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Fibonacci Number Sequence
Leonardo Fibonacci of Pisa was a thirteenth century mathematician who, in my opinion,
really should be more famous than Pythagorous, Leonardo de Vinci... or even Britney Spears!
For starters, Fibonacci introduced the decimal system to Europe in his Book of
Calculation (Liber Abacci). Prior to that, everyone had a very difficult time
using the Greek and Roman values of I, V, X, L, C, D and M in mathematics.
What a momentous breakthrough that was!
The new system was the leading mathematical discovery since the fall of Rome 700 years
earlier and laid the foundation for great developments in higher mathematics and physics,
astronomy and engineering.
In his day, Fibonacci was very famous: Frederick II, the Emperor of the Holy Roman
Empire, the King of Sicily and Jerusalem, and descendant of two of the noblest families in
Europe, travelled to Pisa in Italy to meet Fibonacci in 1225 AD. Here, Fibonacci solved
many mathematical problems in front of the Prince, many of which are shown in the revised
version of his Book of Calculation (1228 AD).
Now, though, the only monuments to Fibonacci are a small statue across the river from
the Leaning Tower of Pisa (of which Fibonacci helped to design) and two street names that
bare his name: one in Florence and one in Pisa.
The Fibonacci Sequence of Numbers
Fibonacci’s other major discovery was actually a rediscovery that he made on his
travels to Asia. But this set of numbers is the one that now bears his name: The Fibonacci
Sequence.
In Fibonacci’s Book of Calculation, a problem is given on solving the reproductory
numbers of two rabbits: How many rabbits would there be after one year if the rabbits, and
each new pair of rabbits, gave birth to a new pair every month?
Each pair needs two month’s time to mature, but once in production, each pair
conceives a new pair every month.
The sequence begins with 1, 1 for the first two months because the first pair of
rabbits need time to mature, and this staggering in maturity can be seen below in Figure
174. There are 2 pairs of rabbits at the start of the third month; 3 pairs at the start of
the fourth month; 5 pairs at the start of the fifth month; 8 pairs at the start of the
sixth month; 13 in the seventh month; 21 in the eighth month; 34 in the ninth month, etc.
After 100 months, the numbers are huge with 354,224,848,179,261,915,075 pairs of
rabbits!
Figure 174: Table illustrating the production of rabbits from Fibonacci’s book.
The number sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. By the eighth month, there are
34 pairs of rabbits.
The Fibonacci sequence of numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
etc.) resulting from the rabbit problem posed in the book have many interesting and UNIQUE
properties.
For example, the sum of the two adjacent numbers in the sequence forms the
next higher number in the sequence:
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
13 + 21 = 34
21 + 34 = 55
34 + 55 = 89
55 + 89 = 144 … and so on to infinity.
After the first few numbers in the sequence, the ratio of any number to the next
number higher is approximately 0.618 to 1…
And the ratio of any number to the next lower number is approximately 1.618 to 1.
(The higher one goes in the sequence, the closer the ratio becomes to this ideal
ratio.)
This set of numbers is unique in having this interrelationship!

1 
2 
3 
5 
8 
13 
21 
34 
55 
89 
144 
1 
1 
2 
3 
5 
8 
13 
21 
34 
55 
89 
144 
2 
0.5 
1 
1.5 
2.5 
4 
6.5 
10.5 
17 
27.5 
44.5 
72 
3 
0.333 
0.666 
1 
1.667 
2.667 
4.333 
8 
11.3 
18.33 
29.67 
48 
5 
0.2 
0.4 
0.6 
1 
1.6 
2.6 
4.2 
6.8 
11 
17.8 
28.8 
8 
0.125 
0.25 
0.375 
0.625 
1 
1.625 
2.625 
4.625 
6.875 
11.13 
18 
13 
0.077 
0.154 
0.231 
0.385 
0.615 
1 
1.615 
2.615 
4.231 
6.846 
11.07 
21 
0.048 
0.095 
0.143 
0.238 
0.381 
0.619 
1 
1.619 
2.619 
4.238 
6.857 
34 
0.029 
0.059 
0.088 
0.147 
0.235 
0.382 
0.618 
1 
1.618 
2.618 
4.235 
55 
0.018 
0.036 
0.055 
0.091 
0.145 
0.236 
0.382 
0.618 
1 
1.618 
2.618 
89 
0.011 
0.022 
0.034 
0.056 
0.090 
0.146 
0.236 
0.382 
0.618 
1 
1.618 
144 
0.007 
0.014 
0.021 
0.035 
0.056 
0.090 
0.146 
0.236 
0.382 
0.618 
1 
Figure 175: Table shows the major unique relationship between the adjacent numbers in
the Fibonacci sequence: After the first few numbers, they have the ratio of 0.618 to 1 or
1.618 to 1.
Have you ever heard of an irrational number – a number that has no end? You are
almost certainly familiar with Pi? Remember this equation from your school days:
2 pi r
Where the circumference of a circle is 2 x Pi x the radius of the circle. Pi,
you will recall is also an irrational number that has no end: 3.147…. to infinity.
The ratio between the Fibonacci sequence of numbers is known by the Latin name of Phi.
It is also an irrational number with no ending: 0.618034… to infinity. Phi
is also known as the Golden Ratio.
Other Consistent Fibonacci Relationships
Between alternate numbers in the Fibonacci sequence, the ratio is
approximately 0.382 to 1.
(Note: 1 – 0.618 = 0.382.)
The inverse of which is 2.618 to 1.
(Note: 2.618 – 1.618 = 1.)
