Virahāṅka–Fibonacci numbers, and the poetry-counting problem that produced them

0:00One, two, three, five, eight, thirteen, twenty-one, thirty-four. You have probably met them before — in flowers, in spirals, in pine cones. And the usual story is that somebody stared at the list until they noticed each is the two before it added together. That is not what happened. The rule was not spotted. It was worked out — as the answer to a question that was not about numbers.

0:29The question came from poetry. In Sanskrit and Prakrit verse, and in Tamil, Telugu, Marathi and Malayalam too, you do not count a line's syllables. You count how long it takes to say. Count that way, and you land on a question whose answer is a number. Here is all the poetry you need. Every syllable is either short or long. A short one is held for one beat. A long one is held for exactly twice as long. Two beats.

1:03So a line of verse is a row of these, and how long it lasts is the total. Short, long, short is one beat, then two, then one. Four beats altogether. And the order is part of the line. Short-then-long and long-then-short take the same time, but they are not the same line, and no poet mixes them up. Hold on to that. It is the thing that makes this problem worth asking.

1:32Now the question the poets actually asked. You want a line lasting exactly eight beats. How many different ways are there to build one? Here is one. Four long syllables. Two, and two, and two, and two. Here is another. Eight short ones in a row. Here is a third. Short, long, long, short, long. And a fourth. Long, long, short, short, long. All four last exactly eight beats, and all four are different lines.

2:04So how many altogether? Right now it probably feels as though the only way to find out is to write them all out. Take the poetry away and what is left is arithmetic. Write one for a short syllable and two for a long one. Those four lines become four sums. Two plus two plus two plus two. One, eight times over. One plus two plus two plus one plus two. Two plus two plus one plus one plus two.

2:37Every one of them comes to eight. So here is the bare question. In how many ways can you write eight as a string of ones and twos? And the order counts. One plus two and two plus one are two different answers, not one answer written twice. That decision does more work than it looks. Ignore the order and the counts run one, two, two, three, three — a duller list, and not ours.

3:07Hard questions get answered by easy ones, so start at the bottom. One beat. There is only one way to fill it: a single short syllable. One. Two beats. A long syllable on its own, or two shorts. Two ways. Three beats. Short, short, short. Short then long. Long then short. Three ways. Four beats. Now there are five. Four shorts; a long in each of the three places it can sit; two longs.

3:38One, two, three, five. You are itching to guess the next one already. Please do not. Guessing is how this ends up remembered as a trick instead of understood as a fact. Five beats. And we are going to answer it without writing down one new line. Take every four-beat line — all five — and put a short syllable at the front of each. Each one now lasts five beats, because we added one beat to four.

4:09That is five five-beat lines, and it took no thought at all. Now take every three-beat line — all three — and put a long syllable at the front. Each of those lasts five beats too, because we added two beats to three. Three more. Five, and three. Eight five-beat lines, built entirely out of answers we already had. But wait. How do we know that is all of them?

4:38We have made eight. Maybe there is a ninth we never built. Here is why there is not, and this is the part everything else rests on. Take any five-beat line at all — one you have never seen, handed to you in a sealed envelope. It begins with a syllable, and that syllable is either short or long. There is no third possibility. If it begins short, take that short away: what is left lasts four beats, so it is in our first pile.

5:10If it begins long, take that long away: what is left lasts three beats, so it is in our second pile. Nothing escapes, because everything has a beginning. And nothing is counted twice, because no line begins in two ways at once. Now look at what that argument did not depend on. It never really used the number five. So run it again. Six beats. A short in front of every five-beat line: eight. A long in front of every four-beat line: five.

5:45Eight and five. Thirteen. We have not written down a single six-beat line, and we know there are thirteen of them. Seven beats. Thirteen and eight. Twenty-one. Eight beats. Twenty-one and thirteen. Thirty-four. One, two, three, five, eight, thirteen, twenty-one, thirty-four. There is the list — and you know why it is that list and not another. Somebody had to do this for the first time. The method is credited to Virahanka, a scholar of Prakrit, working around the year seven hundred.

6:24He was studying metre, and he set the rule down as a poem — a pleasing shape for a rule about poetry to arrive in. He was building on older work. Pingala, writing in Sanskrit, came about a thousand years before him. Gopala wrote about these numbers around eleven thirty-five, and Hemachandra around eleven fifty. And Fibonacci wrote about them in twelve hundred and two. That is roughly five hundred years after Virahanka, and getting on for fifteen hundred after Pingala.

7:01The numbers usually travel under Fibonacci's name. They are older than it by half a thousand years, and they came out of a question about verse. So, back to where we started. Eight beats. Count along the list. One, two, three, five, eight, thirteen, twenty-one, thirty-four. Thirty-four. Thirty-four ways to build an eight-beat line. Listing them would cost you an afternoon — and you would still not be sure you had them all.

7:33One warning about counting along that list, though. It begins one, two, three. Not one, one, two, three. That is because the first entry counts the one-beat lines, and there is one; the second counts the two-beat lines, and there are two. You will meet these numbers written with an extra one at the front. Both are in use. Know which you are reading, or every position you name is a place out.

8:04Here is a question you can answer without adding anything at all. Is the hundredth number on this list odd or even? Tag them as they go. One, odd. Two, even. Three, odd. Five, odd. Eight, even. Thirteen, odd. Twenty-one, odd. Thirty-four, even. Odd, even, odd. Odd, even, odd. It goes round every three. And it has to, because of the way each one is made. Odd and even make odd. Odd and odd make even. Even and odd make odd.

8:40So once you have an odd followed by an even, the next three are forced — and the three after those, for as long as you like. The even ones land at positions two, five, eight, eleven — every third place. A hundred is not one of those, so the hundredth is odd. And I added nothing. One last thing, and it is the best of them. Look back at the argument. It never mentioned poetry.

9:10It needed two facts. That everything is built from ones and twos, and that everything has a first one. So. Here is a staircase of eight steps. You can take them one at a time or two at a time, in any mixture. How many ways up? You already know. Thirty-four. The same problem in different clothes. A short syllable is a single step, a long one a stride over two, a line of verse a route up the stairs.

9:40You will also hear that petals tend to come in these numbers. Thirteen, twenty-one, thirty-four. Often they do — but tend to is as far as anyone should push it. The staircase is the honest one. Not a coincidence in nature, but the same argument twice over — because two things counted the same way come to the same count.