The Fibonacci Sequence

A statue of Fibonacci in Pisa

1. A Little History First
Leonardo of Pisa (c1170 - 1250), known as Fibonacci, "the son of Bonacci," wrote a book called Liber Abaci . In this book he introduced the Arabic numbers into Western Culture. The book also posed a now well-known story problem which involved the reproductive habits of rabbits in an enclosed area:

A man put a pair of rabbits in a cage. During the first month the rabbits produced no offspring, but each month thereafter produced one new pair of rabbits. If each new pair thus produced reproduces in the same manner, how many pairs of rabbits will there be at the end of one year?

The resulting answer, though only theoretically correct, became known as "The Fibonacci Sequence." Fibonacci is noted as a "brilliant mathematician" but it seems unlikely that he could have ever imagined how prevalent his series would prove itself to be in our world. The Fibonacci sequence shows itself in innumerable situations, both natural as well as technological. Situations that Fibonacci could have never imagined. Computers and the supercooled liquid that creates plate glass, things that were a bit before Fibonacci's time. There are even firms that teach stock market strategies through the Fibonacci prinicples of Fibonacci math.

2. Natural Fibonacci
The Fibonacci sequence shows up in innumerable places in nature. Their occurrence is so frequent that it is difficult to know where to begin the descriptions. Since Fibonacci's original problem involving rabbits doesn't really work out (rabbits don't breed quite the way he imagined) does that mean that we can completely disregard his knowledge in small animal husbandry? The answer is "no." Fibonacci seemed to know what those rabbits and even bees were up to when they thought nobody was watching! It turns out that the numerical sequence described by the rabbit problem, is exactly the sequence of a bee's genealogy.
A male bee develops from an unfertilized egg, so it has only one parent. A female bee, on the other hand, develops from a fertilized egg, so she has two parents. Looking at the "Bee Family Tree" you will see the Fibonacci Sequence occurring in the male, female and total ancestors lines. The family tree below begins tracing the family tree with a female bee.

The number of petals on flowers as well as leaves on plants often occur as the Fibonacci Sequence. Phyllotaxis, the arrangement of leaves on a stem and in relation to each other, is not an absolute in all cases. It is definite in some plants (like iris and buttercups) and a strong likelihood in others. Here are a few examples:

3 - Lily, iris 5 - Buttercup, wild rose, larkspur 8 - Delphiniums, cosmos 13 - Ragwort, corn marigold, mayweed 21 - Aster, black-eyed susan 34 - Plantain, pyrethrums

The pattern of leaves, as they spiral up a stem, quite often occurs in the Fibonacci sequence. If the leaves of any plant grew in straight lines - directly above and below each other - then any leaf, or branch that was not at the top would have it's sunlight obscured by the leaves or branches above it. Expressed in the language of math, leaves do not occur 1/1, or every full circular rotation, but more often in phylloctactic ratios. Also called Fibonacci ratios, as their numerator and denominator are both generally Fibonacci numbers. Some example are:

2/3 - Grasses, elm
1/3 - Blackberry, hazel, fiddleneck
2/5 - Apple, plum, cherry, apricot
3/8 - Weeping willow, pear
5/13 - Bottlebrush, almond
One other place that plant life reflects the Fibonacci sequence is in the seed heads of numerous plants. The spiral pattern of the seed heads in both the clockwise, and counterclockwise direction, are quite often Fibonacci numbers. It appears that the reason for this formation is to allow the seedheads to pack the maximum number of seed in the given area. Some final examples:
5/8 - Pinecone
8/13 - Pineapple (the "eyes")
21/34 - Sunflower
A picture might make it a bit easier.

3.Playing with the Numbers
One of the more interesting things about the Fibonacci sequence is the interplay between the numbers; how they interrelate and their "behaviors." Some of the interactions are simple fun tricks, such as those in the children's book where I originally discovered the sequence to more complex paradoxes.
A few examples:

Divisible by 11
The sum of any ten consecutive Fibonacci numbers is always evenly divisible by 11.

More divisibility
Here's another trick. As the consecutive integers (Fn) increase, note how they are divisible by consecutive Fibonacci numbers:
Every 3rd Fibonacci number is divisible by 2.
Every 4th Fibonacci number is divisible by 3.
Every 5th Fibonacci number is divisible by 5.
Every 6th Fibonacci number is divisible by 8.
Every 7th Fibonacci number is divisible by 13.
Every 8th Fibonacci number is divisible by 21.

