[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:13.74,0:00:15.84,Default,,0000,0000,0000,,Pretty much everyone loves eating pizza,
Dialogue: 0,0:00:15.84,0:00:17.57,Default,,0000,0000,0000,,but it can be a messy business.
Dialogue: 0,0:00:17.57,0:00:21.38,Default,,0000,0000,0000,,Pizza is soft and bendable, so how can you stop all that cheese from falling off?
Dialogue: 0,0:00:21.38,0:00:22.89,Default,,0000,0000,0000,,You might know some of the tricks:
Dialogue: 0,0:00:22.89,0:00:23.82,Default,,0000,0000,0000,,you can use two hands,
Dialogue: 0,0:00:23.82,0:00:24.76,Default,,0000,0000,0000,,not so classy,
Dialogue: 0,0:00:24.76,0:00:26.09,Default,,0000,0000,0000,,or you can use a paper plate
Dialogue: 0,0:00:26.09,0:00:28.79,Default,,0000,0000,0000,,and allow only the tip of the pizza to peak out.
Dialogue: 0,0:00:28.79,0:00:29.90,Default,,0000,0000,0000,,There is one other trick, though:
Dialogue: 0,0:00:29.90,0:00:33.22,Default,,0000,0000,0000,,holding the crust, you can sort of fold the slice down the middle.
Dialogue: 0,0:00:33.22,0:00:35.14,Default,,0000,0000,0000,,Now the tip of the pizza isn't falling over,
Dialogue: 0,0:00:35.14,0:00:37.96,Default,,0000,0000,0000,,and you can eat it without getting tomato sauce all over yourself
Dialogue: 0,0:00:37.96,0:00:40.77,Default,,0000,0000,0000,,or accidentally biting off some of that paper plate.
Dialogue: 0,0:00:40.77,0:00:43.90,Default,,0000,0000,0000,,But, why should the tip stay up just because you bent the crust?
Dialogue: 0,0:00:43.90,0:00:46.24,Default,,0000,0000,0000,,To understand this, you need to know two things:
Dialogue: 0,0:00:46.24,0:00:48.31,Default,,0000,0000,0000,,a little bit about the math of curved shapes,
Dialogue: 0,0:00:48.31,0:00:50.89,Default,,0000,0000,0000,,and a little bit about the physics of thin sheets.
Dialogue: 0,0:00:50.89,0:00:51.96,Default,,0000,0000,0000,,First, the math.
Dialogue: 0,0:00:51.96,0:00:54.04,Default,,0000,0000,0000,,Suppose I have a flat sheet made out of rubber.
Dialogue: 0,0:00:54.04,0:00:57.48,Default,,0000,0000,0000,,It's really thin and bendable, so it's easy to roll into a cylinder.
Dialogue: 0,0:00:57.48,0:01:00.42,Default,,0000,0000,0000,,I don't need to stretch the sheet at all, just bend it.
Dialogue: 0,0:01:00.42,0:01:03.33,Default,,0000,0000,0000,,This property where one shape can be transformed into another
Dialogue: 0,0:01:03.33,0:01:06.36,Default,,0000,0000,0000,,without stretching or crumpling is called isometry.
Dialogue: 0,0:01:06.36,0:01:10.24,Default,,0000,0000,0000,,A mathematician would say that a flat sheet is isometric to a cylinder.
Dialogue: 0,0:01:10.24,0:01:11.100,Default,,0000,0000,0000,,But, not all shapes are isometric.
Dialogue: 0,0:01:11.100,0:01:14.74,Default,,0000,0000,0000,,If I try to turn my flat sheet into part of a sphere,
Dialogue: 0,0:01:14.74,0:01:16.02,Default,,0000,0000,0000,,there is no way I can do it.
Dialogue: 0,0:01:16.02,0:01:17.100,Default,,0000,0000,0000,,You can check this for yourself by trying to fit
Dialogue: 0,0:01:17.100,0:01:19.98,Default,,0000,0000,0000,,a flat sheet of paper onto a soccer ball
Dialogue: 0,0:01:19.98,0:01:21.84,Default,,0000,0000,0000,,without stretching or crumpling the paper.
Dialogue: 0,0:01:21.84,0:01:23.29,Default,,0000,0000,0000,,It's just not possible.
Dialogue: 0,0:01:23.29,0:01:24.50,Default,,0000,0000,0000,,So a mathematician would say
Dialogue: 0,0:01:24.50,0:01:27.78,Default,,0000,0000,0000,,that a flat sheet and a sphere aren't isometric.
