Leonardo Fibonacci (1175-1250) and Fibonacci Numbers

Leonardo Fibonacci was born about 1175 in Pisa, Italy. He was known also as Leonard da Pisa. That was the great time for Western civilization. The Crusades awakened the Europe people and brought them in contact with the more advanced intellect of the East. Universities of Naples, Padua, Paris, Oxford, and Cambridge were established. The long struggle between the Papacy and the Empire was culminated, Commerce was flourishing in the Mediterranean world and adventurous travelers such as Marco Polo were penetrating far beyond the known world of Europeans. Leonard Fibonacci went to Algeria when he was about 20, where he began to learn Indian numerals and Arabic calculating methods. Fibonacci used his experience to improve on the commercial computing techniques he knew and to extend the work of classical mathematical writers, such as Greek Mathematician Diophantus and Euclid. His writings on recreational mathematics series, such as Fibonacci sequence.

Leonard wrote an arithmetic and algebra book, Liber Abaci, in 1202 introducing Europe to Arabic numerals from North Africa and the ZERO from India, making calculation much easier than with Roman numerals. He advocated the use of the Hindu-Arabic decimal system, the one we use today, over the Roman numerical system used in Europe in his time. Multiplication and division were so complicated using Roman numerals that it required a college degree to master these skills.

In the second edition of the book (1228), he gave the solution to the famous Rabbit Problem. A pair of rabbits is placed in a pen to find out how many offspring will be produced by this pair in one year, if each pair of rabbits gives birth to a new pairs of rabbits each month starting with the second month of its life; it is assured that deaths do not occur. He traced the progress and answered the question like the following table.

month / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12
pairs / 2 / 3 / 5 / 8 / 13 / 21 / 34 / 55 / 89 / 144 / 233 / 377

That gives the Fibonacci sequence.

  1. Recursive Formula

The Fibonacci sequence is generated by recursion. The Recursive Formula is given by

  1. Golden Ratio

Hence, solve the equation , we have a positive solution

Fibonacci numbers are used to speed binary searches by repeatedly dividing a set of data into groups in accordance with successfully smaller pairs of numbers in the Fibonacci sequence. For example, a data set of 34 items would be divided into one group of 21 and another of 13. If the item being sought in the group 13, then the group of 21 would be discard, the group of 13 would be divided into 5 and 8; then the search would continue until the item was located. The ratio of two consecutive terms in the Fibonacci sequence converges on the Golden Ratio.

  1. The Magic Number r in Geometry: Golden Cut and Stars


The side of an inscribed 10-regular polygon to a unit circle is the magic number . Actually, it is easy to see the value of can be found by using the fact . Apply double – angle formula for sine, and triple - angle formula for cosine, we have

Each side is divided by , then we have the following equation

Solve for a positive solution, we have

=2r

The following figure shows the process of the famous Golden Cut.

From Golden Cut, it is easy to draw an inscribed pentagon and a star to the unit circle.

  1. Curve of a Nautilus Shell (Fibonacci Spiral)

The Golden Spiral is a mystical shape that is an absolute in both abstract mathematics and chaotic nature. It was first discovered by Pythagoras, a failed Greek messiah and mathematical cult leader in the 5th century B.C. The spiral is derived via the golden rectangle, a unique rectangle which has the golden ratio. When squared, it leaves a smaller rectangle behind, which has the same golden ratio as the previous rectangle. The squaring can continue indefinitely with the same result. No other rectangle has this trait. When you connect a curve through the corners of these concentric rectangles, you have formed the golden spiral. The Pythagoreans loved this shape for they found it everywhere in nature: the Nautilus Shell, Ram's horns, milk in coffee, and the face of a Sunflower, your fingerprints, our DNA, and the shape of the Milky Way.


The Golden mean is a ratio, discovered by the Greeks, that is self-mirroring. It is approximately .6 to 1.

If you divide a straight line so that about 61% of it is on one side and 39% on the other, you will find that the ration of the large portion to the small is the same as the ratio of the overall line to the large portion. Rectangles made with these proportions can be subdivided endlessly. This self-mirroring proportion was essential to the art and architecture of the Greeks; it is very pleasing to the mind's eye and was used extensively, including in the design of the Parthenon.

