(音樂)
How many times can you fold a piece of paper? Assume that one had a piece of paper that was very fine, like the kind they typically use to print the Bible. In reality, it seems like a piece of silk. To qualify these ideas, let's say you have a paper that's one-thousandth of a centimeter in thickness. That is 10 to the power of minus three centimeters, which equals .001 centimeters. Let's also assume that you have a big piece of paper, like a page out of the newspaper. Now we begin to fold it in half. How many times do you think it could be folded like that? And another question: If you could fold the paper over and over, as many times as you wish, say 30 times, what would you imagine the thickness of the paper would be then? Before you move on, I encourage you to actually think about a possible answer to this question. OK. After we have folded the paper once, it is now two thousandths of a centimeter in thickness. If we fold it in half once again, the paper will become four thousandths of a centimeter. With every fold we make, the paper doubles in thickness. And if we continue to fold it again and again, always in half, we would confront the following situation after 10 folds. Two to the power of 10, meaning that you multiply two by itself 10 times, is one thousand and 24 thousandths of a centimeter, which is a little bit over one centimeter. Assume we continue folding the paper in half. What will happen then? If we fold it 17 times, we'll get a thickness of two to the power of 17, which is 131 centimeters, and that equals just over four feet. If we were able to fold it 25 times, then we would get two to the power of 25, which is 33,554 centimeters, just over 1,100 feet. That would make it almost as tall as the Empire State Building. 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, 25 times would give us a paper almost a quarter of a mile. What do we learn? This type of growth is called exponential growth, and as you see, just by folding a paper we can go very far, but very fast too. Summarizing, if we fold a paper 25 times, the thickness is almost a quarter of a mile. 30 times, the thickness reaches 6.5 miles, which is about the average height that planes fly. 40 times, the thickness is nearly 7,000 miles, or the average GPS satellite's orbit. 48 times, the thickness is way over one million miles. Now, if you think that the distance between the Earth and the Moon is less than 250,000 miles, then starting with a piece of Bible paper and folding it 45 times, we get to the Moon. And if we double it one more time, we get back to Earth.
一張紙可以對摺幾次? 假設有一張很薄很薄的紙 像是印聖經的那種紙 實際上,就像是一根絲那麼薄 為了要把這個想法量化 假設紙的厚度是 1/1000 公分 也就是 10 的 -3 次方公分 相當於 0.001 公分 再假設這張紙夠大 像報紙那麼大 我們開始將它對摺 你覺得我們可以對摺幾次? 另一個問題是: 如果你能將一張紙 折疊任意多次 比方說 30 次 你想那時紙會有多厚? 在開始之前 我鼓勵你好好想想 這問題可能的答案 好,在對摺一次後 厚度變成 2/1000 公分 如果再對摺一次 這張紙就會變成 4/1000 公分 每摺一次,紙張厚度加倍 如果我們不斷對摺 十次以後 我們就會見到這種情景 2 的 10 次方 也就是將 2 自乘 10 次 就是 1024/1000 公分 比 1 公分多一點 如果我們繼續對摺 會發生什麼事? 如果對摺 17 次 厚度就是 2 的 17 次方 大約 131 公分 超過 4 英尺 (1.3 公尺) 如果我們可以對摺 25 次 就會得到 2 的 25 次方 大約 33,554 公分 超過 1100 英尺 (335 公尺) 差不多跟帝國大廈一樣高 現在很值得停下來反思一下 就算紙張跟聖經一樣薄,對摺了 25 次後,厚度會接近 1/4 英里 我們學到什麼? 這樣的成長稱作「指數型成長」 就如你所看到的,只是把紙對摺 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 厚度大約是 1/4 英里 30 次的厚度,接近 6.5 英里 (10 公里) 大約是飛機的平均飛行高度 40 次的厚度,將近 7,000 英里 (一萬公里) 是 GPS 衛星軌道 平均高度的一半 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 還不到 25 萬英里 (約 38 萬公里) 用一張聖經紙 摺 45 次,我們就可以到達月球 如果再多摺一次 我們就可以回到地球了!