Wednesday, December 19, 2007

Wednesday Math, Vol. 7: Leonardo of Pisa

The subject of today's math chat is Leonardo of Pisa, born in Pisa sometime around 1170 A.D. who died sometime around 1250. After his death, people started to refer to him as Fibonacci, short for "Son of Bonacci", his dad's nickname, which either means kind-hearted or simple. During his life and in his own writings, he was usually referred to as Leonardo Pisano.

His greatest work is the book Liber Abaci, and the longest lasting effect of that is his championing of the number system we use today, which some people call Arabic numerals and some people call Hindu-Arabic numerals, over the Roman numeral system which was in widespread use in Europe when Leonardo was alive. Arab traders brought the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and the place holder system to Europe, but they first saw the system on trips to India where it was invented. Leonardo knew this, and called it modus Indorum, the Indian method . He learned this on a trip to North Africa in his youth, where the educated classes were already getting rid of Roman numerals, which look pretty but are difficult to use for addition, subtraction, multiplication and division.

The place where the name Fibonacci still lives is the Fibonacci numbers, taken from a problem he posed in Liber Abaci which he actually learned from the ancient Indian text. He posed the problem as way that rabbits breed, which is completely hypothetical. It turns out, oddly enough, to be completely applicable to the way cells divide, and so Fibonacci numbers turn up in many ways in nature. This is long before anyone on any continent has ever guessed about the nature of cells in living organisms.

The simplest way to explain the Fibonacci number is to say the first two numbers on the list are 0 and 1, and the next Fibonacci number is always the sum of the last two on the list.

0 + 1 = 1
1 + 1 = 2
1 + 2 = 3 (don't worry, it starts getting good soon)
2 + 3 = 5
3 + 5 = 8...

So the list of Fibonacci numbers goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... and continues on forever. There are lots of interesting patterns in this sequence, and the numbers show up as answers to lots of odd questions from any number of fields in math.

One is in biology.

The way it works with cell growth is that when cells split, one of the new cells is completely ready to split again at the next time interval, while the other cell needs that time to grow to be ready to split in the time interval after next. Here's a short sequence of intervals to show the pattern and why it's a Fibonacci sequence. Here's what will happen when the time interval for splitting comes.

One ready cell becomes two cells: one ready, one not ready
One not ready cell becomes a ready cell.

Start with a ready single cell. Assume that no cell dies.

Interval #1: 2 cells, 1 ready, 1 not ready
Interval #2: 3 cells, 2 ready, 1 not ready
Interval #3: 5 cells, 3 ready, 2 not ready
Interval #4: 8 cells, 5 ready, 3 not ready

One of the places this shows up in nature is counting spiral patterns on objects like pine cones, pineapples, artichokes, etc. The spirals in one direction are always "wrapped tighter" than they are in the other direction. In this picture, the counterclockwise spirals are marked with gold lines, except the "start" which is blue. In that direction there are eight spirals. In the clockwise direction the spirals are marked in cyan, with the "start" in purple. Here there are 13 spirals, the next biggest Fibonacci number after 8. On a pineapple, if you cut far enough away the stem, there might be a lot more spirals in both directions as you count all the way around a cut made perpendicular to the height of the pineapple, but both the counterclockwise and clockwise counts of spirals will be adjacent Fibonacci pairs, like 21 and 34, or 34 and 55, etc.

There are many examples in math where someone poses a problem because it's "fun" or "looks pretty" when there isn't really an underlying scientific need to know the answer, and centuries or decades later, it turns out to be the answer to a real-life application. Some folks get mystical about this. On my wander around the 'net researching this, one blog went into a reverie saying the Fibonacci numbers in nature are a proof of intelligent design. When the subject is taught without an example of cell growth, this kind of magical thinking is easy to understand, but often there is a scientific explanation to the beautiful patterns we see. I do not profess to know the ultimate nature of the universe and its origin, but the proofs of God's existence because of the existence of pretty things always sets off my more skeptical nature.

Here endeth the lesson, but keep your notes. The Fibonacci numbers will show up next week.


CDP said...

This is the best explanation of the Fibonacci numbers that I've ever seen; very instructive for the mathematically challenged (I include myself among them). Very nice 'splainin.

dguzman said...

For a second, I read his book's title as "Liberace" hee hee.

I've heard many times that math is the language in which the universe is written; I think you've just proven that.

Beautiful work, Professor.

Distributorcap said...

so this guy didnt paint the Mona Lisa?


you are bringing back my childhood

Matty Boy said...

Actually, DCap, the Mona Lisa shows up next week in the next related math post.

FranIAm said...

franiam so sad and so dumb at the maths.

so little in life intimidates me. this does.

deep sigh.