Mathematical

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Mathematical Connections
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Table of Contents
The Fibonacci Series
The Golden Section
The Relationship Between Fibonacci and The Golden
Section
References
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The Fibonacci Sequence
The greatest European Mathematician of all
ages.
Leonardo of Pisa of Fibonacci Born 1175 AD
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The Fibonacci Sequence
He gave us our 10 digit number system! He
recognized a series of numbers that often occur
in nature. These are now called the Fibonacci
numbers. The series starts with 0 1. All
following numbers are the sum of the 2 previous
numbers!
Create A Fibonacci Seq.
Leonardo of Pisa or Fibonacci Born 1175 AD
Next
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Create A Fibonacci Series
Remember, just add the 2 previous numbers
together to get the next number... Here's how it
starts... 0,1,1,2,3,5..... So, click on the
button that has the correct choice for the next
several numbers in the Fibonacci series!
8,13
7,15
9,17
Back
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The Fibonacci Spiral
This computer drawing was created using Fibonacci
Numbers. This is called a Fibonacci Spiral
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The Fibonacci Spiral
Remember , the series is 0,1,1,2,3,5,8,13,21.....
Place squares with those lengths for each side as
shown in the drawing ( excluding 0). The two
squares where each side has a length of 1 are in
the center. Find them and click on them!
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The Fibonacci Spiral
On top of the two squares with sides of length 1
is a square where each side has a length of 2.
To the right of those three squares is a square
with a side of length 3. Continue with this,
then draw a quarter circle in each square as
shown!
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The Fibonacci Spiral
In this drawing, side E will have a length of 3
2 5. See if you can figure out what the length
of the sides marked F, G and H are!
Side F is 8
7 9 Side G is
10 13 11
Side H is 15 18
21
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The Fibonacci Spiral
Example of the Fibonacci Spiral in Nature....
Click to see the relationship!
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The Fibonacci Spiral
Example of the Fibonacci Spiral in Nature....
Can you see the spiral in this pine cone? Many
seeds and seed heads have the Fibonacci Spiral in
them! Click to see the relationship!
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The Fibonacci Spiral
Example of the Fibonacci Spiral in Nature....
The Fibonacci Spiral can be seen in the seed head
of a sunflower seed too! Can you see it? Click
to see the relationship! This concludes the
Fibonacci portion... care to explore the Golden
Section?
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The Golden Section
The Golden Section is defined as the ratio most
pleasing to the eye.
M
B
A
Here line AB is divided at point M. The ratio of
AB to MB is the same as the ratio of AM to MB.
This means the line is divided into golden
sections!
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The Golden Section
This ratio is the same as the ratio between 1 and
the number phi (1.6180339887). By constructing a
rectangle where the sides have the golden ratio,
we create a golden rectangle!
Most people would select the middle one, the
"golden rectangle, as most pleasing to the eye.
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The Golden Section
Leonardo da Vinci 1452-1519. Italian painter,
engineer, musician, and scientist. The most
versatile genius of the Renaissance, Leonardo
used the Golden Rectangle in his paintings.
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The Golden Section
Leonardo's famous Mona Lisa reflects the artists
use of the Golden Section. The rectangle
around her face represents a Golden rectangle.
If you subdivide the rectangle at the eyes the
vertical side of the rectangle is divided by the
golden ratio. Click To See!
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The Golden Section
This painting by George Seurat, "La Parade" was
designed by dividing the canvas into the Golden
Section. Click to see the artist's planning
sketch!
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The Golden Section
Even in the time of the ancient Greeks, the
golden rectangle was considered a pleasing
shape. It appears in many of the proportions of
the Parthenon in Athens, Greece.
Continue on to explore the relationship between
the Fibonacci Numbers and the Golden Section.
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The Golden Section Fibonacci Numbers
Remember the spiral we created using the
Fibonacci Numbers? Look closely at the rectangle
made by any adjacent squares.
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The Golden Section Fibonacci Numbers
By doing the math we can see that the larger our
rectangles get, the closer to the perfect golden
section the proportions are!
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The Golden Section Fibonacci Numbers
The sides on the 6th and 7th squares are lengths
13 21. 21 / 13 1.61538... and this result
gets ever closer to the ratio between 1 and the
number ? as the rectangles grow!
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The Golden Section Fibonacci Numbers
The End!!
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References
http//www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
acci/fibonat.html http//www.mos.org/sln/Leonardo
http//www.atrsnet.getty.edu http//www.criba.edu.
ar/girasol http//www.rit.edu/sxa7544/structural.
htm http//share3.esdros.wednet.edu/2/lbohl/golden
sppiral.html http//pass.math.org..uk/issue3/fibon
acci/alt.html http//mongo.tvi.cc.nm.us/art_math/g
olden.html http//sunsite.unc.edu/wm/paint/auth/vi
nci http//familiar.sph.umich.edu/cjackson/s http
//www.notam.uio.no/oyvindha/loga.html http//www.
summum.org/phi.htm