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Section 5-5
The Fibonacci Sequence and the Golden
Ratio
Lesson Objectives
• Work with the Fibonacci sequence.
• Understand the golden ratio.
• See relationships between the Fibonacci
sequence and the golden ratio.
• See the golden ratio as a ratio on a line segment.
Who Was Fibonacci?
~ Born in Pisa, Italy in 1175 AD
~ Full name was Leonardo Pisano
~ Grew up with a North African education under the Moors
~ Traveled extensively around the Mediterranean coast
~ Met with many merchants and learned their systems of
arithmetic
~ Introduced the Hindu-Arabic number system into Europe
The Fibonacci Sequence
The solution of the rabbit problem leads to the
Fibonacci sequence. Here are the first thirteen
terms of the sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233
Notice the pattern. After the first two terms (both
1), each term is obtained by adding the two
previous terms.
Recursive Formula for Fibonacci Sequence
If Fn represents the Fibonacci number in the
nth position in the sequence, then
F1  1
F2  1
Fn  Fn2  Fn1 , for n  3.
The Fibonacci Numbers in Nature
Lilies and irises = 3 petals
Corn marigolds = 13 petals
Buttercups and wild roses = 5 petals
Black-eyed Susan’s = 21 petals
The Fibonacci Numbers in Nature
• The Fibonacci numbers can be found in pineapples
and bananas
• Bananas have 3 or 5 flat sides
• Pineapple scales have Fibonacci spirals in sets of 8,
13, 21
The Golden Ratio: Ratio of Fibonacci
Consider the quotients of successive Fibonacci
numbers and notice a pattern.
1
2
3
5
 1,
 2,
 1.5,
 1.66...,
1
1
2
3
8
13
21
 1.6,
 1.625,
 1.615384
5
8
13
These quotients seem to go toward 1.618.
1 5
In fact, they approach
.
2
This number is known as the golden ratio.
The Golden Ratio: Ratio of Fibonacci
The Golden Ratio: Ratio of Line Segment
Let b=1. Solve for a.
Fibonacci and Golden Ratio
• Take your graph paper and draw a 1 x 1 square in
the middle of the page.
• Draw another 1 x 1 square to its right.
• Draw a 2 x 2 square adjacent to and below the
rectangle.
• Draw a 3 x 3 square adjacent to and to the left of
that rectangle.
• Continue drawing squares in a clockwise direction
until your paper runs out of space
Fibonacci and Golden Ratio
Your drawing should look like this:
Constructing Golden Rectangles
The first three rectangles you should measure are
highlighted in yellow below.
The Golden Spiral
How is the Golden Spiral constructed?
Example of Spiral in Nature:
Shell of Chambered Nautilus
Golden Rectangles Are All Around Us!
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The aspect ratio of YouTube videos
Ratio of length over width of photographs
Computer screens
Smartphone screens
Tablets
Credit Cards
Playing Cards
Human Body
Examples of the Golden Ratio
• On the next pages you will see examples of
the Golden Ratio c
• Many of them have a gauge, called the
Golden Mean Gauge, superimposed over the
picture.
• This gauge was developed by Dr. Eddy Levin
DDS, for use in dentistry and is now used as
the standard for the dental profession.
Golden Mean Gauge: Invented by Dr. Eddy Levin DDS
The Bagdad City Gate
Dome of
St. Paul:
London,
England
The Great Wall of China
Windson Castle
……and more!!!