0:00:13.741,0:00:15.841 Pretty much everyone loves eating pizza, 0:00:15.841,0:00:17.567 but it can be a messy business. 0:00:17.567,0:00:21.385 Pizza is soft and bendable, so how can you stop all that cheese from falling off? 0:00:21.385,0:00:22.886 You might know some of the tricks: 0:00:22.886,0:00:23.821 you can use two hands, 0:00:23.821,0:00:24.758 not so classy, 0:00:24.758,0:00:26.090 or you can use a paper plate 0:00:26.090,0:00:28.792 and allow only the tip of the pizza to peak out. 0:00:28.792,0:00:29.899 There is one other trick, though: 0:00:29.899,0:00:33.222 holding the crust, you can sort of fold the slice down the middle. 0:00:33.222,0:00:35.137 Now the tip of the pizza isn't falling over, 0:00:35.137,0:00:37.964 and you can eat it without getting tomato sauce all over yourself 0:00:37.964,0:00:40.770 or accidentally biting off some of that paper plate. 0:00:40.770,0:00:43.897 But, why should the tip stay up just because you bent the crust? 0:00:43.897,0:00:46.240 To understand this, you need to know two things: 0:00:46.240,0:00:48.314 a little bit about the math of curved shapes, 0:00:48.314,0:00:50.886 and a little bit about the physics of thin sheets. 0:00:50.886,0:00:51.965 First, the math. 0:00:51.965,0:00:54.044 Suppose I have a flat sheet made out of rubber. 0:00:54.044,0:00:57.475 It's really thin and bendable, so it's easy to roll into a cylinder. 0:00:57.475,0:01:00.423 I don't need to stretch the sheet at all, just bend it. 0:01:00.423,0:01:03.326 This property where one shape can be transformed into another 0:01:03.326,0:01:06.362 without stretching or crumbling is called isometry. 0:01:06.362,0:01:10.244 A mathematician would say that a flat sheet is isometric to a cylinder. 0:01:10.244,0:01:11.995 But, not all shapes are isometric. 0:01:11.995,0:01:14.745 If I try to turn my flat sheet into part of a sphere, 0:01:14.745,0:01:16.021 there is no way I can do it. 0:01:16.021,0:01:17.998 You can check this for yourself by trying to fit 0:01:17.998,0:01:19.975 a flat sheet of paper onto a soccer ball 0:01:19.975,0:01:21.843 without stretching or crumbling the paper. 0:01:21.843,0:01:23.292 It's just not possible. 0:01:23.292,0:01:24.497 So a mathematician would say 0:01:24.497,0:01:27.778 that a flat sheet and a sphere aren't isometric. 0:01:27.778,0:01:30.010 There is one more familiar shape that isn't isometric 0:01:30.010,0:01:31.914 to any of the shapes we have seen so far: 0:01:31.914,0:01:32.893 a potato chip. 0:01:32.893,0:01:35.496 Potato chip shapes aren't isometric to flat sheets. 0:01:35.496,0:01:38.615 If you want to get a flat piece of rubber into the shape of a potato chip, 0:01:38.615,0:01:39.915 you need to stretch it, 0:01:39.915,0:01:42.148 not just bend it, but stretch it as well. 0:01:42.148,0:01:43.537 So, that's the math. 0:01:43.537,0:01:44.750 Not so hard, right? 0:01:44.750,0:01:45.944 Now for the physics. 0:01:45.944,0:01:47.714 It can be summed up in one sentence: 0:01:47.714,0:01:50.655 thin sheets are easy to bend but hard to stretch. 0:01:50.655,0:01:52.361 This is really important. 0:01:52.361,0:01:55.486 Thin sheets are easy to bend but hard to stretch. 0:01:55.486,0:01:58.523 Remember when we rolled our flat sheet of rubber into a cylinder? 0:01:58.523,0:01:59.813 That wasn't hard, right? 