Fibonacci’s Sequence

A Natural Occurrence

by

Brooke Norman

What is the Fibonacci sequence?

Fibonacci is a nickname for Leonardo Pisano.  He is the man who discovered the pattern of adding two previous numbers together will result in the next number of the pattern, starting at 0 and then 1.   The general formula for this sequence is fn+1 = fn + fn-1,   f1 = f2 =1.  The resulting sequence looks like the following.  1, 1, 2, 3, 5, 13, 21, 34, 55, 89, 144, …

Here is an example of the first 25 numbers:

(this was done on Microsoft Excel)

 0 0 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 10 55 11 89 12 144 13 233 14 377 15 610 16 987 17 1597 18 2584 19 4181 20 6765 21 10946 22 17711 23 28657 24 46368

Another part of the Fibonacci sequence is the spiral that it makes.  This is made by using the golden ratio.   To better see the relationship the sequence has in a more geometrical way, we can look at it as squares. If we start with a square of sides 1, we add another square of sides 1. The result is two.  This is the new square with sides 2.  If the squares of sides 1 and 2 are added, we get the square of sides 3.  This is continued for adding the square of sides 2 and the square of sides 3 to get a square of sides 5.  This continues on and we continue to get a spiral.

Here is an example.

What many people don’t know is that this sequence is found in nature.  It occurs quite often and is sometimes referred to as a "law of nature."

Click the following links to see different examples in nature.

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·      Pinecones and Sunflowers

·      Honeycombs

Fibonacci would illustrate his sequence by asking people to solve his “Rabbit Problem.”

How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?