1. The Golden Ratio and Fibonacci Numbers in Nature
By: Mary Catherine Clark

2. History
Leonardo Fibonacci was the most outstanding mathematician of the European Middle Ages.
He was known by other names including Leonardo Pisano or Leonard of Pisa.
Little was know about his life except for the few facts given in his mathematical writings.

3. History cont. Fibonacci was born around 1170.
Received his early education from a Muslim schoolmaster.
His first book, published in 1202, was called Liber Abaci. This book is devoted to arithmetic and elementary algebra and with Arabic numerals.
His next book, Practica Geometriae, he wrote in This book presents geometry and trigonometry with Euclidean methods.

4. History cont.
He employed algebra to solve geometric problems and geometry to solve algebraic problems. This was radical in Europe at the time.
He also wrote the Liber Quadratorum, a book that earned him the reputation as a major number theorist.

5. History cont.
In the Liber Abaci he introduce the Hindu-Arabic number system to Europe- the system used today.
The Liber Abaci was instrumental in the process of replacing the Roman numeration system with the Arabic system.
Today a statue of Fibonacci stands in a garden across the Arno River, near the Leaning Tower of Pisa.

6. Fibonacci’s rabbits
One of the problems in The Liber Abaci was: Suppose a newly born pair of rabbits (a male and female) are put in a field. The rabbits are able to mate at the age of one month so that at the end of the second month a female can produce another pair of rabbits. Assuming that the rabbits never die and the female always produces a new pair every month from the second month on. How many pairs will there be in one year?
Answer: 144 rabbits

7. Fibonacci numbers
The sequence in which each number is the sum of the two preceding numbers.
This sequence is defined by the linear recurrence equation.
and by the definition above
 The Fibonacci numbers for n=1,2 are 1, 1, 2, 3, 5, 8, 13, 21,…
Before Fibonacci wrote his work, these numbers had already been investigated by Indian Scholars interested in rhythmic patterns of syllables.

8. What is the golden ratio?

9. The golden ratio
The golden ratio is an irrational number defined to be
This has a value of and is sometimes denoted by φ after the mathematician Phidias who studied its properties.
From the definition above we can see that the golden ratio a/b satisfies the equation 1+ 1/x = x.

10. The golden rectangle
The golden rectangle can be constructed from these line segment so that the length to width ratio is φ.
At each step of this construction, the golden rectangle may be divided into a square and a smaller golden rectangle. Let us verify that on the board!
The ancient Greeks believed that a rectangle constructed in this manner was the most aesthetically pleasing of all rectangles and they incorporated this shape into a lot of their art and architectural designs.

11. What does this have to do with fibonacci numbers?
The ratio of successive Fibonacci numbers is something you might be surprised by!
Theorem: As n increases, the ratio of approaches the golden ratio, i.e., =

12. Proof.
If the limit of Fn/Fn-1 exists, it is the same as the limit of Fn+1/Fn . So when n is large,
Fn/Fn-1 = Fn+1/Fn + e(1)
e(1) means that the difference of these two formulas goes to zero when n goes to infinity.
But Fn+1= Fn-1+ Fn, and so, Fn+1/Fn = 1+Fn-1/Fn .
If we let a= Fn and b= Fn-1, from the formula above we have, up to a small error,
a/b= 1+b/a
Which is the definition of the golden ratio.

13. Fibonacci numbers and the golden rectangle
Let us construct a quarter of a circle in each square going from one corner to the opposite.
This is not a true mathematical spiral.

14. Fibonacci numbers in nature
Look at any seed head, and you will notice what look like spiral patterns curving out form the center left and right.
If you count these spirals you will find a Fibonacci number. If you look at the spirals to the left and then the right you will notices these are two consecutive Fibonacci numbers.

15. fibonacci numbers in nature
These can also be seen in pinecones, pineapples, cauliflower, and much more!

16. More fibonacci numbers in nature
Most of the time, the number of pedals on a flower is a Fibonacci number!
1 pedal-calla lily
2 pedals-euphorbia
5 pedals-columbine
3 pedals-trillium
13 pedals-black eyed susan
8 pedals-bloodroot

17. Works cited
Dunlap, Richard A. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific, 1997.
Koshy, Thomas. Fibonacci and Lucas Numbers with Applications. New York: John Wiley & Sons, inc., 2001.
Vorobiev, Nicolai N. Fibonacci Numbers. 6th. Basel: Birkhauser Verlag, 1992.

18. Works cited Http://www.world-mysteries.com/sci_17.htm