In the 15th century an Italian named Fibonacci discovered that if you add 1 to itself, then 2 to 1, then the sum 3 to 2, and the sum 5 to 3, etc., you end up with a series of numbers 1, 1, 2, 3, 5, 8, 13,…,etc. The ratios of these, one to another, dance around and approach more and more closely the golden mean of .6 to 1.

These ratios describe the most efficient way of packaging spirals about themselves IN TWO DIMENSIONS; you will see them in the center of a sunflower. If you count the spirals going one way and they add up to13, there will be either 8 or 21 spirals going the other way. Moving your perspective in our out to about 2/3rds of the original size will move you to the next level of spirals.

I experimented with adding up more than two dimensions, that is, with the numbers the universe would use to construct 3, 4, 5 and more dimensions. I discovered that at 8 we will find resonanceif we keep adding only from the second and third terms, and that the ratios of the terms viz-a-viz each other (n different sequences of numbers) will approach the golden mean, just at they do within sequences. This tells me the universe naturally resonates RIGHT AT THE CHANGE from the 3rd to the 4th dimension, AND IT RESONATES IN STEPS OF EIGHT.

This is why we hear octaves, I suppose, and why chemical properties return to similarity after additions of eight electrons (as seen in the periodic table). It also is very relevant to the IChing, and therefore to this list, and Terence's idea of a "resonance" in Time made of it's own self-mirrored "modular hierarchies" (since termed fractals).

As a side note, these additions of eight steps to themselves are self-created, and the clue to how this could be lies in the backward sort of vision that a self-mirroring system must utilize... starting with nothing it must declare itself to be whole, to be, or, as we would term it, the number 1. It can only do this by what it adds to itself later... thus 1+1=2, etc., so if it keeps adding in steps of two, as in the original Fibonacci series, it "creates" a two-dimensional vision, as seen in a sunflower. If it adds in longer series, what it is saying is that these additions of 3, 4, 5 or more AGAIN equal MYSELF, the number 1... you see, the thing that mirrors itself in order to exist is constantly, renewingly defining the number 1... defining itself. So life continues, the mirror expands (which we see as space expanding) and all of it is both discreet and continuous because conflicting and self-resonating definitions of the number 1, of itself-seeing-itself, create dichotomies and unresolved conflicts.

These conflicts are always resolved in the next dimension up... i.e., just as you and I, being 3-dimensional beings, see everything in 2-dimensional tableaus, so our 3-dimensional world is created by a 4-dimensional "vision". We see the fourth dimension as "time", being as we cannot see it otherwise; it appears as a very strange set of connections, just as a three dimensional world would appear to a hypothetical inhabitant of a two-dimensional "Flatland". It seems, however, as physicists are discovering, that at eight dimensions the system has a sort of recursive break (read perhaps Michio Kaku's "Hyperspace") and this is the foundation of the discoveries related to String Theory... and Ramanujan's mathematical musings, mentioned earlier on this list. They claim that god had no choice, that the world had to have 8 dimensions (they actually put thenumber at 10, in order to restore symmetry in relativistic equations).

Remember however that these "higher dimensional visions"are all self-created, not a function of a higher god so-to-speak but a function of itself. This directly relates to the ESCHATON in my view, in that as WE discover OURSELVES, i.e., the whole human race, or rather, AS HISTORICAL BEINGS and in so doing define ourselves as the number 1 (that is, become one through self-postulated vision) we will break through a barrier... this to me is a mechanical process, a sort of discovery of a system of organization akin to the one nature uses to see itself---self-mirroring on a human, world-wide, historical scale. It requires seeing ourselves over and over, one more time than we are used to, TRANSCENDING, as it were, to the next dimension up... it is the higher view.

The GOLDEN MEAN gives me hope in this enterprise... it tells us that opposites are not equal, that all does not spiral down to meaninglessness necessarily, as Timothy Leary described to me (what was to him) the Zen view of the world shortly before his death. It tells me that we can see one as greater than the other, and I suspect it is up to us to see hope defeating despair. I suspect it is not "inherent" except inasmuch as we are inherent.