Phi is the only number that when added to 1 produces its inverse.
For example,
0.618 + 1 = 1.618.
Other relationships can be found from multiplying the number by itself (squaring,
cubing, etc.).
For example:
Alternatively, using the inverse:
Some other interrelated properties include:
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Besides the numbers 1 and 2, any Fibonacci number multiplied by 4, when added to a
selected Fibonacci number, gives another Fibonacci number:
ILLUSTRATION ONLY AVAILABLE TO COURSE PURCHASERS
This sequence of numbers illustrates the ratio between the second Fibonacci numbers
away from the Golden Mean on Figure 175; namely those of 4.236 and .236.
How uncanny is all of this?
Remember that this set of numbers (in the Fibonacci sequence) is the only set of
numbers that contain this special interrelationship….
Spooky!
The Golden Section
1.618 or 0.618 is known as the Golden Ratio, Golden Section, Divine Section or the
Golden Mean.
Any length can be divided into the Phi ratio of 0.618.
For example:
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Figure 176: A line of 1 unit long, divided into the Golden Section of 0.618.
The Golden Rectangle
The sides of a Golden Rectangle are in the Phi proportion of 1.618 to 1. To
construct a Golden Rectangle start by drawing a square of 2 units by 2 units, then drawing
a line from the midpoint of one side of the square to one of the corners on the opposite
side of the square.
For example:
ILLUSTRATION ONLY AVAILABLE TO COURSE PURCHASERS
Figure 177: Draw a line from point E (which is the midpoint of line CD) to one of the
opposite corners of the square.
As the triangle BDE is also a rightangled triangle, the square of the
hypotenuse (X) is equal to the sum of the squares of the other two sides: X squared = 2
squared + 1 squared; or X squared = 5 squared = 2.236 units. (Remember your Pythagorous
lessons?)
Measure this length from point E and B and then measure this from point E to extend
line CD. This makes EG equal to the square root of 5 (the square root of 5 = 2.236 units
in length).
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Figure 178: EG is equal in length to X. As X is the hypotenuse of a right angled
triangle, then using Pythagorous, X squared = 1 squared + 2 squared = 5 squared. 5 squared
= 2.236 which is a Fibonacci ratio.
The completed rectangle is in the proportion of the Golden Ratio. Both rectangles
AFGC and BFGD are Golden Rectangles!
Appearances of the Golden Ratio in History
The magic relationship of Phi was noted thousands of years ago by Plato, who
considered Phi and the Golden Section to be "the most binding of all
mathematical relations that is the key to the physics of the cosmos."
The ancient Egyptians recorded their knowledge of Phi thousands of years ago and
used the 0.618 relationship in their architecture, including the pyramids, giving the
sloping faces a slope height of 1.618 times half the base. The Egyptian scientists went so
far as to use Phi and Pi to square the circle and cube the square making
them of equal area and volume – a feat which was not replicated for over 4000 years.
They considered Phi not as just a number but as "a symbol of creative function
or a reproduction of an endless series. To them it represented the rational order of the
universe, an imminent natural law, a life giving force behind all things, the universal
structure governing and permeating the world."
Fibonacci Relationships in Geometry
Pythagorous used the fivepointed star as the "symbol of his order". Each
segment of the fivepointed star is in a Golden Ratio to the next smaller segment.
Joining the points of a pentagram (to make a pentagon) illustrates Fibonacci
relationships between the lines. Also note that both these shapes are constructed around
nature’s most natural shape – a circle. (Remember constructing a pentagram in
your geometry lessons?)
Figure 179: A fivesided star contains lines that are related by Fibonacci numbers 5, 8
and 13.
Figure 180: Expanding on the fivepointed star with each star having many Fibonacci
relationships to each other.
Some more interesting geometric Fibonacci and Phi relationships can be found at:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html
Other famous "fans" of Fibonacci were mathematician Jacob Bernoulli, who had
the Golden Spiral etched into his headstone, and Isaac Newton who had the Golden Spiral
engraved in the headboard of his bed. Any fellow Brits out there who have watch
the BBC2 program, QI (Quite Interesting) with Stephen Fry may have noticed the
Golden Spiral on the set behind the contestants. See figure 182 below.
The Golden Section in Art
Using the Golden Ratio has enhanced many works of art. During the Renaissance, Leonardo
da Vinci – aware of the proportions of the Golden Section – used them to enhance
his paintings’ appeal. He said, "If a thing does not have the right look, it
does not work."
For this reason, many paintings often use a rectangular canvas with a Golden Ratio
because it has a better look than a different shaped canvas such as a square. For example,
a portrait picture will often be approximately 1.618 times as high as it is wide or a
landscape picture will be 1.618 times as long as it is high.
In music, the scale is based on an 8note octave. The piano keyboard has 8 white keys
and 5 black keys making 13 keys in total.
As a guitar player, I regularly play bar chords that contain the 1^{st}, 3^{rd}
and 5^{th} notes of the scale (1, 3, 5 Fibonacci numbers) as they create the
sweetest tonality. Triad chords make up a vast majority of popular tunes and they are made
with the 1^{st}, 3^{rd} and 5^{th} notes of a scale. For example,
a D major chord contains the notes A, D and F sharp – the 5^{th}, 1^{st}
and 3^{rd} notes of the D major scale.
The cochlea of the inner ear is shaped in a Fibonacci logarithmic spiral, which
explains why it sounds so good to us.
In architecture and design, buildings, windows, picture frames, books and
cemetery crosses often illustrate signs of the Golden Ratio.......
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