Wednesday, July 17, 2013

Another Reason To Love Pisa: Leonardo Fibonacci (Or, "Impress Your Friends, Clients, And Colleagues With Today's Math Lesson")

If you knew me during high school -- or if we competed at the Long Island Math Fair at any point during 1982-1985 -- then you surely remember the Fibonacci Sequence.  This famous mathematical sequence begins with the following numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.....  You can see the clear pattern, si?  Each number in the sequence is the sum of the preceding two.  In mathematical terms, it's simple:

        f(n+2) = f(n+1) + f(n)

        where n begins with 1, f(1) = 1, and f(2) =1

The series is named for Leonardo Fibonacci, an Italian mathematician who first posed a mathematical problem (based on reproduction patterns in rabbits) the answer to which is the sequence that would later bear his name.  (See, generally,  Fibonacci lived from around 1170 to 1250, and he is widely considered to be the greatest western mathematician of the Middle Ages.  He haled from Pisa -- hence his nickname: Leonardo of Pisa.

Well, the Fibonacci Sequence is absolutely everywhere.  It proves, definitively, that if God is not a lawyer, s/he is a mathematician.  I need one quick detour to back up that conclusion.  It concerns the "Golden Ratio."

If you take two consecutive Fibonacci numbers and divide the later one by the earlier one, a pattern quickly emerges:

     1/1 = 1
     2/1 = 2
     3/2 = 1.5
     5/3 = 1.666666....
     8/5 = 1.6
     13/8 = 1.625
     21/13 = 1.615384...

What you see is that the ratios approach a special number called the Golden Ratio, which is approximately 1.618.  (The actual number is 1 plus the square root of 5 where the entire sum is divided by 2.)  A rectangle where the ratio of length to width is 1.618 is considered to be most pleasing to the eye and is called a "golden rectangle."  The Greeks knew this, so the Parthenon is a "golden" rectangle.  Da Vinci knew this, so he used golden rectangles everywhere including in the Mona Lisa.  The same goes for other famous works of art that are too numerous to mention here.

We find the golden ratio and Fibonacci numbers in music too, most famously in the work of Bela Bartok.

And nature.  The spiral of the chambered nautilus spins out in a path that tracks the golden ratio and the Fibonacci Sequence.  (See, e.g.,  Likewise for the spirals on pine cones and on the faces of sunflowers.  (See, e.g.,   Reproductive patterns in some species -- like bees -- follow the Fibonacci Sequence.  (Id.)  Even the patterns of spots on Dalmatians are organized according to Fibonacci numbers.  (OK, I totally made that last one up; just wanted to see if anyone was still reading.)

But, seriously, other examples from the arts, nature, aesthetics, sex, computer science, physics, and other branches of mathematics abound.  I could go on and on.  And I did -- this was a project for three straight years in high school, and even in one class during college.

So, the point here is that Pisa remains proud of its greatest mathematical son.  In the Camposanto, which is one of the 4 buildings that constitute the Campo dei Miracoli, stands a statue of Fibonacci.  Seeing it live was a thrill for me and I'm sure it would be the same for you.  The crowd around the statue of Fib was not quite as huge as the line for the Leaning Tower.  But it was close.

There is actually more.  According to a website I read on the train ride up to Pisa, as well as the map on my iPhone, there is a street in Pisa named after Fibonacci.  It's a little bit out of the way and completely on the other side of town from the Leaning Tower.  But we trekked over there.  I mean, how many times will you get to see the intersection of 2 streets where one is named after Galileo and one is named after Fibonacci?!?!?   The only problem is that the street sign for Rue de Fibonacci was missing.  It was cute to see Galileo Avenue though.

This is where the street sign for Leonardo Fibonacci is supposed to go.
There endeth the lesson.


  1. There must be a Fibonacci Dalmatian out there somewhere. Keep looking.

  2. I'm sure you're right, Bryan. I will start my hunt now. Even if you're wrong, this would be a good sub-plot for a Da Vinci Code-styled novel.

  3. Loved this! N used to run the Fibonacci sequence to entertain himself at long boring restaurant dinners.

  4. Brings back memories...I am told that Oreo Cookies are made using the Golden Ratio of crème to cookie...truly everywhere.

  5. Also when you slice a pizza, you should always follow the golden ratio..

  6. I use the Golden Ratio to make my gin and tonics. Tonic/gin = 1.618.... Nice drink.

  7. Hi Jeff: that sounds great. The next time I make a gin & tonic, I'm going to try that! Ciao.