Maths: Fibonacci’s serious rabbits

An unlucky rabbit.

The early 1200s were very much the Middle Ages, but even then, steps were being taken that would lead, in time, to the Renaissance. Most of those steps involved trade, which led to new goods being introduced to new places, and along with them, new ideas.

Very few people have heard of Leonardo Pisano Bigollo by that name, but a lot more know a bit about him under the name Fibonacci. He was the son of a merchant who carried the nickname ‘Bonaccio’ which you can take to mean either ‘good-natured’ or ‘simple’, according to taste. I prefer to believe that Bonaccio, like his son, was far from simple.

Anyhow, the father ran an Italian trading post in what is now Algeria, and he produced a very bright son, who joined him there in the late 1100s, where the boy learned the Arabic (or Hindu) system of writing numbers. Leonardo knew a good thing when he saw it, and so he travelled around the Mediterranean, studying with various Arab scholars. Then he published his Liber Abaci in 1202. Literally ‘the book of the abacus’, this work introduced the new counting system in terms that tradesmen and academics could both understand, and young Leo gave practical examples.

It was by no means the first book to mention Hindu-Arabic notation, but it took off. This must have been due, at least in part, to the practical examples, like one in which he examined the way rabbits breed like, well, rabbits. Everybody knew how fast rabbit populations grew, but what was the mathematics of it?

He assumed that rabbits take a month to mature, then breed and produce two young, a male and a female, after one more month, and that rabbits live for 12 months. From that, he wondered how many rabbits there would be at the end of this time, starting with just two new-born rabbits. At the end of the first month, the rabbits mate, and there is still just one pair. At the end of the second month, the doe gives birth and there are two pairs. The parents mate again, and at the end of the third month, there is a third pair.

The first of the new young and the parents both mate and produce young at the end of the month. Now there are five pairs, three ready to breed, and two immature. At the end of the next month, there are eight pairs, and so on. The total number of rabbit pairs, taking the start as Month 0 increases like this:

The rabbity origin of the series.

The point of this small puzzle was for young Leo to show how much easier and quicker it was to add 89 and 144 than to add LXXXIX and CXLIV, the same numbers in Roman numerals.

At a Greek restaurant in Palm Cove in Queensland, I photographed this tiled table, to the puzzlement of a waiter. I have added numbers that may help you to work out why I was delighted. There are no other hints on this, but read on.

The new numbers caught on with the mob. It did no harm that there is a somewhat mystic number, f (phi), known as the Golden Ratio or Golden Mean, defined by the simple equation f–1 = 1/f. (phi-1 = 1/ph)

This value turns up in art, in Greek architecture, even in the proportions of normal sheets of paper, and when you divide any term in the Fibonacci series by the previous one, you obtain values that get closer and closer to f (phi), about 1.618. That really impressed people.

After he died, Leonardo came to be known as Bonacci’s boy, filius Bonacci in mediaeval Italian, or Fibonacci for short, and that is why the extended sequence of numbers is known today as the Fibonacci series. Of course, if you wanted a continuing series, you needed to assume that the rabbits were immortal, but given his other assumptions about the rabbits, what was wrong with that?

Though maybe we should celebrate Leonardo, the Golden Mean and his randy, speedy, rabbits by calling it the Phibonacci series, a comment that I will now explain.

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