron,
Dialogue: 0,0:01:26.63,0:01:28.34,Default,,0000,0000,0000,,this entire cube.
Dialogue: 0,0:01:28.34,0:01:32.35,Default,,0000,0000,0000,,So now let's think about\Nnets of polyhedron.
Dialogue: 0,0:01:32.35,0:01:35.92,Default,,0000,0000,0000,,So what is a net\Nof a polyhedron?
Dialogue: 0,0:01:35.92,0:01:39.05,Default,,0000,0000,0000,,Well, one way to think about it\Nis if you kind of viewed this
Dialogue: 0,0:01:39.05,0:01:42.39,Default,,0000,0000,0000,,as made up of cardboard, and you\Nwere to unfold it in some way
Dialogue: 0,0:01:42.39,0:01:44.04,Default,,0000,0000,0000,,so it would become\Nflat, or another way
Dialogue: 0,0:01:44.04,0:01:45.50,Default,,0000,0000,0000,,of thinking about\Nit is if you were
Dialogue: 0,0:01:45.50,0:01:47.22,Default,,0000,0000,0000,,to cut out some\Ncardboard or some paper,
Dialogue: 0,0:01:47.22,0:01:49.61,Default,,0000,0000,0000,,and you wanted to fold it up\Ninto one of these figures,
Dialogue: 0,0:01:49.61,0:01:51.72,Default,,0000,0000,0000,,how would you go about doing it?
Dialogue: 0,0:01:51.72,0:01:55.31,Default,,0000,0000,0000,,And each of these polyhedra\Nhas multiple different nets
Dialogue: 0,0:01:55.31,0:01:58.41,Default,,0000,0000,0000,,that you could create so\Nthat it can be folded up
Dialogue: 0,0:01:58.41,0:02:00.68,Default,,0000,0000,0000,,into this\Nthree-dimensional figure.
Dialogue: 0,0:02:00.68,0:02:02.55,Default,,0000,0000,0000,,So let's take an example.
Dialogue: 0,0:02:02.55,0:02:05.41,Default,,0000,0000,0000,,And maybe the simplest example\Nwould be a cube like this.
Dialogue: 0,0:02:05.41,0:02:06.70,Default,,0000,0000,0000,,And I'm going to color code it.
Dialogue: 0,0:02:06.70,0:02:12.81,Default,,0000,0000,0000,,So let's say that the bottom of\Nthis cube was this green color.
Dialogue: 0,0:02:12.81,0:02:15.79,Default,,0000,0000,0000,,And so I can represent\Nit like this.
Dialogue: 0,0:02:15.79,0:02:17.50,Default,,0000,0000,0000,,That's the bottom of the cube.
Dialogue: 0,0:02:17.50,0:02:19.81,Default,,0000,0000,0000,,It's that green color.
Dialogue: 0,0:02:19.81,0:02:25.76,Default,,0000,0000,0000,,Now, let's say that this back\Nsurface of the cube is orange.
Dialogue: 0,0:02:25.76,0:02:28.26,Default,,0000,0000,0000,,Well, I could\Nrepresent it like this.
Dialogue: 0,0:02:28.26,0:02:31.47,Default,,0000,0000,0000,,And notice, I've kind\Nof folded it out.
Dialogue: 0,0:02:31.47,0:02:32.67,Default,,0000,0000,0000,,I'm folding it out.
Dialogue: 0,0:02:32.67,0:02:36.46,Default,,0000,0000,0000,,And so if I were to flatten it\Nout, it would look like this.
Dialogue: 0,0:02:36.46,0:02:38.86,Default,,0000,0000,0000,,It would look like that.
Dialogue: 0,0:02:38.86,0:02:43.06,Default,,0000,0000,0000,,Now, this other backside,\NI'll shade it in yellow.
Dialogue: 0,0:02:43.06,0:02:46.31,Default,,0000,0000,0000,,This other backside right over\Nhere, I could fold it backwards
Dialogue: 0,0:02:46.31,0:02:52.40,Default,,0000,0000,0000,,and keep it connected along\Nthis edge, fold it backwards.
Dialogue: 0,0:02:52.40,0:02:54.44,Default,,0000,0000,0000,,It would look like this.
Dialogue: 0,0:02:54.44,0:02:55.97,Default,,0000,0000,0000,,It would look like that.
Dialogue: 0,0:02:55.97,0:02:57.55,Default,,0000,0000,0000,,I think you get the\Ngeneral idea here.
Dialogue: 0,0:02:57.55,0:03:00.87,Default,,0000,0000,0000,,And just to be clear,\Nthis edge right over here
Dialogue: 0,0:03:00.87,0:03:03.22,Default,,0000,0000,0000,,is this edge right over there.
Dialogue: 0,0:03:03.22,0:03:08.49,Default,,0000,0000,0000,,Now I have to worry\Nabout this top part.
Dialogue: 0,0:03:08.49,0:03:12.14,Default,,0000,0000,0000,,Maybe it is in-- let me\Ndo it in a pink color.
