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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 inter-relationship!

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:

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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 inter-relationship….

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:

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Figure 177: Draw a line from point E (which is the mid-point of line C-D) to one of the opposite corners of the square.

As the triangle BDE is also a right-angled 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 C-D. This makes E-G equal to the square root of 5 (the square root of 5 = 2.236 units in length).

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Figure 178: E-G 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 five-pointed star as the "symbol of his order". Each segment of the five-pointed 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 five-sided star contains lines that are related by Fibonacci numbers 5, 8 and 13.

Figure 180: Expanding on the five-pointed 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 8-note 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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