Post on 24-Feb-2016
description
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 5 3
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 = 1F2 = 1F3 = 2F4 = 3F5 = 5F6 = 8etc…
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 .6182.236
F
10122.966 .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!