0:01:59.813,0:02:01.868 But imagine how hard you would you have pull on the sheet 0:02:01.868,0:02:04.008 to increase its area by 10%. 0:02:04.008,0:02:05.587 It would be pretty difficult. 0:02:05.587,0:02:09.292 The point is that bending a thin sheet takes a relatively small amount of force, 0:02:09.292,0:02:12.735 but stretching or crumbling a thin sheet is much harder. 0:02:12.735,0:02:15.426 Now, finally, we get to talk about pizza. 0:02:15.426,0:02:18.322 Suppose you go down to the pizzeria and buy yourself a slice. 0:02:18.322,0:02:21.303 You pick it up from the crust, first, without doing the fold. 0:02:21.303,0:02:24.075 Because of gravity, the slice bends downwards. 0:02:24.075,0:02:25.758 Pizza is pretty thin, after all, 0:02:25.758,0:02:28.305 and we know that thin sheets are easy to bend. 0:02:28.305,0:02:29.515 You can't get it into your mouth, 0:02:29.515,0:02:31.293 cheese and tomato sauce are dripping everywhere, 0:02:31.293,0:02:32.425 it's a big mess. 0:02:32.425,0:02:33.414 So you fold the crust. 0:02:33.414,0:02:36.660 When you do that, you force the pizza into something like a taco shape. 0:02:36.660,0:02:38.109 That's not hard to do. 0:02:38.109,0:02:41.920 After all, this shape is isometric to the original pizza, which was flat. 0:02:41.920,0:02:44.864 But imagine what would happen if the pizza were to droop down 0:02:44.864,0:02:46.329 while you are bending it. 0:02:46.329,0:02:48.224 Now it looks like a droopy taco. 0:02:48.224,0:02:49.940 And what does a droopy taco look like? 0:02:49.940,0:02:51.047 A potato chip! 0:02:51.047,0:02:54.878 But we know that potato chips are not isometric to flat pieces of rubber, 0:02:54.878,0:02:56.337 or flat pizzas, 0:02:56.337,0:02:58.805 and that means that in order to get into the shape it's in now, 0:02:58.805,0:03:00.925 the slice of pizza had to stretch. 0:03:00.925,0:03:03.593 Since the pizza is thin, this takes a lot of force, 0:03:03.593,0:03:05.221 compared to the amount of force it takes 0:03:05.221,0:03:06.965 to bend the pizza in the first place. 0:03:06.965,0:03:08.993 So, what's the conclusion? 0:03:08.993,0:03:10.708 When you fold the pizza at the crust, 0:03:10.708,0:03:14.095 you make it into a shape where a lot of force is needed to bend the tip down. 0:03:14.095,0:03:16.974 Often gravity isn't strong enough to provide this force. 0:03:16.974,0:03:18.411 That was kind of a lot of information, 0:03:18.411,0:03:20.369 so let's do a quick backwards recap. 0:03:20.369,0:03:22.140 When the pizza is folded at the crust, 0:03:22.140,0:03:24.033 gravity isn't strong enough to bend the tip. 0:03:24.033,0:03:24.648 Why? 0:03:24.648,0:03:26.481 Because stretching a pizza is hard, 0:03:26.481,0:03:27.705 and to bend the tip downwards, 0:03:27.705,0:03:29.250 the pizza would have to stretch. 0:03:29.250,0:03:29.872 Why? 0:03:29.872,0:03:31.622 Because the shape that the pizza would be in, 0:03:31.622,0:03:32.818 the droopy taco shape, 0:03:32.818,0:03:35.210 is not isometric to the original flat pizza. 0:03:35.210,0:03:35.812 Why? 0:03:35.812,0:03:37.372 Because of math. 0:03:37.372,0:03:38.772 As the pizza example shows, 0:03:38.772,0:03:42.337 we can learn a lot by looking at the mathematical properties of different shapes. 0:03:42.337,0:03:45.315 And it's especially nice when those shapes happen to be pizza slices.