WEBVTT 00:00:13.580 --> 00:00:15.907 Pretty much everyone loves eating pizza, 00:00:15.931 --> 00:00:17.645 but it can be a messy business. 00:00:17.669 --> 00:00:19.163 Pizza is soft and bendable. 00:00:19.187 --> 00:00:21.697 So how can you stop all that cheese from falling off? 00:00:21.721 --> 00:00:23.023 You might know some tricks: 00:00:23.047 --> 00:00:24.888 you can use two hands -- not so classy, 00:00:24.912 --> 00:00:26.271 or you can use a paper plate 00:00:26.295 --> 00:00:28.636 and allow only the tip of the pizza to peek out. 00:00:28.660 --> 00:00:30.215 There's one other trick, though: 00:00:30.239 --> 00:00:33.364 holding the crust, you can sort of fold the slice down the middle. 00:00:33.388 --> 00:00:35.460 Now the tip of the pizza isn't falling over, 00:00:35.484 --> 00:00:38.577 and you can eat it without getting tomato sauce all over yourself 00:00:38.601 --> 00:00:41.060 or accidentally biting off some of that paper plate. 00:00:41.084 --> 00:00:44.090 But why should the tip stay up just because you bent the crust? 00:00:44.114 --> 00:00:46.366 To understand this, you need to know two things: 00:00:46.390 --> 00:00:48.460 a little bit about the math of curved shapes 00:00:48.484 --> 00:00:50.777 and a little about the physics of thin sheets. 00:00:50.801 --> 00:00:51.954 First, the math. 00:00:51.978 --> 00:00:54.183 Suppose I have a flat sheet made out of rubber. 00:00:54.207 --> 00:00:57.451 It's really thin and bendable, so it's easy to roll into a cylinder. 00:00:57.475 --> 00:01:00.399 I don't need to stretch the sheet at all, just bend it. 00:01:00.423 --> 00:01:03.328 This property where one shape can be transformed into another 00:01:03.352 --> 00:01:06.338 without stretching or crumpling, is called isometry. 00:01:06.362 --> 00:01:10.160 A mathematician would say that a flat sheet is isometric to a cylinder. 00:01:10.184 --> 00:01:11.971 But not all shapes are isometric. 00:01:11.995 --> 00:01:14.615 If I try to turn my flat sheet into part of a sphere, 00:01:14.639 --> 00:01:15.932 there's no way I can do it. 00:01:15.956 --> 00:01:17.511 You can check this for yourself, 00:01:17.535 --> 00:01:20.221 by trying to fit a flat sheet of paper onto a soccer ball 00:01:20.245 --> 00:01:22.253 without stretching or crumpling the paper. 00:01:22.277 --> 00:01:23.428 It's just not possible. 00:01:23.452 --> 00:01:24.815 So a mathematician would say 00:01:24.839 --> 00:01:27.633 that a flat sheet and a sphere aren't isometric. 00:01:27.657 --> 00:01:30.103 There's one more familiar shape that isn't isometric 00:01:30.127 --> 00:01:32.869 to any of the shapes we've seen so far: a potato chip. 00:01:32.893 --> 00:01:35.557 Potato chip shapes aren't isometric to flat sheets. 00:01:35.581 --> 00:01:39.073 If you want to get a flat piece of rubber into the shape of a potato chip, 00:01:39.097 --> 00:01:42.284 you need to stretch it -- not just bend it, but stretch it as well. 00:01:42.308 --> 00:01:43.467 So, that's the math. 00:01:43.491 --> 00:01:44.657 Not so hard, right? 00:01:44.681 --> 00:01:45.832 Now for the physics. 00:01:45.856 --> 00:01:47.751 It can be summed up in one sentence: 00:01:47.775 --> 00:01:50.656 Thin sheets are easy to bend but hard to stretch. 00:01:50.680 --> 00:01:52.236 This is really important. 00:01:52.260 --> 00:01:55.371 Thin sheets are easy to bend but hard to stretch. 00:01:55.395 --> 00:01:58.458 Remember when we rolled our flat sheet of rubber into a cylinder? 