Exponential growth: How folding paper can get you to the Moon

折纸送你上月球 - 阿德里安潘萨 (Adrian Paenza)

翻译人员: Shu Yang 校对人员: Qiwen Lu

（音乐）

一张纸可以对折几次？

How many times can you fold a piece of paper?

假设有一张很薄很薄的纸

Assume that one had a piece of paper that was very fine,

像是印圣经的那种纸

like the kind they typically use to print the Bible.

实际上，就像是一根丝那么薄

In reality, it seems like a piece of silk.

为了要把这个想法量化

To qualify these ideas,

let's say you have a paper

假设纸的厚度是 1/1000 厘米

that's one-thousandth of a centimeter in thickness.

也就是 10 的 -3 次方厘米

That is 10 to the power of minus three centimeters,

which equals .001 centimeters.

0...............为 0.001 厘米

再假设这张纸够大

Let's also assume that you have a big piece of paper,

像报纸那么大

like a page out of the newspaper.

Now we begin to fold it in half.

我们开始将它对折

你觉得我们可以对折几次？

How many times do you think it could be folded like that?

And another question:

另一个问题是：

If you could fold the paper over and over, as many times as you wish,

如果你能将一张纸 折叠任意多次

say 30 times,

比方说 30 次 你想那时纸会有多厚？

what would you imagine the thickness of the paper would be then?

在开始之前

Before you move on,

I encourage you to actually think about a possible answer to this question.

我鼓励你好好想想 这问题可能的答案

好，在对折一次后

OK.

After we have folded the paper once,

厚度变成 2/1000 厘米

it is now two thousandths of a centimeter in thickness.

如果再对折一次

If we fold it in half once again,

the paper will become four thousandths of a centimeter.

这张纸就会变成 4/1000 厘米

With every fold we make, the paper doubles in thickness.

每折一次，纸张厚度加倍

如果我们不断对折

And if we continue to fold it again and again,

十次以后

always in half, we would confront the following situation

after 10 folds.

我们就会见到这种情景

Two to the power of 10,

2 的 10 次方

meaning that you multiply two by itself 10 times,

也就是将 2 自乘 10 次

就是 1024/1000 厘米

is one thousand and 24 thousandths of a centimeter,

which is a little bit over one centimeter.

比 1 厘米多一点

如果我们继续对折

Assume we continue folding the paper in half.

会发生什么事？

What will happen then?

If we fold it 17 times,

如果对折 17 次

we'll get a thickness of two to the power of 17,

厚度就是 2 的 17 次方

which is 131 centimeters,

大约 131 厘米

and that equals just over four feet.

超过 4 英尺 (1.3 米)

If we were able to fold it 25 times,

如果我们可以对折 25 次

then we would get two to the power of 25,

就会得到 2 的 25 次方

which is 33,554 centimeters,

大约 33,554 厘米

超过 1100 英尺 (335 米)

just over 1,100 feet.

差不多跟帝国大厦一样高

That would make it almost as tall as the Empire State Building.

现在很值得停下来反思一下

It's worthwhile to stop here and reflect for a moment.

Folding a paper in half, even a paper as fine as that of the Bible,

就算纸张跟圣经一样薄，对折了

25 times would give us a paper almost a quarter of a mile.

25 次后，厚度会接近 1/4 英里

What do we learn?

我们学到什么？

This type of growth is called exponential growth,

这样的增长称作「指数型增长」

and as you see, just by folding a paper

就如你所看到的，只是把纸对折

we can go very far, but very fast too.

就会变得很厚，而且变厚得很快

总之，如果我们将纸对折 25 次

Summarizing, if we fold a paper 25 times,

厚度大约是 1/4 英里 (0.4公里)

the thickness is almost a quarter of a mile.

30 次的厚度，接近 6.5 英里 (10 公里)

30 times, the thickness reaches 6.5 miles,

大约是飞机的平均飞行高度

which is about the average height that planes fly.

40 times, the thickness is nearly 7,000 miles,

40 次的厚度，将近 7,000 英里 (一万公里)

or the average GPS satellite's orbit.

是 GPS 卫星轨道 平均高度的一半

48 times, the thickness is way over one million miles.

折 48 次，厚度就超过一百万英里 (160万公里)

你想想，地球到月球的距离

Now, if you think that the distance between the Earth and the Moon

还不到 25 万英里 (约 38 万公里)

is less than 250,000 miles,

用一张圣经纸

then starting with a piece of Bible paper

and folding it 45 times, we get to the Moon.

折 45 次，我们就可以到达月球

如果再多折一次

And if we double it one more time,

我们就可以回到地球了！

we get back to Earth.

课程: Adrian Paenza 讲解: Adrian Paenza 动画: TED-ED 团队