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

摺紙送你上月球 - 阿德里安潘薩 (Adrian Paenza)

譯者: Jephian Lin 審譯者: Allen Li

（音樂）

一張紙可以對摺幾次？

How many times can you fold a piece of paper?

假設有一張很薄很薄的紙

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

像是印聖經的那種紙

like the kind they typically use to print the Bible.

實際上，就像是一根絲那麼薄

In reality, it seems like a piece of silk.

為了要把這個想法量化

To qualify these ideas,

let's say you have a paper

假設紙的厚度是 1/1000 公分

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

也就是 10 的 -3 次方公分

That is 10 to the power of minus three centimeters,

which equals .001 centimeters.

相當於 0.001 公分

再假設這張紙夠大

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

像報紙那麼大

like a page out of the newspaper.

Now we begin to fold it in half.

我們開始將它對摺

你覺得我們可以對摺幾次？

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

And another question:

另一個問題是：

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

如果你能將一張紙 折疊任意多次

say 30 times,

比方說 30 次 你想那時紙會有多厚？

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

在開始之前

Before you move on,

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

我鼓勵你好好想想 這問題可能的答案

好，在對摺一次後

OK.

After we have folded the paper once,

厚度變成 2/1000 公分

it is now two thousandths of a centimeter in thickness.

如果再對摺一次

If we fold it in half once again,

the paper will become four thousandths of a centimeter.

這張紙就會變成 4/1000 公分

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

每摺一次，紙張厚度加倍

如果我們不斷對摺

And if we continue to fold it again and again,

十次以後

always in half, we would confront the following situation

after 10 folds.

我們就會見到這種情景

Two to the power of 10,

2 的 10 次方

meaning that you multiply two by itself 10 times,

也就是將 2 自乘 10 次

就是 1024/1000 公分

is one thousand and 24 thousandths of a centimeter,

which is a little bit over one centimeter.

比 1 公分多一點

如果我們繼續對摺

Assume we continue folding the paper in half.

會發生什麼事？

What will happen then?

If we fold it 17 times,

如果對摺 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,

大約 131 公分

and that equals just over four feet.

超過 4 英尺 (1.3 公尺)

If we were able to fold it 25 times,

如果我們可以對摺 25 次

then we would get two to the power of 25,

就會得到 2 的 25 次方

which is 33,554 centimeters,

大約 33,554 公分

超過 1100 英尺 (335 公尺)

just over 1,100 feet.

差不多跟帝國大廈一樣高

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

現在很值得停下來反思一下

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,

就算紙張跟聖經一樣薄，對摺了

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

25 次後，厚度會接近 1/4 英里

What do we learn?

我們學到什麼？

This type of growth is called exponential growth,

這樣的成長稱作「指數型成長」

and as you see, just by folding a paper

就如你所看到的，只是把紙對摺

we can go very far, but very fast too.

就會變得很厚，而且變厚得很快

總之，如果我們將紙對摺 25 次

Summarizing, if we fold a paper 25 times,

厚度大約是 1/4 英里

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,

大約是飛機的平均飛行高度

which is about the average height that planes fly.

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

40 次的厚度，將近 7,000 英里 (一萬公里)

or the average GPS satellite's orbit.

是 GPS 衛星軌道 平均高度的一半

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

摺 48 次，厚度就超過一百萬英里 (280 萬公里)

你想想，地球到月球的距離

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

還不到 25 萬英里 (約 38 萬公里)

is less than 250,000 miles,

用一張聖經紙

then starting with a piece of Bible paper

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

摺 45 次，我們就可以到達月球

如果再多摺一次

And if we double it one more time,

我們就可以回到地球了！

we get back to Earth.

課程: Adrian Paenza 講解: Adrian Paenza 動畫: TED-ED 團隊