Question

Dear Mitch,

Can you explain to me in very simple terms what Fibonacci numbers are and why they are so famous? And what in the world do they have to do with nature?

From Marissa S.

Berkeley, CA

Answer

Dear Marissa S.,

Sure!

The series of numbers known as the "Fibonacci Numbers" or the "Fibonacci sequence" is a pattern of real numbers, integers that were discovered by an Italian mathematician in the 1200's. His real name was Leonardo of Pisa, and he wrote a book about this fascinating arrangement he discovered, signing the book "Fibonacci".

In his book he told stories to make math concepts come alive (and he was one of the first known mathematicians to do this). His famous story was about a pair of rabbits. He posed the question, How many pairs of rabbits will there be each month if you start with that one pair and that pair produces a pair of babies every month. The rabbits start having babies when they are two months old, he said, and their babies also have their first set of babies at the age of two months.

The pattern that emerges is this:

Beginning:1 pair.

After Month 1: 1 pair,

After Month 2: 2 pairs,

After Month 3: 3 pairs,

After Month 4: 5 pairs,

After Month 5: 8 pairs,

After Month 6: 13 pairs,

After Month 7: 21 pairs

After Month 8: 34 pairs

After Month 9: 55 pairs,

After Month 10: 89 pairs

After Month 11: 144 pairs

And if you write the numbers of the Fibonacci sequence out from left to right, you might notice one of its very special properties. Take a look:

1 1 2 3 5 8 13 21 34 55 89 144

And what you might notice from looking at these first 12 numbers of the Fibonacci sequence of numbers is that after the first two, the others can be made by adding together the two numbers that immediately precede it. In other words, after the first two numbers (1, 1) you can get the next number in the sequence and the next and the next by adding together the one before the number you'd like to find with the number preceding that number. For example, if you were only given the first five numbers, 11235, and you wanted to come up with the next number, all you would have to do is add together the last two you were given, the 3 and the 5, and you would get 8. And although that is interesting, it becomes absolutely amazing when you do just a bit of investigating and discover that this pattern Fibonacci discovered appears all over the place in life—such as art, architecture, and -- most significantly -- in nature!

The petals of a flower, for example, usually occur in a Fibonacci number. Pine needles sprout in groups (called "bundles") and the bundles almost always come in groups of 1 or 2 or 3 or 5 needles. Pine cones have clockwise spirals and counterclockwise spirals, and if you look at a pine cone from its base after using two different colored pieces of chalk to color one of the spirals one color and another of the spirals that goes the other way another color, the number of complete spirals will be different numbers when you count them, BUT, miraculously, each of these numbers will be a Fibonacci number!

Here's another example: The number of spirals of diamond-shaped scales on the outside of a pineapple will always be a Fibonacci number!

There are many, many other ways that Fibonacci numbers show up in nature, but I think you probably have gotten the idea by now.

Can you guess what I find to be the most amazing part?

THIS:

No one really knows *why* nature works so consistently in this particular mathematical pattern. It is still one of the great mysteries that perplex us!

By the way, as miraculous as this discovery was, I think Fibonacci should be recognized first for being the one to convince the people of the world to switch from Roman numerals to the numerals used in the Arabic world. It is because of him that we no longer have to write a bunch of X's and V's and L's and I's for every two-digit or three-digit number!

Hope this helps you get an idea for the magic and mystery of math and nature,

Mitch