Factors of Fibonacci
Another interesting characteristic of the Fibonacci sequence is that no two consecutive Fibonacci numbers have any common factors. Like this:

* The 12th Fibonacci number (144) is the square of 12.
It is also the only square number in the entire sequence
well, as long as you don't count the number 1.

4. Fibonacci goes Gold
The Golden Ratio
1.618033988749894842... The Golden Ratio is a "special" number that goes on forever, like Pi, without ever repeating. It is often symbolized as the greek letter Phi.

The Golden Rectangle
The golden Rectangle is believed to be the most aesthetically pleasing proportion. It is a rectangle in which the length to width ratio is the golden ratio. If the Length of the rectangle is equal to Width x1.62 (or Phi-the Golden Ratio) then it is a golden rectangle. Should I try that agian another way? Trudi Hammel Garland, in Fascinating Fibonaccis explains it this way:

If the small part is called S and the large part is called L, the proportions can be mathematically stated as follows:

S/L = L/S+L
That is about as far as my mathematics descriptions will take me, but if you need more, or a precise visual, then here's the place to go for some "Golden Geometry"

Fibonacci Squares
A Fibonacci square will work a bit differently, but with a similar result. If you begin with a square that is one by one (unit, cenitmeter, inch...no matter) and add another that is the same size you create a rectangle. If you continue adding squares whose sides are the length of the rectangle - that longer side will always be a Fibonacci number. If you keep adding squares, you will end up with a rectangle that gets closer and closer to the "Golden Rectangle".

This is my aproximation that bares only a symbolic, but nowhere near technically correct, representation of the above described process.

Golden Triangles
In Fascinating Fibonaccis by Trudi Hammel Garland, Golden Triangles are explained thusly:
It is an isosceleles triangle with one short side in golden proportion to each of the two longer, equal sides. Fibonacci numbers can be used to construct such triangles... Among the interesting properties of the golden triangle is the fact that the bisector of a base angle (which is always 72o) cuts the side opposite it into the golden proportions. That bisector also cuts the triangle into two new isosceles triangles...whose areas are in golden proportion to each other. The process can be repeated endlessly.

Again, I have done my best to demonstrate this process, it is not a prefect representation:

Aesthetically Pleasing Fibonacci?

5. Architecture
Is there something about the Fibonacci sequence that appeals to people, that makes it pleasing to us, even if we are unaware of it's existence? Everything I have read so far indicates that there is. And there is quite a bit of evidence. The golden rectangle seems to appear throughout history in architecture. From the Parthenon, the United Nations Building and even the burial chamber of Ramses IV, the golden rectangle and ratio are there.

6. Art
It would be difficult for most people to articulate exactly what it is that they find pleasing in art and other things of beauty. Proportion would be an easy place to begin when attempting that description. This was especially true of the Greek whose art including urns, vases, statues and building frequently reveals the golden ratio and rectangle. There are numerous examples of art in many different forms and cultures where the golden ratio or golden rectangle appears. In less than scientific term, it is a work where the focal point appears to fit of fill the shape and proportions of the golden rectangle.

7. Music at the Fibonacci series

The Fibonacci series appears in the foundation of aspects of art, beauty and life. Even music has a foundation in the series, as: 13 notes separate each octave of 8 notes in a scale, of which the 5th and 3rd notes create the basic foundation of all chords, and are based on whole tone which is 2 steps from the root tone, that is the 1st note of the scale.

Note too how the piano keyboard scale of 13 keys has 8 white keys and 5 black keys, split into groups of 3 and 2.

8. The Formulas
The Golden Ratio

.Where Phi has the value 1.618..

Fibonacci Sequence
But you have to know the previous number!
F(1) = 1

F (2)= 1

F (n) = F (n-1) + F (n-2) for n > 2

And for those not faint of heart

.Where a and b are the roots of the quadratic equation

Calculating the Golden Ratio/Fibonnacci Sequence

9. History of Phi

.While the proportion known as the Golden Mean has always existed in mathematics and in the physical universe, it is unknown exactly when it was first discovered and applied by mankind. It is reasonable to assume that it has perhaps been discovered and rediscovered throughout history, which explains why it goes under several names.

Uses in architecture date to the ancient Egyptians and Greeks
The Egytians used both pi and phi in the design of the Great Pyramids. The Greeks, who called it the Golden Section, based the entire design of the Parthenon on this proportion.
Phidias (500 BC - 432 BC), a Greek sculptor and mathematician, studied phi and applied it to the design of sculptures for the Parthenon.

Plato (circa 428 BC - 347 BC), in his views on natural science and cosmology presented in his "Timaeus," considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos.

