Pizza is soft and bendable, so how can you stop all that cheese from falling off?

披薩又軟又容易彎折，那要怎麼阻止所有的起司滑落呢？

You might know some of the tricks:

你可能知道一些小訣竅：

You can use two hands, not so classy, or you can use a paper plate and allow only the very tip of the pizza to peek out.

你可以不太優雅地兩手並用，或是利用紙盤盛裝，只讓披薩的最尖端露出來。

There is one other trick, though, holding the crust, you can sort of fold the slice down the middle.

不過還有另一個妙招，就是握著餅皮，沿中線把披薩對折。

Now the tip of the pizza isn't falling over, and you can eat it without getting tomato sauce all over yourself or accidentally biting off some of that paper plate.

現在披薩尖端不再往下掉，你也不會吃得到處都是番茄醬，或不小心把紙盤吃下肚。

But why should the tip stay up just because you bent the crust?

不過，為什麼折了餅皮後，尖端就不會往下掉呢？

To understand this, you need to know two things:

要了解這個原理，你必須知道兩件事：

A little bit about the math of curved shapes, and a little bit about the physics of thin sheets.

一點點關於彎曲形狀的數學原理以及薄紙張的物理原理。

First, the math.

首先是數學原理。

Suppose I have a flat sheet made out of rubber.

假設我有一張橡皮做的薄板。

It's really thin and bendable, so it's easy to roll into a cylinder.

它很薄而且可彎曲，所以可以很輕易被彎曲為圓柱。

I don't need to stretch the sheet at all; just bend it.

我完全不需要拉伸薄板，只要彎曲就可以。

This property where one shape can be transformed into another without stretching or crumpling is called isometry.

這種不需要拉伸或揉捏就能將一個形狀轉變為另一個性狀的特性稱為「等距同構」。

A mathematician would say that a flat sheet is isometric to a cylinder.

數學家會說一個平板跟一個圓柱是等距同構的。

But not all shapes are isometric.

但並非所有形狀都屬等距同構。

If I try to turn my flat sheet into part of a sphere, there is no way I can do it.

如果我嘗試要把平板轉變為球體的一部分，我是絕對做不到的。

You can check this for yourself by trying to fit a flat sheet of paper onto a soccer ball without stretching or crumpling the paper.

你可以透過嘗試將一張平滑的紙，在不拉伸或揉捏的狀況下覆蓋足球，來進行驗證。

It's just not possible.

這是不可能的。

So a mathematician would say that a flat sheet and a sphere aren't isometric.

所以數學家會說，一個平板跟一個球體並非等距同構。

There's one more familiar shape that isn't isometric to any of the shapes we have seen so far : a potato chip.

還有一個我們熟悉的形狀，跟目前所看到的所有形狀都不是等距同構：洋芋片。

Potato chip shapes aren't isometric to flat sheets.

洋芋片的形狀跟平面不是等距同構。

If you want to get a flat piece of rubber into the shape of a potato chip, you need to stretch it, not just bend it, but stretch it as well.

如果你想要把一片橡皮轉變為洋芋片的形狀，你需要拉伸它，不只是彎曲，還要拉伸。

So, that's the math.

數學原理就是這樣。

Not so hard, right?

不是太難吧？

Now for the physics.

現在來談物理原理。

It can be summed up in one sentence : Thin sheets are easy to bend but hard to stretch.

這可以用一句話總結：薄片很容易彎曲但難以拉伸。

This is really important.

這一點很重要。

Thin sheets are easy to bend but hard to stretch.

薄片很容易彎曲但難以拉伸。

Remember when we rolled our flat sheet of rubber into a cylinder?

還記得我們把一片橡皮彎成圓柱時嗎？

That wasn't hard, right?

那不難，對吧？

But imagine how hard you'd have pull on the sheet to increase its area by 10%.

但想像一下，如果要將面積拉伸 10%，需要花多少力氣。

It would be pretty difficult.

應該會蠻困難的。

The point is that bending a thin sheet takes a relatively small amount of force, but stretching or crumpling a thin sheet is much harder.

重點是，彎曲一張薄紙需要相對小的力量，但是拉伸或揉捏一張薄紙則困難多了。

Now, finally, we get to talk about pizza.

接著我們終於可以討論披薩了。

Suppose you go down to the pizzeria and buy yourself a slice.

假設你去假設你去披薩點買了一小片披薩。

You pick it up from the crust first, without doing the fold.

你先從後方餅皮處把它拿起來，不做折疊。

Because of gravity, the slice bends downwards.

因為地心引力的關係，那片披薩會往下彎。

Pizza is pretty thin, after all, and we know that thin sheets are easy to bend.

畢竟披薩蠻薄的，而我們都知道薄片很容易彎曲。

You can't get it in your mouth, cheese and tomato sauce are dripping everywhere... It's a big mess!

你披薩放不進嘴巴裡，起司和番茄醬滴得到處都是... 真的亂成一團了！

So you fold the crust.

於是你把餅皮對折。

When you do that, you force the pizza into something like a taco shape.

當你這麼做的時候，你硬是把披薩轉變成類似塔可餅的形狀。

That's not hard to do.

那不難做到。

After all, this shape is isometric to the original pizza, which was flat.

畢竟這個形狀跟披薩原本平的形狀是等距同構。

But imagine what would happen if the pizza were to droop down while you are bending it.

但想像一下，如果披薩在你對折的同時下垂，會發生什麼事。

Now it looks like a droopy taco.

它現在看起來就像軟垂的塔可餅。

And what does a droopy taco look like?

那軟垂的塔可餅看起來樣什麼呢？

A potato chip!

就是洋芋片！

But we know that potato chips are not isometric to flat pieces of rubber, or flat pizzas, and that means that in order to get into the shape it's in now, the slice of pizza had to stretch.

但我們知道，洋芋片和平的橡皮或披薩並非等距同構，這意味著，要變成現在的形狀，這片披薩必須拉伸。

Since the pizza is thin, this takes a lot of force, compared to the amount of force it takes to bend the pizza in the first place.

因為披薩很薄，所以相較於一開始對折披薩，這會耗上更多力氣。

So, what's the conclusion?

那麼結論是什麼？

When you fold the pizza at the crust, you make it into a shape where a lot of force is needed to bend the tip down.

當你從餅皮位置對折披薩，你把它變成一個需要耗費很大力氣才能彎曲尖端的形狀。

Often gravity isn't strong enough to provide this force.

地心引力通常沒有大到能夠提供這股力量。

That was kind of a lot of information, so let's do a quick backwards recap.

提供的資訊量有點龐大，我們簡單地倒退複習吧。

When the pizza is folded at the crust, gravity isn't strong enough to bend the tip.

當披薩從餅皮處對折時，地心引力沒有強大到得以彎曲尖端。

Why?

為什麼呢？

Because stretching a pizza is hard, and to bend the tip downwards, the pizza would have to stretch.

因為拉伸披薩很難，偏偏要把尖端往下彎的話，必須拉伸披薩。

Why?

為什麼呢？

Because the shape that the pizza would be in, the droopy taco shape, isn't isometric to the original flat pizza.

因為披薩此時的軟垂塔可形狀跟原本的扁皮披薩並非等距同構。

Why?

為什麼呢？

Because of math.

因為數學原理。

As the pizza example shows, we can learn a lot by looking at the mathematical properties of different shapes.

如同披薩這個範例顯示，我們可以透過不同形狀的數學特性學到很多。

And it's especially nice when those shapes happen to be pizza slices.

而當這些形狀恰好是披薩時，就更好了。