Fibonacci Numbers in Nature & the Golden Ratio

The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.

Fibonacci Numbers

The sequence, in which each number is the sum of the two preceding numbers is known as the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ... (each number is the sum of the previous two).

The ratio of successive pairs is so-called golden section (GS) - 1.618033989 . . . . .
whose reciprocal is 0.618033989 . . . . . so that we have 1/GS = 1 + GS.

The Fibonacci sequence, generated by the rule f1 = f2 = 1 , fn+1 = fn + fn-1,
is well known in many different areas of mathematics and science.

Pascal's Triangle and Fibonacci Numbers

The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui (about 500 years earlier, in fact) and the Persian astronomer-poet Omar Khayyám.

Pascal's Triangle is described by the following formula:


is a binomial coefficient.



It is quite amazing that the Fibonacci number patterns occur so frequently in nature(flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal "laws of nature". Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves The Fibonacci numbers are also found in the family tree of honeybees.

Fibonacci and Nature

Plants do not know about this sequence - they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144. Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks.

Why do these arrangements occur? In the case of leaf arrangement, or phyllotaxis, some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. In the case of close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of space. This is well described in several books listed here >>

So nature isn't trying to use the Fibonacci numbers: they are appearing as a by-product of a deeper physical process. That is why the spirals are imperfect.
The plant is responding to physical constraints, not to a mathematical rule.

The basic idea is that the position of each new growth is about 222.5 degrees away from the previous one, because it provides, on average, the maximum space for all the shoots. This angle is called the golden angle, and it divides the complete 360 degree circle in the golden section, 0.618033989 . . . .

Examples of the Fibonacci sequence in nature.

Petals on flowers*

Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number:

3 petals: lily, iris
5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
8 petals: delphiniums
13 petals: ragwort, corn marigold, cineraria,
21 petals: aster, black-eyed susan, chicory
34 petals: plantain, pyrethrum
55, 89 petals: michaelmas daisies, the asteraceae family
Some species are very precise about the number of petals they have - e.g. buttercups, but others have petals that are very near those above, with the average being a Fibonacci number.

One-petalled ... White calla lily



Two-petalled flowers are not common.... Euphorbia



Three petals are more common.... Trillium




Five petals - there are hundreds of species, both wild and cultivated, with five petals....



Eight-petalled flowers are not so common as five-petalled, but there are quite a number of well-known species with eight..... Bloodroot




Thirteen, ... Black-eyed Susan



Twenty-one and thirty-four petals are also quite common. The outer ring of ray florets in the daisy family illustrate the Fibonacci sequence extremely well. Daisies with 13, 21, 34, 55 or 89 petals are quite common.... Shasta Daisy



Ordinary field daisies have 34 petals ...
a fact to be taken in consideration when playing "she loves me, she loves me not". In saying that daisies have 34 petals, one is generalizing about the species - but any individual member of the species may deviate from this general pattern. There is more likelihood of a possible under development than over-development, so that 33 is more common than 35........



Flower Patterns and Fibonacci Numbers

Why is it that the number of petals in a flower is often one of the following numbers: 3, 5, 8, 13, 21, 34 or 55? For example, the lily has three petals, buttercups have five of them, the chicory has 21 of them, the daisy has often 34 or 55 petals, etc. Furthermore, when one observes the heads of sunflowers, one notices two series of curves, one winding in one sense and one in another; the number of spirals not being the same in each sense. Why is the number of spirals in general either 21 and 34, either 34 and 55, either 55 and 89, or 89 and 144? The same for pinecones : why do they have either 8 spirals from one side and 13 from the other, or either 5 spirals from one side and 8 from the other? Finally, why is the number of diagonals of a pineapple also 8 in one direction and 13 in the other?