Mr. McKenna's eschaton is not then an automatic thing in any sort of certainty. It is an historical event he speaks of, and should then be created by us. I spoke at one of his events of "engineering the eschaton" and that is still what I am pretending to be doing...

"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."

-- Johannes Kepler (1571-1630)

Any objective observation we make must include a discussion of proportion for it is the rule of proportion in the examination of nature that causes us to observe an organized universe and a universe in chaos, rational and irrational numbers, harmony and discord, truth and falsity. These descriptions are merely proportional effects of the opposition that is inherent in all things.

We see harmony expressed by those emotions, feelings, and characteristics present within ourselves. This harmony is viewed within nature as the Divine Proportion. The Divine Proportion ascribed to our collective state of observation has been expressed, "For of three magnitudes, if the greatest (AB) is to the mean (CB) a the mean (CB) is to the least (AC), they therefore all shall be one."

AB/CB = CB/AC = 1.618... =1/r

The Divine Proportion was closely studied by the Greek sculpture, Phidias, hence, it was given the name Phi. Also known as the Golden Mean, the Magic Ratio, the fibonacci Series, etc., Phi can be found throughout the universe; from the spirals of galaxies to the spiral of a Nautilus seashell; from the harmony of music to the beauty in art. A botanist will find it in the growth patterns of flowers and plants, while the zoologist sees it in the breeding of rabbits. The entomologist views it in the genealogy of a bee, and the physicist observes it in the behavior of light and atoms. A Wall Street analyst finds it in the rising and falling patterns of a market, the mathematician in the examination of the pentagram.

Throughout history, Phi has been observed to evoke emotion or aesthetic feelings within the us. The ancient Egyptians used it in the construction of the great pyramids and in the design of hieroglyphs found on tomb walls. At another time, thousands of miles away, the ancients of Mexico embraced Phi while building the Sun Pyramid at Teotihuacan. The Greeks studied Phi closely through their mathematics and used it in their architecture. The Parthenon at Athens is a classic example of the use of the Golden Rectangle. Plato in his Timaeus considered it the most binding of all mathematical relations and makes it the key to the physics of the cosmos. During the Renaissance, Phi served as the "hermetic" structure on which some of the great masterpieces were composed. Notable artists such as Michelangelo, Raphael, and Leonardo da Vinci made use of it for they knew of its appealing qualities.

Phi must be considered in its relation to the human psyche since it is the psyche that interprets this phenomena. Although Phi appears to be fixed in nature, it actually is not. The only reason it seems fixed is because it is fixed within our own minds. This proportion corresponds to the mental vibrations that are within us and dictate our sense of pleasure and pain, beauty and ugliness, love and hate, etc. The result is we are held captive by these memories fixed by both body and mind. For if we were to view nature from an altered state of consciousness, the proportion would also be altered.

Therefore, the Divine Proportion presents itself in the very physical nature of Creation. It is seen as the beauty and organization within the cosmos. It is the harmony and glue that holds the unity of the universe.

1... 2... 3... 5... 8... 13... 21... 34... 55... 89... 144... 233...377...

Leonardo da Vinci understood that Man [*Canon of Man] was intended in the proportion of Phi and indeed much of the natural world was Phi based. In the anatomy of Man, the spinal vertebrae are relative to each other in the Phi ratio. The Nautilus shell spirals in the Phi ratio. Plants and trees grow in the Phi ratio. The Earth and Moon have this same relationship. The sunflower is a wonderful example of the spiraling effect of Phi. Look at a pine cone and find the same relationship of Phi...in two directions at the same time.

track _Phi (sheik mix)_ MP3 by Paragliders off of _Oasis_ ep

the film _Pi_ references the golden section as well as Leonardo Da Vinci

release _The Golden Section_ by System 7 1997

System 7 is Steve Hillage and Youth_The Rite of Spring_ Don Corleone_ reference to the film _The Godfather_ directed by Francis Ford Coppola _Y2k (Beatnik Mix)_ _Ring Of Fire_

_Exdreamist_

_Wave Bender_ 6:04

  1. Proportions of playing cards
  1. Parthenon in Athens, Greece