Paper folding: the growth that outruns intuition

0:00Take a sheet of paper and fold it in half. Fold it again. And again. How many times can you actually do that? Most people get to six or seven and then the paper simply refuses. Try it, if you have a sheet nearby — it is a real limit and it is worth meeting once. Now suppose the paper did not refuse. Suppose it went on folding as many times as you liked.

0:28Someone once said that forty-six folds would reach the Moon. Someone else said that was ridiculous, that a few zeros must have gone missing after the forty-six. Neither of them can settle that by arguing. So let us settle it by counting. First, be clear about what is growing. Folding does not make the paper longer. Each fold halves the area and doubles the thickness. The sheet gets smaller and smaller across; the stack gets taller and taller.

1:02It is the height of that stack we are following, and nothing else. Start with a sheet one hundredth of a millimetre thick. Zero point zero zero one centimetres. Thin enough that a single sheet is not something you would notice at all. And every fold does exactly one thing to that number. It doubles it. Fold one takes it to zero point zero zero two centimetres. Fold two, zero point zero zero four.

1:35Fold three, zero point zero zero eight. Nothing is happening. After six folds the stack is zero point six four millimetres. Six folds, and it has not reached one millimetre. Fold seven is the first one that does. Then zero point one two eight, zero point two five six, zero point five one two. And fold ten is one point zero two four centimetres. Ten folds to cross a single centimetre.

2:07So if you think growth this fast must have started big, look at the start again. The first six folds were invisible. Seventeen folds. One hundred and thirty-one point zero seven two centimetres. That is over a metre and a third — a stack of paper taller than a small child, from a sheet you cannot see the edge of. And one word about how a number like that gets written.

2:36You will sometimes see this fold as about one hundred and thirty-one centimetres, and sometimes as one hundred and thirty-one point zero seven two. Those are the same quantity, rounded two different ways. The wavy equals sign means about. It is a claim about size, not an exact value, and nothing here disagrees with anything. Three more folds and the stack is ten point four eight five seven six metres.

3:09Now it starts to hurt. Twenty-six folds: six hundred and seventy-one metres. Set that against the tallest building anyone has put up, which stands eight hundred and thirty metres. Twenty-six folds is not quite there. One more fold — one — and the stack is one point three four kilometres. From four-fifths of that building to more than one and a half times its height, in a single fold. And nothing was brought in from outside to do it.

3:41The stack only sat down on top of itself. Thirty folds: ten point seven kilometres. That is roughly the height a passenger plane cruises at. Now turn it upside down. The deepest point in any ocean is about eleven kilometres down. Thirty folds does not reach it. The stack is a quarter of a kilometre short. Then fold thirty-one arrives and the stack is twenty-one point four seven kilometres. It does not merely clear that trench.

4:17It overshoots by more than ten kilometres, which is more than the whole depth of the trench. One fold. The step you take is as big as everything you already have. So: forty-six folds, and the Moon. The Moon is three hundred and eighty-four thousand, four hundred kilometres away. Fold forty-five puts the stack at about three hundred and fifty-two thousand kilometres. Close, and not there. Fold forty-six doubles that to about seven hundred and four thousand kilometres.

4:54It passes the Moon, and it passes it by a long way, on the fold straight after the one that fell short. So the first claim was right. And the objection — that some zeros had gone missing — was not just wrong, it was enormous. Drop a zero from forty-six and you are talking about fold four, which is zero point zero one six centimetres. Here is why nobody's intuition survives this.

5:25Ask how much the stack gains over ten folds. From fold zero to fold ten it gains one point zero two three centimetres. From ten to twenty it gains about ten and a half metres. From twenty to thirty, about ten point seven kilometres. From thirty to forty, about eleven thousand kilometres. The same ten folds every time. Four gains that do not share an order of size — each one is over a thousand times the one before it.

5:59If somebody told you the stack gains a fixed amount per fold, what amount would you write down? There is no such number. These differences are not merely uneven. They are useless. Now take those same four intervals and do something else to them. Instead of subtracting, divide. Fold ten divided by fold zero: one thousand and twenty-four. Fold twenty divided by fold ten: one thousand and twenty-four. Twenty to thirty: one thousand and twenty-four.

6:36Thirty to forty: one thousand and twenty-four. Four numbers that had nothing in common when we subtracted are one number when we divide. And not approximately one number. Exactly, with nothing left over, every time. Every ten-fold interval anywhere in this table does the same. The steadiness was there all along. We were looking at it with the wrong operation. And one thousand and twenty-four is not a coincidence waiting to be discovered.

7:11Start at one and double it ten times. Two, four, eight, sixteen, thirty-two, sixty-four, one hundred and twenty-eight, two hundred and fifty-six, five hundred and twelve, one thousand and twenty-four. Ten doublings multiply by that, whatever you began with. Which means you can watch the same thing happen in miniature. Three folds multiply by eight, because doubling three times is two times two times two. One fold doubles. Down at fold four, zero point zero one six becomes zero point zero three two.

7:52Up at fold nine, zero point five one two becomes one point zero two four. The same operation, low on the table and high on it. To see what that costs, suppose the stack had grown the other way. Suppose every fold added what the first fold added — one thousandth of a centimetre, fixed, every time. Forty-six of those leaves the stack at zero point zero four seven centimetres.

8:23Not the Moon. Under half a millimetre. Give it eighty folds and it still has not reached one centimetre. Same sheet, same number of folds. The only change is that a fixed amount was added instead of the whole stack. That is the entire difference between adding and doubling, and the size of it is the distance to the Moon. Growth like this has a name. When each step multiplies by a fixed number, it is called exponential growth, or multiplicative growth.

9:00Two names for one behaviour. The fixed number is the growth factor, and here it is two. Notice what the name is not about. It is not about the numbers being large; they started invisibly small. It is not about the steps being big; the first step was a thousandth of a centimetre. It is about the step being a fixed multiple instead of a fixed amount. Get that one thing right and the whole table is a single sentence.

9:31Each fold doubles it. Everything else is bookkeeping.