Dialogue: 0,0:03:12.14,0:03:15.68,Default,,0000,0000,0000,,This top part of the cube\Nis in this pink color,
Dialogue: 0,0:03:15.68,0:03:18.34,Default,,0000,0000,0000,,and it needs to be attached\Nto one of these sides.
Dialogue: 0,0:03:18.34,0:03:20.70,Default,,0000,0000,0000,,I could attach it to\Nthis side or this side.
Dialogue: 0,0:03:20.70,0:03:22.20,Default,,0000,0000,0000,,Let's attach it over here.
Dialogue: 0,0:03:22.20,0:03:24.63,Default,,0000,0000,0000,,So let's say it's attached to\Nthat yellow side back here.
Dialogue: 0,0:03:24.63,0:03:26.66,Default,,0000,0000,0000,,So then when we\Nfold it out, when
Dialogue: 0,0:03:26.66,0:03:30.14,Default,,0000,0000,0000,,we really unpack the thing,\Nso we folded that yellow part
Dialogue: 0,0:03:30.14,0:03:32.29,Default,,0000,0000,0000,,back, then we're\Nfolding this part back,
Dialogue: 0,0:03:32.29,0:03:33.95,Default,,0000,0000,0000,,then it would be\Nright over here.
Dialogue: 0,0:03:37.48,0:03:43.03,Default,,0000,0000,0000,,And then we could fold this\Nfront face right over here.
Dialogue: 0,0:03:43.03,0:03:46.90,Default,,0000,0000,0000,,We could fold that\Nout along this edge,
Dialogue: 0,0:03:46.90,0:03:49.73,Default,,0000,0000,0000,,and it would go\Nright over there.
Dialogue: 0,0:03:49.73,0:03:50.94,Default,,0000,0000,0000,,It would go right over there.
Dialogue: 0,0:03:50.94,0:03:54.34,Default,,0000,0000,0000,,And then we have one\Nface of the cube left.
Dialogue: 0,0:03:54.34,0:03:58.07,Default,,0000,0000,0000,,We have this side\Nright over here.
Dialogue: 0,0:03:58.07,0:04:00.17,Default,,0000,0000,0000,,Well, we could do,\Nactually, several things.
Dialogue: 0,0:04:00.17,0:04:02.41,Default,,0000,0000,0000,,We could fold it\Nout along this edge.
Dialogue: 0,0:04:02.41,0:04:06.31,Default,,0000,0000,0000,,And then we would draw the\Nsurface right over there.
Dialogue: 0,0:04:06.31,0:04:09.20,Default,,0000,0000,0000,,Or if we want to do\Nsomething interesting,
Dialogue: 0,0:04:09.20,0:04:14.01,Default,,0000,0000,0000,,we could fold it\Nout along the edge
Dialogue: 0,0:04:14.01,0:04:16.12,Default,,0000,0000,0000,,that it shares with the\Nyellow, that backside.
Dialogue: 0,0:04:16.12,0:04:18.10,Default,,0000,0000,0000,,So we could fold\Nit out like this.
Dialogue: 0,0:04:18.10,0:04:19.47,Default,,0000,0000,0000,,So if we folded\Nit out like this,
Dialogue: 0,0:04:19.47,0:04:22.33,Default,,0000,0000,0000,,it would be connected to the\Nyellow square right over here.
Dialogue: 0,0:04:22.33,0:04:26.07,Default,,0000,0000,0000,,So you see that\Nthere's many, many ways
Dialogue: 0,0:04:26.07,0:04:29.30,Default,,0000,0000,0000,,to construct a net\Nor a net that when
Dialogue: 0,0:04:29.30,0:04:33.32,Default,,0000,0000,0000,,you fold it all back up will\Nturn into this polyhedron,
Dialogue: 0,0:04:33.32,0:04:34.83,Default,,0000,0000,0000,,in this case, a cube.
Dialogue: 0,0:04:34.83,0:04:35.91,Default,,0000,0000,0000,,Let's do one more example.
Dialogue: 0,0:04:35.91,0:04:38.16,Default,,0000,0000,0000,,Let's do the rectangular\Npyramid, because all of these
Dialogue: 0,0:04:38.16,0:04:39.30,Default,,0000,0000,0000,,had rectangles.
Dialogue: 0,0:04:39.30,0:04:43.21,Default,,0000,0000,0000,,Or in particular, these had\Nsquares as our surfaces.
Dialogue: 0,0:04:43.21,0:04:45.25,Default,,0000,0000,0000,,Now, the most\Nobvious one might be
Dialogue: 0,0:04:45.25,0:04:48.52,Default,,0000,0000,0000,,to start with your\Nbase right over here.
Dialogue: 0,0:04:48.52,0:04:52.93,Default,,0000,0000,0000,,Start with your base and\Nthen take the different sides
Dialogue: 0,0:04:52.93,0:04:54.98,Default,,0000,0000,0000,,and then just fold\Nthem straight out.
