Thursday, January 19, 2012

Leonardo Pisano Bigollo

It's very unlikely that you’ve heard of Leonardo Pisano Bigollo; I hadn’t until I started looked for some background information for this blog.  He was born in 1170 and died some 80 years later, he was an intellectual friend of Emperor Fredrick II, and he is regarded as perhaps the greatest western mathematician of the middle ages.  He also made an enormous contribution to western culture by recognizing and popularizing a concept that changed the way we deal with numbers to an extent equivalent to what computers have meant in our era.

But he is remembered today because he gave his name (or at least one of his names) to one of the most intriguing mathematical phenomena in nature.  Part of the name problem is that he seemed to go under a variety of names – frowned upon nowadays! – but in the intellectual world he was often referred to as Leonardo Fibonacci or, most usually, just Fibonacci.

Statue of Fibonacci in Pisa

Fibonacci’s great contribution was that he recognized the value of the Hindu-Arabic number system, and popularized it in the west through a book Liber Abaci.  At that time the Roman number system was in use.  We still find it quaint to use it on clocks or at the end of movies to code the year, but imagine doing arithmetic in it. XII + XII = ?  Easy rules enable us to work out 12 + 12 but how would you go about doing the Roman calculation?  (No cheating by converting any of the Roman characters to Hindu-Arabic numerals.)  And after you’ve worked that out you might like to try XII x XII = ?  The only odd thing is that it took more than six hundred years from the invention of the Hindu-Arabic system for someone to get the point in the west.  Of course, communication was very different in the twelfth century, and no doubt there was a bit of prejudice about foreign ideas.  (Not everything’s changed in the last 800 years.)

Fibonacci did some really deep research of his own in number theory in an era where there were few established mathematical tools.  But he's remembered today for a particular sequence of numbers which is now called the Fibonacci sequence, although it too goes back to the Hindu mathematicians.  Fibonacci was interested in the numbers because of an idealized rabbit breeding problem.  (I’m not going to go into that here – idealized rabbit breeding sounds like an oxymoron to me.)  This is how the sequence works.

Fibonacci Chimney by Mario Merz

Each number in the sequence is obtained by adding the previous two numbers together.  The first two numbers are 1 and 1.  Thus the third number is 1+1=2. The fourth number is 1+2 = 3.  The fifth is 2+3 = 5, the next 3+5 = 8, then 13, 21, 34 and so on.  The first few terms of the sequence are displayed on the chimney of Turku Energia in Turku, Finland, in two meter high neon lights.
Fibonacci's Dream by MartinaSchettina
What is fascinating about the Fibonacci numbers is that they occur in all sorts of surprising places in nature.  They lead to particularly attractive spiral curves, they predict the way leaves and flower petals are arranged.  And those spirals are clearly visible in the seed heads of sunflowers and many others.  The arrangement of plant primordia in this way even has a name - phyllotaxis. 

A Fibonacci Spiral
Each square's side
is a Fibonacci number
They seem to have an almost mystical fascination.  That's how they arose originally in connection with Sanskrit prosody.  There is a formula which allows one to calculate the numbers, and it involves the Golden Ratio beloved of the ancient Greeks for the design of temples.  For some reason there is a feeling of perfect proportion about this ratio.  When we see it in art or architecture we immediately feel a satisfaction with the proportions.  Where does that come from?  Take a look at the Parthenon below. 

Modern artists use the numbers themselves, and they made a bit appearance in The Da Vinci Code.  There is a journal published quarterly devoted specifically to their properties and applications.  Half an hour with Google shows that they are used to argue everything from numerological conspiracies to the existence of God.

However, their ubiquitous occurrence in nature turns out to be a nod in Darwin’s direction.  The arrangement of leaves in this particular spiral allows the best average exposure to sunlight.  The Fibonacci spirals of the sunflower seed head can be shown to be the best way of packing seeds that are generated at the center and then move outwards.  Here’s a link to a YouTube simulation which shows how the process might work. Sunflower

So Fibonacci will be remembered forever, albeit maybe for the wrong reason.  Not bad for a mess of rabbits.

Michael – Thursday. 

PS Stanley and I just learned that our book DEATH OF THE MANTIS has been shortlisted for the Edgar Award for best paperback original.  We're absolutely thrilled and just want to thank all our friends and readers out there for all your enthusiasm, encouragement and support.



  1. I can't believe I finally understood this. You write so clearly. However, the number of numbers is still overwhelming to me :)

  2. Good point! For the number of numbers can't itself be a number! :)

  3. I'm so confused. The only calculation I'm sure of at the moment is MichaelxStanley=Edgar.

  4. I've got to get to my TBR pile. Good luck to you guys!