Pizza physics (New York-style) - Colm Kelleher

薄片披薩的物理 - 科爾姆凱勒赫 (Colm Kelleher)

Transcriber: Andrea McDonough Reviewer: Bedirhan Cinar

譯者: Jephian Lin 審譯者: Allen Li

Pretty much everyone loves eating pizza,

很多人都愛吃披薩

but it can be a messy business.

但有時也很麻煩

Pizza is soft and bendable.

披薩又軟又可以彎折 要怎樣起司才不會掉下去呢？

So how can you stop all that cheese from falling off?

You might know some tricks:

你可能會用幾種方法：

你可以用兩手拿

you can use two hands -- not so classy,

但少了點氣質

or you can use a paper plate

你可以用紙盤裝

and allow only the tip of the pizza to peek out.

只把尖端露出來

There's one other trick, though:

不過還有一種辦法：

抓住餅皮，沿中線折起來

holding the crust, you can sort of fold the slice down the middle.

Now the tip of the pizza isn't falling over,

這樣披薩的尖端就不會往下掉

and you can eat it without getting tomato sauce all over yourself

你就不會吃得滿臉都是蕃茄醬

或不小心吃到紙盤

or accidentally biting off some of that paper plate.

不過為什麼折起來後 尖端就不會往下掉呢？

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

想了解原因，先要知道兩件事：

To understand this, you need to know two things:

a little bit about the math of curved shapes

一點關於曲面的數學

and a little about the physics of thin sheets.

和一點關於薄片的物理

First, the math.

首先是數學

Suppose I have a flat sheet made out of rubber.

假設我有一個橡膠做的平面

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

很薄、而且可彎曲 可以輕易捲成圓柱狀

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

我並沒有拉長，只是捲起來

This property where one shape can be transformed into another

一個形狀可以不經過拉長或壓皺

without stretching or crumpling, is called isometry.

變換成另一個形狀 這性質叫等距同構 (isometry)

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

數學家會說平面和圓柱面是 等距同構

But not all shapes are isometric.

但並非所有形狀都是等距同構

If I try to turn my flat sheet into part of a sphere,

若我想把平面彎成球面的一部分

there's no way I can do it.

那我一定做不到

You can check this for yourself,

你可以自己試試看

by trying to fit a flat sheet of paper onto a soccer ball

一張紙不拉長或壓皺

想密合在足球表面上

without stretching or crumpling the paper.

是不可能的

It's just not possible.

So a mathematician would say

所以數學家說

that a flat sheet and a sphere aren't isometric.

平面和球面不是等距同構

There's one more familiar shape that isn't isometric

還有一種常見的形狀 和我們剛剛看過的任何形狀

to any of the shapes we've seen so far: a potato chip.

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 和平面不是等距同構

If you want to get a flat piece of rubber into the shape of a potato chip,

如果你想把橡膠平面 彎成洋芋片的形狀

一定要拉長

you need to stretch it -- not just bend it, but stretch it as well.

不只是彎曲，同時也要拉長

So, that's the math.

以上是數學的部分

Not so hard, right?

不算太難，對吧？

Now for the physics.

現在改談物理

It can be summed up in one sentence:

可以歸納為一句話：

Thin sheets are easy to bend but hard to stretch.

薄片容易彎曲，但較難拉長

This is really important.

這非常重要

Thin sheets are easy to bend but hard to stretch.

薄片容易彎曲，但較難拉長

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

還記得我們把橡膠平面捲成圓柱嗎？

That wasn't hard, right?

這不太難，對吧？

But imagine how hard you'd have pull on the sheet

但想像一下，如果想把它拉長

讓面積增加一成，會有多困難

to increase its area by 10 percent.

It would be pretty difficult.

非常困難

The point is that bending a thin sheet

問題在於彎折薄片，需力較少

takes a relatively small amount of force,

but stretching or crumbling a thin sheet is much harder.

拉長或壓皺薄片比較困難

Now, finally, we get to talk about pizza.

最後回來談披薩

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

假設你買了一塊披薩

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

你從餅皮的部分拿起來，沒有對折

Because of gravity, the slice bends downwards.

由於重力，這塊披薩就會向下彎

Pizza is pretty thin, after all,

畢竟披薩蠻薄的

and we know that thin sheets are easy to bend.

而我們知道薄片是容易彎曲的

You can't get it in your mouth,

你沒辦法把它送進嘴裡

cheese and tomato sauce dripping everywhere -- it's a big mess.

因為起司和蕃茄醬會滴滿地

很麻煩

So you fold the crust.

所以你把它對折

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

此時你強迫披薩變成 塔可餅 (taco) 的形狀

這並不太難

That's not hard to do -- after all, this shape is isometric

畢竟這形狀和原本的平面披薩 是等距同構的

to the original pizza, which was flat.

But imagine what would happen if the pizza were to droop down

但想像披薩要向下掉時 會發生什麼事

如果你有對折的話

while you're bending it.

Now it looks like a droopy taco.

它會像一個下垂的塔可餅

And what does a droopy taco look like? A potato chip!

而下垂的塔可餅跟什麼很像？

就是洋芋片！

But we know that potato chips are not isometric to flat pieces of rubber

但我們知道洋芋片和橡膠平面 或者說「披薩平面」

or flat pizzas,

並非等距同構

and that means that in order to get into the shape it's in now,

這表示要變成這樣的形狀

披薩一定要拉長

the slice of pizza had to stretch.

Since the pizza is thin, this takes a lot of force,

因為披薩是薄片，需要很大的力量

compared to the amount of force it takes

相對於一開始

to bend the pizza in the first place.

對折的力量

所以結論是？

So, what's the conclusion?

When you fold the pizza at the crust,

當你對折披薩，它變成另一種形狀

you make it into a shape where a lot of force is needed to bend the tip down.

需要很大力量才能把尖端彎下來

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

通常重力不足以達到這種力量

That was kind of a lot of information,

好像一次講太多了

so let's do a quick backwards recap.

我們來個快速重點複習

When pizza is folded at the crust,

當披薩對折時

gravity isn't strong enough to bend the tip.

重力不足以把尖端往下彎

Why? Because stretching a pizza is hard.

為什麼？

因為要把披薩拉長並不容易

And to bend the tip downwards, the pizza would have to stretch,

而要把尖端往下彎

就一定要拉長

because the shape the pizza would be in,

為什麼？

因為那樣披薩的形狀

the droopy taco shape,

就會像下垂的塔可餅一樣

isn't isometric to the original flat pizza.

和原本平面的披薩並非等距同構

Why? Because of math.

為什麼？

因為數學

As the pizza example shows, we can learn a lot

像是披薩這個例子

我們可從不同形狀的 數學性質中學到很多

by looking at the mathematical properties of different shapes.

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

若形狀剛好和披薩一樣，那就更好了