Dialogue: 0,0:01:27.78,0:01:30.01,Default,,0000,0000,0000,,There is one more familiar shape that isn't isometric
Dialogue: 0,0:01:30.01,0:01:31.91,Default,,0000,0000,0000,,to any of the shapes we have seen so far:
Dialogue: 0,0:01:31.91,0:01:32.89,Default,,0000,0000,0000,,a potato chip.
Dialogue: 0,0:01:32.89,0:01:35.50,Default,,0000,0000,0000,,Potato chip shapes aren't isometric to flat sheets.
Dialogue: 0,0:01:35.50,0:01:38.62,Default,,0000,0000,0000,,If you want to get a flat piece of rubber into the shape of a potato chip,
Dialogue: 0,0:01:38.62,0:01:39.92,Default,,0000,0000,0000,,you need to stretch it,
Dialogue: 0,0:01:39.92,0:01:42.15,Default,,0000,0000,0000,,not just bend it, but stretch it as well.
Dialogue: 0,0:01:42.15,0:01:43.54,Default,,0000,0000,0000,,So, that's the math.
Dialogue: 0,0:01:43.54,0:01:44.75,Default,,0000,0000,0000,,Not so hard, right?
Dialogue: 0,0:01:44.75,0:01:45.94,Default,,0000,0000,0000,,Now for the physics.
Dialogue: 0,0:01:45.94,0:01:47.71,Default,,0000,0000,0000,,It can be summed up in one sentence:
Dialogue: 0,0:01:47.71,0:01:50.66,Default,,0000,0000,0000,,thin sheets are easy to bend but hard to stretch.
Dialogue: 0,0:01:50.66,0:01:52.36,Default,,0000,0000,0000,,This is really important.
Dialogue: 0,0:01:52.36,0:01:55.49,Default,,0000,0000,0000,,Thin sheets are easy to bend but hard to stretch.
Dialogue: 0,0:01:55.49,0:01:58.52,Default,,0000,0000,0000,,Remember when we rolled our flat sheet of rubber into a cylinder?
Dialogue: 0,0:01:58.52,0:01:59.81,Default,,0000,0000,0000,,That wasn't hard, right?
Dialogue: 0,0:01:59.81,0:02:01.87,Default,,0000,0000,0000,,But imagine how hard you would you have pull on the sheet
Dialogue: 0,0:02:01.87,0:02:04.01,Default,,0000,0000,0000,,to increase its area by 10%.
Dialogue: 0,0:02:04.01,0:02:05.59,Default,,0000,0000,0000,,It would be pretty difficult.
Dialogue: 0,0:02:05.59,0:02:09.29,Default,,0000,0000,0000,,The point is that bending a thin sheet takes a relatively small amount of force,
Dialogue: 0,0:02:09.29,0:02:12.74,Default,,0000,0000,0000,,but stretching or crumbling a thin sheet is much harder.
Dialogue: 0,0:02:12.74,0:02:15.43,Default,,0000,0000,0000,,Now, finally, we get to talk about pizza.
Dialogue: 0,0:02:15.43,0:02:18.32,Default,,0000,0000,0000,,Suppose you go down to the pizzeria and buy yourself a slice.
Dialogue: 0,0:02:18.32,0:02:21.30,Default,,0000,0000,0000,,You pick it up from the crust, first, without doing the fold.
Dialogue: 0,0:02:21.30,0:02:24.08,Default,,0000,0000,0000,,Because of gravity, the slice bends downwards.
Dialogue: 0,0:02:24.08,0:02:25.76,Default,,0000,0000,0000,,Pizza is pretty thin, after all,
Dialogue: 0,0:02:25.76,0:02:28.30,Default,,0000,0000,0000,,and we know that thin sheets are easy to bend.
Dialogue: 0,0:02:28.30,0:02:29.52,Default,,0000,0000,0000,,You can't get it into your mouth,
Dialogue: 0,0:02:29.52,0:02:31.29,Default,,0000,0000,0000,,cheese and tomato sauce are dripping everywhere,
Dialogue: 0,0:02:31.29,0:02:32.42,Default,,0000,0000,0000,,it's a big mess.
Dialogue: 0,0:02:32.42,0:02:33.41,Default,,0000,0000,0000,,So you fold the crust.
Dialogue: 0,0:02:33.41,0:02:36.66,Default,,0000,0000,0000,,When you do that, you force the pizza into something like a taco shape.