Euclid (365 BC - 300 BC), in "Elements," referred to dividing a line at the 0.6180399... point as dividing a line in the extreme and mean ratio. This later gave rise to the use of the term mean in the golden mean. He also linked this number to the construction of a pentagram.

The Fibonacci Series was discovered around 1200 AD
Leonardo Fibonacci, an Italian born in 1175 AD (2) discovered the unusual properties of the numerical series that now bears his name, but it's not certain that he even realized its connection to phi and the Golden Mean. His most notable contribution to mathematics was a work known as Liber Abaci, which became a pivotal influence in adoption by the Europeans of the Arabic decimal system of counting over Roman numerals. (3)

It was first called the "Divine Proportion" in the 1500's
Da Vinci provided illustrations for a dissertation published by Luca Pacioli in 1509 entitled "De Divina Proportione" (1), perhaps the earliest reference in literature to another of its names, the "Divine Proportion." This book contains drawings made by Leonardo da Vinci of the five Platonic solids. It was probably da Vinci who first called it the "sectio aurea," which is Latin for golden section.

The Renaissance artists used the Golden Mean extensively in their paintings and sculptures to achieve balance and beauty. Leonardo Da Vinci, for instance, used it to define all the fundamental proportions of his painting of "The Last Supper," from the dimensions of the table at which Christ and the disciples sat to the proportions of the walls and windows in the background.

Johannes Kepler (1571-1630), discoverer of the elliptical nature of the orbits of the planets around the sun, also made mention of the "Divine Proportion," saying this about it:
"Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel."

The term "Phi" was not used until the 1900's
It wasn't until the 1900's that American mathematician Mark Barr used the Greek letter phi to designate this proportion. By this time this ubiquitous proportion was known as the golden mean, golden section and golden ratio as well as the Divine proportion. Phi is the first letter of Phidias (1), who used the golden ratio in his sculptures, as well as the Greek equivalent to the letter "F," the first letter of Fibonacci. The character for phi, however, also has some interesting theological implications.

Recent appearances of Phi in math and physics
Phi continues to open new doors in our understanding of life and the universe. It appeared in Roger Penrose's discovery in the 1970's of "Penrose Tiles," which first allowed surfaces to be tiled in five-fold symmetry. It appeared again in the 1980's in quasi-crystals, a newly discovered form of matter.

Phi as a door to understanding life
The description of this proportion as Golden and Divine is fitting perhaps because it is seen by many to open the door to a deeper understanding of beauty and spirituality in life. That's an incredible role for a single number to play, but then again this one number has played an incredible role in human history and in the universe at large.

10. Theology

Is there meaning hidden in , the symbol for the Golden Number?
The use of the Greek letter Phi to represent the golden number 1.618... is generally said to acknowledge Phidias, a 5th century B.C. sculptor and mathematician of ancient Greece, who studied phi and created sculptures for the Parthenon and Olympus.

The message from scripture of all the major monotheistic religions is that God is One, Who created the universe from nothing, splitting nothingness into offsetting forces and elements. Today we understand the universe to consist of positive and negative atomic and subatomic particles and charges, matter and anti-matter, all coming from a singularity in what we term the "Big Bang."

Curiously, the mathematical constant of 1.618... that is found throughout creation is represented by the symbol F, which is the symbol 0 for nothing split in two by the symbol 1 for unity and one. Could this be the true meaning behind the symbol ?

Nothing	Unity / God	Nothing split by Unity is Phi, the constant of creation

 

Adding Unity to nothingness produces the Fibonacci series, which converges on
Now ADD God to the void, or Unity to Nothing. In other words, add 0 plus 1 to get 1, and then follow this pattern to the Infinite. This is the Fibonacci series. The ratio of each number in the series to the one before it converges on as you move towards infinity, !

Number in the series	0	1	1	2	3	5	8	13	...
Ratio of each number in the series to the previous number in the series	.		1	2	1.5	1.66...	1.6000	1.625	...

The Golden Proportion is analogous to God's relationship to creation
The Golden Section, or Phi, found throughout nature, also applies in understanding the relationship of God to Creation. In the golden section, we see that there is only one way to divide a line so that its parts are in proportion to, or in the image of, the whole:
The ratio of the larger section (B) to the whole line (A) is the same as the ratio as the smaller section (C) to the large section (B):

Only "tri-viding" the whole preserves the relationship to the whole
And so it is with our understanding of God, that we are created in His image. Not by dividing the whole, but only by tri-viding the whole does each piece retain its unique relationship to the whole. Only here do we see three that are two that are one.

