Fibonacci

The sequence was known before Fibonacci's time

Fibonacci credits the Indians (knowledge brought to Europe by the Persians and the Arabs) for much of what he presents in his first book. The sequence that bears his name is a good example. The Indian mathematician Hemachandra, while studying the structure of Sanskrit poetry, explained the solution was what we call the Fibonacci sequence, but he published around 1150, before Fibonacci was born!

Interesting properties of the sequence

Numbers are relatively prime

Adjacent numbers in the sequence are relatively prime. That is, they do not have common factors. For example, 21=3x7, 34=2x17 and 55=5x11, none of 21, 34 or 55 is prime, but none of their factors are the same. This property is very important when it comes to understanding why the spirals of the sunflower, for example are adjacent numbers in the Fibonacci sequence.

Numbers grow at the rate of ϕ

Adjacent numbers quickly approach a constant ratio equal to the "golden ratio" ϕ (approximately 1.618). It's worth mentioning here that:

ϕxϕ = ϕ + 1

1/ϕ = ϕ - 1

The fraction 1/89 is composed of the Fibonacci sequence

Type 1/89 into your calculator and you'll see a familiar sight:

0.011235955...

You'll see the start of the Fibonacci sequence. In fact, the whole sequence is buried there because

1/89 = 1/10² + 1/10³ + 2/10⁴ + 3/10⁵ + 5/10⁶ + 8/10⁷ + 13/10⁸ + ...

Why? Click here for the complete explanation.

The series is found in Pascal's Triangle

The series can be generated from that simple yet fascinating set of numbers, Pascal's triangle.

Click here to see.

Fibonacci's puzzle: The Rabbits

In his first publication, Liber Abaci*, Fibonacci presented the problem:

A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. Because the abovewritten pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs; of these in one month two are pregnant, and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; ...

And so on, he explains each month in turn. Finally, he concludes:

You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth, and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.

So the zero we now place at the start of the sequence arguably doesn't belong, especially considering it wasn't part of Fibonacci's puzzle problem that made the sequence famous.

* Fibonacci wrote in Latin. This translation is quoted from Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation, L. E. Sigler. Springer-Verlag, 2003. pp. 404-405.