Are these numbers the product of chance? No! They all belong to the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (where each number is obtained from the sum of the two preceding). A more abstract way of putting it is that the Fibonacci numbers fn are given by the formula f1 = 1, f2 = 2, f3 = 3, f4 = 5 and generally f n+2 = fn+1 + fn . For a long time, it had been noticed that these numbers were important in nature, but only relatively recently that one understands why. It is a question of efficiency during the growth process of plants.

The explanation is linked to another famous number, the golden mean, itself intimately linked to the spiral form of certain types of shell. Let's mention also that in the case of the sunflower, the pineapple and of the pinecone, the correspondence with the Fibonacci numbers is very exact, while in the case of the number of flower petals, it is only verified on average (and in certain cases, the number is doubled since the petals are arranged on two levels).

Let's underline also that although Fibonacci historically introduced these numbers in 1202 in attempting to model the growth of populations of rabbits, this does not at all correspond to reality! On the contrary, as we have just seen, his numbers play really a fundamental role in the context of the growth of plants.

Human Hand

Every human has two hands, each one of these has five fingers, each finger has three parts which are separated by two knuckles. All of these numbers fit into the sequence. However keep in mind, this could simply be coincidence.



Golden Ratio & Golden Section

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.



Expressed algebraically:

The golden ratio is often denoted by the Greek letter phi (Φ or φ).
The figure of a golden section illustrates the geometric relationship that defines this constant. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

Golden Ratio in Nature

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law. Zeising wrote in 1854:

The Golden Ratio is a universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.


Examples:









Golden Ratio in Architecture and Art

Many architects and artists have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.