The Golden Section as a universal constant of design
The teachings of most religions express the thought that part of God is within each of us and that we are created in His image. The pervasive appearance of phi throughout life and the universe is believed by some to be the signature of God, a universal constant of design used to assure the beauty and unity of His creation.

11. Phi and the Solar System

Certain solar system orbital periods are related to ph

Certain planets of our solar system seem to exhibit a relationship to phi, as shown by the following table of the time it takes to orbit around the Sun:

Saturn's rings are divided at two phi points

The Cassini division in the rings of Saturn falls at the Golden Section of the width of the ring.

A closer look at Saturn's rings reveals a darker inner ring which exhibits the same golden section proportion as the brighter outer ring.

Venus and Earth reveal a phi relationship
Venus and the Earth are linked in an unusual relationship involving phi. Start by letting Mercury represent the basic unit of orbital distance and period in the solar system:

Curiously enough we find:
Period of Venus * Phi = Distance of the Earth

In addition, Venus orbits the Sun in 224.695 days while Earth orbits the Sun in 365.242 days, creating a ratio of 8/13 (both Fibonacci numbers) or 0.615 (roughly phi.) Thus 5 conjunctions of Earth and Venus occur every 8 orbits of the Earth around the Sun and every 13 orbits of Venus.

Mercury, on the other hand, orbits the Sun in 87.968 Earth days, creating a conjunction with the Earth every 115.88 days. Thus there are 365.24/115.88 conjunctions in a year, or 22 conjunctions in 7 years, which is very close to Pi!

Relative planetary distances average to Phi
The average of the mean orbital distances of each successive planet in relation to the one before it approximates phi:

12. "89" and the Fibonacci Series

The reciprocal of 89, a Fibonacci number, is based on the Fibonacci series
This is a little curiousity involving the number 89, one of the Fibonacci series numbers.

If you take each Fibonacci number, divide it by 10 raised to the power of its position in the Fibonacci sequence and add them all together, you get 0.011235955..., the same number as the reciprocal of 89.

Note: You can see the beginning of the Fibonacci sequence in the first 6 digits of the decimal equivalent of 1/89. (i.e., 0,1,1,2,3,5 appears as 0.011235..)

13. Fibonacci numbers define the movements of stocks in Elliott Wave Theory

Fibonacci numbers were used by W.D Gann and R.N. Elliott, pioneers in technical analysis of the stock market. In Elliott Wave Theory, all major market moves are described by a five-wave series. The classic Elliott Wave series consists of an initial wave up, a second wave down (often retracing 61.8% of the initial move up), then the third wave (usually the largest) up again, then another retracement, and finally the fifth wave, which would exhaust the movement. In addition, each of the major waves (1, 3, and 5) could themselves be separated into subwaves, and so on, and exhibit other Fibonacci relationships.

A sample stock price wave analysis could look something like this:

Major, minor and sub waves are shown in RED, YELLOW and GREEN and the total number of increases and decreases (2, 5 or 8) is a Fibonacci number. Note too that the predicted end result is based in the Fibonacci series as well as the end price is 61.8% of the high and 0.618 is equal 1/ and 0.382 is 1/ 2.

14. What is Phi?

Phi ( = 1.618033988749895... ) is simply an irrational number like pi ( = 3.14159265358979... ), but one with many unusual mathematical properties.

Phi is the basis for the Golden Section, Ratio or Mean
The ratio, or proportion, determined by Phi (1.618...) was known to the Greeks as the "Golden Section" and to Renaissance artists as the "Divine Proportion" It is also called the Golden Ratio and the Golden Mean.

Phi, like Pi, is a ratio defined by a geometric construction
Just as pi ( is the ratio of the circumference of a circle to its diameter, phi () is simply the ratio of the line segments that result when a line is divided in one very special and unique way.

Divide a line so that:

the ratio of the length of the entire line (A)
to the length of larger line segment (B)
is the same as
the ratio of the length of the larger line segment (B)
to the length of the smaller line segment (C).

This happens only at the point where:
A is 1.618... times B and B is 1.618... times C.
Alternatively, C is 0.618... of B and B is 0.618... of A.

Phi with an upper case "P" is 1.618033987..., while phi with a lower case "p" is 0.6180339887, the reciprocal of Phi and also Phi minus 1.

.What makes phi even more unusual is that it can be derived in many ways and shows up in relationships throughout the universe.

Phi can be derived through:

Phi appears in: