FIBONACCI, THE GOLDEN

NUMBER, AND SPIRAL GROWTH IN

NATURE

http://www.youtube.com/watch?v=kkGeOWYOFoA

FIBONACCI'S DILEMMA (YEAR 1202)

Original Question:

How fast rabbits can rabbits breed in ideal circumstances?

Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits.

Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was...

How many pairs will there be in one year?

FIBONACCI SEQUENCEEach number created by adding the two previous numbers

What are the next 5 Fibonacci numbers?

34, 55, 89, 144, 233, …

This sequence has fascinated mathematicians for centuries…

1,1,2,3,5,8,13,21,…

FIBONACCI NUMBERS ARE EVERYWHERE!Flower Petals:

MOST flowers have petals that occur in Fibonacci numbers (1, 2, 3, 5, 8, …)

Very few flowers have petals that do not occur in Fibonacci numbers (4, 6, 7, …)

1 petal

2 petal

3 petal

5 petal8 petal

13 petal

21 petal

34 petal

PINEAPPLE SPIRALS

PINECONE SPIRALS

13 spirals

8 spirals

SUNFLOWER SEED SPIRALS

HUMAN HAND BONE MEASUREMENTSNot to mention, we have 2 hands,each with 5 fingers, each with 3 parts!

8 53

2

THE FIBONACCI RECTANGLE:THE GOLDEN SPIRAL A Fibonacci Rectangle (the Golden

Rectangle) is created by taking the Fibonacci numbers and arranging them as shown:

GOLDEN SPIRAL By drawing the curve through the

corners of the boxes, we create something called the golden spiral (or sometimes logarithmic spiral)

GOLDEN SPIRAL:NAUTILUS SHELLThe most classic example of the golden spiral in nature is the cross section of the chambers of the Nautilus Shell.

NAUTILUS SHELLS

THE GOLDEN RATIO:If you start dividing the Fibonacci

numbers backwards, the quotient gets closer and closer to the number 1.6182/1=23/2=1.55/3=1.6678/5=1.613/8=1.62521/13=1.61534/21=1.61955/34=1.61889/55=1.618144/89=1.618

We call this number φIt can be pronounced“Fee” or “Fye”

Φ=1.618… and is called the Golden Ratio

THE GOLDEN RATIO: PHI Φ

GOLDEN RATIOS & GOLDEN RECTANGLES The golden ratio is

considered to be the most aesthetically pleasing ratio to the human eye. It is used in art, architecture, and advertising.

Any rectangle whose length ÷ width ≈ 1.618 is called a golden rectangle.

GOLDEN RECTANGLES

Apple IPOD dimensions are 1:1.67 and is the closest MP3 player to the golden ratio.

CULT OF THE GOLDEN RATIOSome people are obsessed with finding golden ratios in everything they see. The see the shape of cereal boxes, cigarette packages, and note-cards

as a giant conspiracy.

Jack Ruby shoots assassin Lee Harvey Oswald in this famous news photo.The area taken up by Ruby: the area taken up by Oswald = 1.618

FIBONACCI FALSITIES?There are just as many sources that say that finding Fibonacci and the Golden Ratio “EVERYWHERE” is garbage.

Google “Fibonacci Skeptics” to find much discourse on the subject.

FIBONACCI NOTATION You may see

Fibonacci numbers written as Fn

F1 = 1

F2 = 1

F3 = 2

F4 = 3

F5 = 5

F6 = 8

etc…

What is F10? 55

Recursive Definition of Fibonacci Numbers:

FN = FN-1 + FN-2

EXPLICIT DEFINITION OF FIBONACCI NUMBERS:Euler improved another mathematician’s theorem to show that:

1010

10

1 5 1 52 2

5F

1010

10

1.618 .618

2.236F

10

122.966 .008

2.236F

10 54.990F

Not a bad estimate for 55!You don’t have to know the 8th and 9th Fibonacci numbers to find it!

http://www.youtube.com/watch?v=kkGeOWYOFoA