KRYPTOS

Kryptos is a sculpture by American artist Jim Sanborn located on the grounds of the Central Intelligence Agency (CIA) in Langley, Virginia. Since its dedication on November 3, 1990, there has been much speculation about the meaning of the encrypted messages it bears. Of the four sections, three have been solved, with the fourth remaining one of the most famous unsolved codes in the world. The sculpture continues to provide a diversion for some employees of the CIA and other cryptanalysts attempting to decrypt the messages.
Description
The main sculpture is made of red and green slate, white quartz, petrified wood, lodestone, and copper, and is located in the northwest corner of the New Headquarters Building courtyard, outside of the Agency cafeteria.
The name Kryptos comes from the Greek word for "hidden", and the theme of the sculpture is "intelligence gathering." The most prominent feature is a large vertical S-shaped copper screen resembling a scroll, or piece of paper emerging from a computer printer, covered with characters comprising encrypted text. The characters consist of the 26 letters of the standard Roman alphabet and question marks cut out of the copper. This "inscription" contains four separate enigmatic messages, three of which have been solved.
At the same time as the main sculpture was installed, sculptor Sanborn also placed several other pieces around CIA grounds, such as several large granite slabs with sandwiched copper sheets outside the entrance to the New Headquarters Building. Several morse code messages are engraved in the copper, and one of the slabs has an engraved compass rose. Other elements of Sanborn's installation include a landscaped area, a duck pond, and several other seemingly unmarked slabs.
The cost of the sculpture was $250,000.
Encrypted messages
The ciphertext on one half of the main sculpture contains 869 characters in total — in April 2006, however, Sanborn released information stating that an intended letter on the main half of Kryptos was missing. This would bring the total number of characters to 870 on the main portion. The other half of the sculpture comprises a Vigenère encryption tableau, comprising 869 characters, if spaces are counted.
Sanborn worked with a retiring CIA employee named Ed Scheidt, Chairman of the CIA Cryptographic Center, to come up with the cryptographic systems used on the sculpture. Sanborn has revealed that the sculpture contains a riddle within a riddle, which will be solvable only after the four encrypted passages have been decrypted. He said that he gave the complete solution at the time of the sculpture's dedication to then-CIA director William H. Webster. Sanborn said, however, that he had not given Webster the entire solution. He did, however, confirm that where in part 2 it says "Who knows the exact location? Only WW," that "WW" was intended to refer to William Webster. Sanborn also confirmed that should he die before the entire sculpture becomes deciphered that there will be someone able to confirm the solution.
Solvers
The first person to publicly announce solving the first three sections, in 1999, was James Gillogly, a computer scientist from southern California, who deciphered 760 of the characters (772 less 3 question marks and XLAYERTWO). The portion that he couldn't solve, the remaining 97 or 98 characters, is the same part which has stumped the government's own cryptanalysts. After Gillogly's announcement, the CIA revealed that their analyst David Stein had also solved the same sections in 1998, using pencil and paper techniques, though at the time of his solution the information was only disseminated within the intelligence community, and no public announcement was made. The NSA also claimed at that time that they had solvers, but would not reveal names or dates until 2005, when it was learned that an NSA team led by Ken Miller, along with Dennis McDaniels and two other unnamed individuals, had solved parts 1–3 using a computer in late 1992, but that they too had been stumped by the fourth section.
Solution 1
Keywords: Kryptos, Palimpsest
BETWEEN SUBTLE SHADING AND THE ABSENCE OF LIGHT LIES THE NUANCE OF IQLUSION.
Solution 2
Keywords: Kryptos, Abscissa
IT WAS TOTALLY INVISIBLE HOWS THAT POSSIBLE ? THEY USED THE EARTHS MAGNETIC FIELD X THE INFORMATION WAS GATHERED AND TRANSMITTED UNDERGRUUND TO AN UNKNOWN LOCATION X DOES LANGLEY KNOW ABOUT THIS ? THEY SHOULD ITS BURIED OUT THERE SOMEWHERE X WHO KNOWS THE EXACT LOCATION ? ONLY WW THIS WAS HIS LAST MESSAGE X THIRTY EIGHT DEGREES FIFTY SEVEN MINUTES SIX POINT FIVE SECONDS NORTH SEVENTY SEVEN DEGREES EIGHT MINUTES FORTY FOUR SECONDS WEST X LAYER TWO
On April 19, 2006, Sanborn contacted the Kryptos Group (an online community dedicated to the Kryptos puzzle) to inform them that the accepted solution to part 2 was wrong. He said that he made an error in the sculpture by omitting an "X" used to indicate a break for aesthetic reasons, and that the decrypted text which ended "...FOUR SECONDS WEST ID BY ROW S" should actually be "...FOUR SECONDS WEST X LAYER TWO".[5]
Note: The coordinates mentioned in the plaintext: 38°57′6.5″N 77°8′44″W; on Google Maps; analysis of the cited location. The point is about 200 feet southeast of the sculpture itself.
Solution 3
SLOWLY DESPARATLY SLOWLY THE REMAINS OF PASSAGE DEBRIS THAT ENCUMBERED THE LOWER PART OF THE DOORWAY WAS REMOVED WITH TREMBLING HANDS I MADE A TINY BREACH IN THE UPPER LEFT HAND CORNER AND THEN WIDENING THE HOLE A LITTLE I INSERTED THE CANDLE AND PEERED IN THE HOT AIR ESCAPING FROM THE CHAMBER CAUSED THE FLAME TO FLICKER BUT PRESENTLY DETAILS OF THE ROOM WITHIN EMERGED FROM THE MIST X CAN YOU SEE ANYTHING Q (?)
This is a paraphrased and misspelled quotation from Howard Carter's account of the opening of the tomb of Tutankhamun on November 26, 1922, as described in his 1923 book The Tomb of Tutankhamun. The question with which it ends is that posed by Lord Carnarvon, to which Carter (in the book) famously replied "wonderful things". In the actual November 26, 1922 field notes, his reply was, "Yes, it is wonderful."
Solution 4
Part 4 remains publicly unsolved, though there is an active Yahoo! Group[7] (formed in 2003) that coordinates the work of over 1000 members toward decryption of the code.
To solve it, it is crucial to have the first three sections correct, since when commenting on his error in section 2, Sanborn said they contain the clues to the last section.