Dialogue: 0,0:04:54.98,0:04:58.45,Default,,0000,0000,0000,,So, for example, we could take\Nthis side right over here,
Dialogue: 0,0:04:58.45,0:05:03.27,Default,,0000,0000,0000,,fold it out, and it\Nwould look like that.
Dialogue: 0,0:05:03.27,0:05:07.05,Default,,0000,0000,0000,,We could take this\Nside back here,
Dialogue: 0,0:05:07.05,0:05:11.26,Default,,0000,0000,0000,,and once again,\Njust fold it out.
Dialogue: 0,0:05:11.26,0:05:13.12,Default,,0000,0000,0000,,And it would look like that.
Dialogue: 0,0:05:13.12,0:05:15.08,Default,,0000,0000,0000,,It should be the same\Nsize as that orange side,
Dialogue: 0,0:05:15.08,0:05:19.96,Default,,0000,0000,0000,,but I'm hand drawing it, so\Nit's not going to be perfect.
Dialogue: 0,0:05:19.96,0:05:22.34,Default,,0000,0000,0000,,So that's that right over there.
Dialogue: 0,0:05:22.34,0:05:27.72,Default,,0000,0000,0000,,And then you could take this\Nfront side right over here,
Dialogue: 0,0:05:27.72,0:05:30.45,Default,,0000,0000,0000,,and once again, fold\Nit out along this edge.
Dialogue: 0,0:05:30.45,0:05:33.65,Default,,0000,0000,0000,,So it would look like this.
Dialogue: 0,0:05:33.65,0:05:38.61,Default,,0000,0000,0000,,And then finally, you could\Ntake this side right over
Dialogue: 0,0:05:38.61,0:05:43.13,Default,,0000,0000,0000,,here, and once again, fold\Nit out along this edge
Dialogue: 0,0:05:43.13,0:05:46.03,Default,,0000,0000,0000,,and it would go right there.
Dialogue: 0,0:05:46.03,0:05:49.16,Default,,0000,0000,0000,,But this isn't the only net\Nfor this rectangular pyramid.
Dialogue: 0,0:05:49.16,0:05:50.30,Default,,0000,0000,0000,,There's other options.
Dialogue: 0,0:05:50.30,0:05:52.32,Default,,0000,0000,0000,,For example, and just\Nto explore one of them,
Dialogue: 0,0:05:52.32,0:05:57.13,Default,,0000,0000,0000,,instead of folding that\Ngreen side out that way,
Dialogue: 0,0:05:57.13,0:05:59.84,Default,,0000,0000,0000,,instead we might have\Nwanted to fold it out
Dialogue: 0,0:05:59.84,0:06:03.09,Default,,0000,0000,0000,,along this edge with the\Nyellow side that you can't see.
Dialogue: 0,0:06:03.09,0:06:05.05,Default,,0000,0000,0000,,Actually, let's make it\Na little bit different.
Dialogue: 0,0:06:05.05,0:06:07.88,Default,,0000,0000,0000,,Let's fold it out along this\Nside since we can see the edge.
Dialogue: 0,0:06:07.88,0:06:09.95,Default,,0000,0000,0000,,And let me color the edge.
Dialogue: 0,0:06:09.95,0:06:13.21,Default,,0000,0000,0000,,So this is the edge right over\Nhere on the blue triangle.
Dialogue: 0,0:06:13.21,0:06:14.65,Default,,0000,0000,0000,,So this is the edge.
Dialogue: 0,0:06:14.65,0:06:17.74,Default,,0000,0000,0000,,And when you fold the\Ngreen triangle out,
Dialogue: 0,0:06:17.74,0:06:19.38,Default,,0000,0000,0000,,it would look like this.
Dialogue: 0,0:06:19.38,0:06:21.78,Default,,0000,0000,0000,,If you fold it the\Ngreen triangle out,
Dialogue: 0,0:06:21.78,0:06:23.67,Default,,0000,0000,0000,,it would look like this.
Dialogue: 0,0:06:23.67,0:06:27.17,Default,,0000,0000,0000,,So hopefully this gives\Nyou an appreciation.
Dialogue: 0,0:06:27.17,0:06:30.88,Default,,0000,0000,0000,,There's multiple ways to\Nunfold these three-dimensional
Dialogue: 0,0:06:30.88,0:06:33.83,Default,,0000,0000,0000,,figures, these polyhedra,\Nor multiple ways
Dialogue: 0,0:06:33.83,0:06:35.53,Default,,0000,0000,0000,,if you wanted to do\Na cardboard cutout
Dialogue: 0,0:06:35.53,0:06:38.48,Default,,0000,0000,0000,,and then fold things back\Ntogether to construct them.
Dialogue: 0,0:06:38.48,0:06:40.43,Default,,0000,0000,0000,,And these flattened\Nversions of them,
Dialogue: 0,0:06:40.43,0:06:47.26,Default,,0000,0000,0000,,these things, these unpacking of\Nthese polyhedra, we call nets.