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.

 

·      Flowers

·      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?

 

Go ahead. Try it out!

---

 

Return to Brooke’s EMAT 6690 homepage

 

 

 

 

 

  *