Dialogue: 0,0:02:36.66,0:02:38.11,Default,,0000,0000,0000,,That's not hard to do.
Dialogue: 0,0:02:38.11,0:02:41.92,Default,,0000,0000,0000,,After all, this shape is isometric to the original pizza, which was flat.
Dialogue: 0,0:02:41.92,0:02:44.86,Default,,0000,0000,0000,,But imagine what would happen if the pizza were to droop down
Dialogue: 0,0:02:44.86,0:02:46.33,Default,,0000,0000,0000,,while you are bending it.
Dialogue: 0,0:02:46.33,0:02:48.22,Default,,0000,0000,0000,,Now it looks like a droopy taco.
Dialogue: 0,0:02:48.22,0:02:49.94,Default,,0000,0000,0000,,And what does a droopy taco look like?
Dialogue: 0,0:02:49.94,0:02:51.05,Default,,0000,0000,0000,,A potato chip!
Dialogue: 0,0:02:51.05,0:02:54.88,Default,,0000,0000,0000,,But we know that potato chips are not isometric to flat pieces of rubber,
Dialogue: 0,0:02:54.88,0:02:56.34,Default,,0000,0000,0000,,or flat pizzas,
Dialogue: 0,0:02:56.34,0:02:58.80,Default,,0000,0000,0000,,and that means that in order to get into the shape it's in now,
Dialogue: 0,0:02:58.80,0:03:00.92,Default,,0000,0000,0000,,the slice of pizza had to stretch.
Dialogue: 0,0:03:00.92,0:03:03.59,Default,,0000,0000,0000,,Since the pizza is thin, this takes a lot of force,
Dialogue: 0,0:03:03.59,0:03:05.22,Default,,0000,0000,0000,,compared to the amount of force it takes
Dialogue: 0,0:03:05.22,0:03:06.96,Default,,0000,0000,0000,,to bend the pizza in the first place.
Dialogue: 0,0:03:06.96,0:03:08.99,Default,,0000,0000,0000,,So, what's the conclusion?
Dialogue: 0,0:03:08.99,0:03:10.71,Default,,0000,0000,0000,,When you fold the pizza at the crust,
Dialogue: 0,0:03:10.71,0:03:14.10,Default,,0000,0000,0000,,you make it into a shape where a lot of force is needed to bend the tip down.
Dialogue: 0,0:03:14.10,0:03:16.97,Default,,0000,0000,0000,,Often gravity isn't strong enough to provide this force.
Dialogue: 0,0:03:16.97,0:03:18.41,Default,,0000,0000,0000,,That was kind of a lot of information,
Dialogue: 0,0:03:18.41,0:03:20.37,Default,,0000,0000,0000,,so let's do a quick backwards recap.
Dialogue: 0,0:03:20.37,0:03:22.14,Default,,0000,0000,0000,,When the pizza is folded at the crust,
Dialogue: 0,0:03:22.14,0:03:24.03,Default,,0000,0000,0000,,gravity isn't strong enough to bend the tip.
Dialogue: 0,0:03:24.03,0:03:24.65,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:03:24.65,0:03:26.48,Default,,0000,0000,0000,,Because stretching a pizza is hard,
Dialogue: 0,0:03:26.48,0:03:27.70,Default,,0000,0000,0000,,and to bend the tip downwards,
Dialogue: 0,0:03:27.70,0:03:29.25,Default,,0000,0000,0000,,the pizza would have to stretch.
Dialogue: 0,0:03:29.25,0:03:29.87,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:03:29.87,0:03:31.62,Default,,0000,0000,0000,,Because the shape that the pizza would be in,
Dialogue: 0,0:03:31.62,0:03:32.82,Default,,0000,0000,0000,,the droopy taco shape,
Dialogue: 0,0:03:32.82,0:03:35.21,Default,,0000,0000,0000,,is not isometric to the original flat pizza.
Dialogue: 0,0:03:35.21,0:03:35.81,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:03:35.81,0:03:37.37,Default,,0000,0000,0000,,Because of math.
Dialogue: 0,0:03:37.37,0:03:38.77,Default,,0000,0000,0000,,As the pizza example shows,
Dialogue: 0,0:03:38.77,0:03:42.34,Default,,0000,0000,0000,,we can learn a lot by looking at the mathematical properties of different shapes.
Dialogue: 0,0:03:42.34,0:03:45.32,Default,,0000,0000,0000,,And it's especially nice when those shapes happen to be pizza slices.