Pretty much everyone loves eating pizza, but it can be a messy business.
几乎所有人都爱吃比萨，但有时也很麻烦。
Pizza is soft and bendable. So how can you stop all that cheese from falling off?
比萨很柔软、易弯。你该如何避免奶酪掉到地上呢？
You might know some tricks: you can use two hands -- not so classy,
你也许知道一些技巧：你可以使用两只手，这不怎么文明
or you can use a paper plate and allow only the tip of the pizza to peek out.
你可以用纸盘装，只把尖端露出来。
There's one other trick, though: holding the crust, you can sort of fold the slice down the middle.
还有另一个法子，那就是：抓住饼皮，沿中线折起来。
Now the tip of the pizza isn't falling over,
这样披萨的尖端就不会往下掉。
and you can eat it without getting tomato sauce all over yourself
你可以吃着它，同时不担心番茄酱会洒落在身上
or accidentally biting off some of that paper plate.
或不小心吃到那纸盘的边缘。
But why should the tip stay up just because you bent the crust?
但为什么当你折叠饼皮边时，比萨饼尖会保持它原先的位置而不弯曲？
To understand this, you need to know two things:
为了理解这个现象，你必须知道两件事：
a little bit about the math of curved shapes and a little about the physics of thin sheets.
一点儿关于曲面的数学，和一点儿关于薄片的物理学。
First, the math. Suppose I have a flat sheet made out of rubber.
首先是数学。假设我有一个橡胶做的平面。
It's really thin and bendable, so it's easy to roll into a cylinder.
它非常薄，可以弯折，所以它非常容易卷成筒状。
I don't need to stretch the sheet at all, just bend it.
我不须要拉长薄片，就只是弯曲它。
This property where one shape can be transformed into another without stretching or crumpling, is called isometry.
这使一个形状可以转变为另一个形状，却不会拉伸或弄皱其原本的特性，这性质叫等距同构。
A mathematician would say that a flat sheet is isometric to a cylinder.
数学家可能会说薄片就是个等距的圆筒。
But not all shapes are isometric.
但并不是每一种形状都是等距的。
If I try to turn my flat sheet into part of a sphere, there's no way I can do it.
如果我试着将我的薄片弄成球面，这完全不可行。
You can check this for yourself,
你可以自己试着
by trying to fit a flat sheet of paper onto a soccer ball without stretching or crumpling the paper.
将一张薄纸片弄成一个足球，而不拉伸或弯折那张纸。
It's just not possible. So a mathematician would say that a flat sheet and a sphere aren't isometric.
这完全不可能。因此数学家会说薄片和球体不是等距的。
There's one more familiar shape that isn't isometric to any of the shapes we've seen so far: a potato chip.
还有一种常见的形状和我们刚刚看过的任何形状都不是等距同构：薯片。
Potato chip shapes aren't isometric to flat sheets.
薯片的形状对平板来说不是等距的。
If you want to get a flat piece of rubber into the shape of a potato chip,
如果你想要将一个橡胶平面变成一个薯片，
you need to stretch it -- not just bend it, but stretch it as well.
你需要拉长它--而不只是弯曲它，还要拉长它。
So, that's the math. Not so hard, right? Now for the physics.
以上是数学的部分。不难吧，不是吗？现在我们看看物理学。

It can be summed up in one sentence:
它可以总结为一句话：
Thin sheets are easy to bend but hard to stretch. This is really important.
薄片容易弯曲但难以拉长，这很重要。
Thin sheets are easy to bend but hard to stretch.
薄片容易弯曲，但难以拉长。
Remember when we rolled our flat sheet of rubber into a cylinder? That wasn't hard, right?
还记得我们将橡胶薄片卷成圆筒的时刻吗？那不难，是吧？
But imagine how hard you'd have pull on the sheet to increase its area by 10 percent. It would be pretty difficult.
但想象一下你得多用力才能将那薄片增加百分之十的面积。那会很困难的。
The point is that bending a thin sheet takes a relatively small amount of force,
重点是将一张薄片弯曲只需要花很少的力气
but stretching or crumbling a thin sheet is much harder.
但是要拉长或揉捏一张薄片是比较困难的。
Now, finally, we get to talk about pizza.
现在，最后，我们来谈谈比萨。
Suppose you go down to the pizzeria and buy yourself a slice.
假设你去比萨店买一片比萨。
You pick it up from the crust, first, without doing the fold.
你首先拾起它的硬皮边，试着不让它弯折下落。
Because of gravity, the slice bends downwards.
由于重力的原因，这片饼皮会往下弯曲。
Pizza is pretty thin, after all, and we know that thin sheets are easy to bend.
比萨挺薄的，我们知道薄片容易弯曲。
You can't get it in your mouth, cheese and tomato sauce dripping everywhere -- it's a big mess.
你没法儿将它放进嘴里，起司和番茄酱会掉满地，弄得一团糟。
So you fold the crust. When you do, you force the pizza into something like a taco shape.
所以你将饼皮折叠在一起。当你这么做时，你使比萨形成某种玉米薄饼卷的形状。
That's not hard to do -- after all, this shape is isometric to the original pizza, which was flat.
这并不难--毕竟这种形状与原来比萨饼是等距的，它是平的。
But imagine what would happen if the pizza were to droop down while you're bending it.
但想象一下，如果在你弯曲比萨时它掉到地上的话，这会怎样呢？
Now it looks like a droopy taco.
现在，它看起来像一个下垂的玉米薄饼卷。
And what does a droopy taco look like? A potato chip!
一个下垂的玉米薄饼卷看起来像什么呢？一片薯片！
But we know that potato chips are not isometric to flat pieces of rubber or flat pizzas,
但我们知道薯片和橡胶薄片、或是扁平的比萨不是等距的，
and that means that in order to get into the shape it's in now, the slice of pizza had to stretch.
这意味着为了把它弄成现在的形状，我们需要拉长比萨饼皮。
Since the pizza is thin, this takes a lot of force,
由于比萨饼很薄，这需要很多力气，
compared to the amount of force it takes to bend the pizza in the first place.
相较于一开始折叠比萨饼时所需的力气。
So, what's the conclusion? When you fold the pizza at the crust,
所以结论是什么呢？当你在饼皮尖折比萨的时候，
you make it into a shape where a lot of force is needed to bend the tip down.
你必须花上许多力气来弯曲那尖端。
Often gravity isn't strong enough to provide this force.
通常地心引力没有提供那么强烈的力量。
That was kind of a lot of information, so let's do a quick backwards recap.
这堂课包含了许多信息，所以让我们来回顾一下吧。
When pizza is folded at the crust, gravity isn't strong enough to bend the tip.
当比萨饼从饼皮端对折时，地心引力并不强大到把尖端往下弯。
Why? Because stretching a pizza is hard.
为什么？因为要拉长比萨饼是很困难的。
And to bend the tip downwards, the pizza would have to stretch,
为了要弯曲尖端，比萨必须拉长，
because the shape the pizza would be in, the droopy taco shape, isn't isometric to the original flat pizza.
因为那样，披萨的形状就会像下垂的玉米薄饼卷一样，和原本平面的披萨并非等距同构。
Why? Because of math.
为什么呢？因为数学。
As the pizza example shows, we can learn a lot by looking at the mathematical properties of different shapes.
就像那比萨饼的例子所显示的，我们可以借由观察不同的形状的数学属性来学习。
And it's especially nice when those shapes happen to be pizza slices.
若形状刚好和披萨一样，那就更好了。