00:01:58.482 --> 00:01:59.720 That wasn't hard, right? 00:01:59.744 --> 00:02:02.118 But imagine how hard you'd have pull on the sheet 00:02:02.142 --> 00:02:03.984 to increase its area by 10 percent. 00:02:04.008 --> 00:02:05.467 It would be pretty difficult. 00:02:05.491 --> 00:02:07.453 The point is that bending a thin sheet 00:02:07.477 --> 00:02:09.499 takes a relatively small amount of force, 00:02:09.523 --> 00:02:12.758 but stretching or crumbling a thin sheet is much harder. 00:02:12.782 --> 00:02:15.296 Now, finally, we get to talk about pizza. 00:02:15.320 --> 00:02:18.331 Suppose you go down to the pizzeria and buy yourself a slice. 00:02:18.355 --> 00:02:21.279 You pick it up from the crust, first, without doing the fold. 00:02:21.303 --> 00:02:23.994 Because of gravity, the slice bends downwards. 00:02:24.018 --> 00:02:25.786 Pizza is pretty thin, after all, 00:02:25.810 --> 00:02:28.176 and we know that thin sheets are easy to bend. 00:02:28.200 --> 00:02:29.681 You can't get it in your mouth, 00:02:29.705 --> 00:02:32.686 cheese and tomato sauce dripping everywhere -- it's a big mess. 00:02:32.710 --> 00:02:33.861 So you fold the crust. 00:02:33.885 --> 00:02:36.996 When you do, you force the pizza into something like a taco shape. 00:02:37.020 --> 00:02:39.944 That's not hard to do -- after all, this shape is isometric 00:02:39.968 --> 00:02:41.923 to the original pizza, which was flat. 00:02:41.947 --> 00:02:45.008 But imagine what would happen if the pizza were to droop down 00:02:45.032 --> 00:02:46.305 while you're bending it. 00:02:46.329 --> 00:02:48.112 Now it looks like a droopy taco. 00:02:48.136 --> 00:02:50.872 And what does a droopy taco look like? A potato chip! 00:02:50.896 --> 00:02:54.779 But we know that potato chips are not isometric to flat pieces of rubber 00:02:54.803 --> 00:02:56.249 or flat pizzas, 00:02:56.273 --> 00:02:59.247 and that means that in order to get into the shape it's in now, 00:02:59.271 --> 00:03:00.901 the slice of pizza had to stretch. 00:03:00.925 --> 00:03:03.516 Since the pizza is thin, this takes a lot of force, 00:03:03.540 --> 00:03:05.498 compared to the amount of force it takes 00:03:05.522 --> 00:03:07.422 to bend the pizza in the first place. 00:03:07.446 --> 00:03:08.800 So, what's the conclusion? 00:03:08.824 --> 00:03:10.626 When you fold the pizza at the crust, 00:03:10.650 --> 00:03:14.300 you make it into a shape where a lot of force is needed to bend the tip down. 00:03:14.324 --> 00:03:16.951 Often gravity isn't strong enough to provide this force. 00:03:16.975 --> 00:03:18.800 That was kind of a lot of information, 00:03:18.824 --> 00:03:20.545 so let's do a quick backwards recap. 00:03:20.569 --> 00:03:22.205 When pizza is folded at the crust, 00:03:22.229 --> 00:03:24.299 gravity isn't strong enough to bend the tip. 00:03:24.323 --> 00:03:26.376 Why? Because stretching a pizza is hard. 00:03:26.400 --> 00:03:29.389 And to bend the tip downwards, the pizza would have to stretch, 00:03:29.413 --> 00:03:31.598 because the shape the pizza would be in, 00:03:31.622 --> 00:03:32.793 the droopy taco shape, 00:03:32.817 --> 00:03:35.370 isn't isometric to the original flat pizza. 00:03:35.394 --> 00:03:37.204 Why? Because of math. 00:03:37.228 --> 00:03:39.407 As the pizza example shows, we can learn a lot 00:03:39.431 --> 00:03:42.581 by looking at the mathematical properties of different shapes. 00:03:42.605 --> 00:03:46.136 And it's especially nice when those shapes happen to be pizza slices.