Leonardo Pisano and his spiral(*)

The year is 1202. In the east, Genghis Khan was alive and well, conquering most of Asia as well as parts of Europe.In Europe and the nearer East, the Crusades were in full swing. Columbus had not been born yet and in the far West, in America (north and south) the Mayas, Aztecs and Incas were building impressive civilizations. No European had ever heard of tomatoes, potatoes, tobacco, corn (maize) or chocolate. True, some of the Vikings got to America by then, but it was north north America (Canada), and, as far as I know, they found no tomatoes, potatoes or other such stuff there. Joan of Arc had not been born yet. Half of Spain was Muslim country, the kingdom of El Andalus. Italy was just a conglomerate of states, often at war with each other. The Holy Roman Emperor claimed to be the direct descendant of the Roman emperors, and so did the popes. After the fall of the Roman Empire, some 800 years earlier, Europe entered a time in which science did not advance much; in fact it could be said to be going backward. By the year 1200 the great intellectual centers of the world had shifted mostly to the Arab countries, especially to Cordoba in Spain, Baghdad in what is now Irak. Far away in India, Indian mathematicians made one of the greatest advances ever in mathematics: They discovered zero, negative numbers, and the way we write numbers now, in positional notation. The Arabs adopted this discovery, so much so that our numbers are sometimes called arabic numerals.

In the late 1100's, a young Italian man travelled the world and learned about all these wonderful new inventions, not so new by then, and he made it his mission to spread the word about them in Italy and the rest of Europe. His name was Leonardo; because he was born in the beautiful town of Pisa(**), near the even more beautiful city of Florence, he is known as Leonardo Pisano. He is even better known as the son of Bonacci or Fibonacci. In 1202 he published his most famous book, the Liber Abaci.

Imagine you have to add 562 + 738 in Europe, in those long gone days. If you used Roman numerals (the most common way of writing numbers), you had to perform
DLXII + DCCXXXVIII.
Not so easy. Not as easy as
Imagine multiplying or dividing numbers written in Roman notation! Most people couldn't do it without the help of some calculating device; the most common being the abacus
A Roman abacusA Chinese abacus

Liber is Latin for book, and one might think that Liber Abaci means book of the abacus. Not so! People good at calculations became known as abacists and abacus became a word meaning calculation. That is why Fibonacci's great work has to be translated as The Book of Calculations. By this book, Fibonacci was going to teach Europe to calculate without needing an abacus. But the book is so much more! It teaches by posing problems and solving them and we'll look at a few of these problems. And try to solve them.

We'll start with the most famous one of them all, the rabbit problem, where the even more famous Fibonacci Sequence appears for the first time. We'll also see a few more modern problems related to this sequence; it is not just for rabbits anymore.


(*) Well, it isn't really his spiral; it is a spiral based on his sequence known more formally as a logarithmic spiral. It appears a lot in nature.
(**) The construction of the leaning tower of Pisa began in 1173, some three years after Fibonacci's birth. It took 344 years to be completed.