Fibonacci Numbers

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What are Fibonacci numbers? Discovered by Leonardo Fibonacci
(1170-1250 A.D.) who was born in Pisa, Italy, the Fibonacci sequence
is an infinite sequence of numbers, beginning: 1, 1, 2, 3, 5, 8, 13,
21, 34, 55, 89, 144, ... where each number is the sum of the two
preceding it. Thus: 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13 and so on. For
any value larger than 3 in the sequence, the ratio between any two
consecutive numbers is 1:1.618, or the Golden Ratio.

Known by the Greek letter phi, the Golden Ratio is an irrational
number (i.e. one that cannot be expressed as the ratio or fraction of
two whole numbers) with several curious properties. We can define it
as the number that is equal to its own reciprocal plus one: phi = 1/
phi + 1, with its value commonly expressed as 1.618,033,989. It's
digits were calculated to ten million places in 1996, and they never
repeat. It is related to Fibonacci numbers in that if you divide two
consecutive numbers in the Fibonacci sequence, the answer is always an
approximation of phi.

Also known as the Divine Proportion, the Golden Mean, or the Golden
Section, this ratio is found with surprising frequency in natural
structures as well as man-made art and architecture. where the ratio
of length to width of approximately 1:1.618 is seen as visually
pleasing. It is the point dividing a distance into two segments where
the proportion of the smaller segment to the larger segment is the
same as the larger to the whole.

Leaves On A Stem

If we look down on a plant, the leaves are often arranged so that
leaves above do not hide leaves below. This means that each gets a
good share of the sunlight and catches the most rain to channel down
to the roots as it runs down the leaf to the stem.

The Fibonacci numbers occur when counting both the number of times we
go around the stem, going from leaf to leaf, as well as counting the
leaves we meet until we encounter a leaf directly above the starting
one. If we count in the other direction, we get a different number of
turns for the same number of leaves. One estimate is that 90 percent
of all plants exhibit Fibonacci numbers in their leaf patterns.
Some common trees with their Fibonacci leaf arrangement numbers are:

1/2 elm, linden, lime, grasses
1/3 beech, hazel, grasses, blackberry
2/5 oak, cherry, apple, holly, plum, common groundsel
3/8 poplar, rose, pear, willow
5/13 pussy willow, almond

The numerator is # of turns and denominator is # of leaves til we
arrive a a leaf directly under the starting leaf. Each leaf is this
fraction of a turn after the last leaf. Or, there are so many turns
(numerator) for every so many leaves (denominator).

Cactus's spines often show the same spirals as we see on pine cones,
petals and leaf arrangements, but they are much more clearly visible.

Plants illustrate the Fibonacci series in the numbers and arrangements
of petals, leaves, sections and seeds. Plants that are formed in
spirals, such as artichokes, pinecones, pineapples, daisies and
sunflowers, illustrate Fibonacci numbers. Many plants produce new
branches in quantities that are based on Fibonacci numbers.

Calla Lilies have one petal. Euphorbia have two petals. Lilies have
three. Buttercups, Roses, Pinks and Geraniums have five, Delphiniums
and Bloodroot have 8 petals, Marigolds and Ragworts have 13, Asters
and Black-eyed Susans 21, Pyrethrum have 34, and most Daisies have 34,
55, or 89. You don't find any other numbers very often. (The main
exceptions are when those same numbers appear doubled, or when the so-
called anomalous series appears: 3, 4, 7, 11, 18 and so on.). The five-
pointed star is unusually common in nature. Not only in flowers, but
also in the starfish, sand dollar, apple, etcetera. Why? Because Phi
is the ratio of the side of a pentagon to its diagonal (see picture).

The Sunflower, Pinecone, Daisy and Nautilus Shell

Furthermore, the SUNFLOWER has 21 spirals on its head in one direction
and 34 going in the other -- consecutive Fibonacci numbers. Or 34 and
55, or 55 and 89, or 89 and 144. The precise numbers depend on the
species of SUNFLOWER.The same is true with DAISIES. The outside of a
PINECONE has spirals that run clockwise and counterclockwise, and the
ratio of the number of spirals is sequential Fibonacci values. In the
elegant curves of a NAUTILUS shell, each complete new revolution will
be a ratio of 1:1.618, when compared to the distance from the center
of the previous spiral. If the divergence angle of a SUNFLOWER varies
from 137.5 degrees (seen in the pattern below) to either 137.4 or
137.6 degrees, much of the beauty of the spiral is lost as well as its
most compact arrangement. The SUNFLOWER therefore is in a precise
state of perfection. ROMANESQUE BROCCOLI/CAULIFLOWER is Fibonacci
spiral upon spiral. The same can be said of any other plant or animal
containing this Fibonacci sequence. Everything in nature is already in
a state of perfection -- contrary to the theory of Evolution..Where
are the transitional stages? Where are the vestigial organs? Where are
the imperfect spirals? Who told a DAISY to have the same Fibonacci
sequence as a NAUTILUS, or a PINECONE, or a SUNFLOWER, or a RAM'S HORN
or even the appearance of a BREAKING WAVE? None of them have a brain.
None of them are in any way related to the other. Design demands a
Designer. Beauty demands a Master Artist.

The Human Head

Human beauty is based on the Divine Proportion. Many experiments have
been carried out to prove that the proportions of the top models'
faces conform more closely to the Golden Ratio than the rest of the
population. The human face is based entirely on Phi .The head forms a
golden rectangle with the eyes at its midpoint. The mouth and nose
are each placed at golden sections of the distance between the eyes
and the bottom of the chin. The ear reflects the shape of a Fibonacci
spiral. Even the dimensions of our teeth are based on phi. The front
two incisor teeth form a golden rectangle, with a phi ratio in the
heighth to the width. The ratio of the width of the first tooth to the
second tooth from the center is also phi. The ratio of the width of
the smile to the third tooth from the center is phi as well.

The Human Finger Joints and Hand

Each segment of each finger, from the tip to the base of the wrist, is
larger than the preceding one by about the Fibonacci ratio of 1.618.
In other words, the smallest segment is 2, the next 3, the next 5, and
the back of your hand is 8. By this scale, your fingernail is 1 unit
in length. You also have 2 hands, each with 5 digits, and your 8
fingers are each comprised of 3 sections. All Fibonacci numbers! Your
hand creates a golden section in relation to your forearm, as the
ratio of your forearm to your hand is also 1.618, the Divine
Proportion. Furthermore, your hand and forearm together are a golden
section in relation to the entire length of your arm.

The Human Body

The ventricles of the human heart reset themselves at the Golden Ratio
point of the heart's rhythmic cycle.

Leonardo Da Vinci found that the total height of the human body, from
toes to top of head, and the height from the toes to the navel
depression are in Golden Ratio. Also the height from toes to navel and
from the navel to the top of the head are in Golden Ratio. This has
been confirmed by measuring 207 students at the Pascal Gymnasium in
Munster Germany, where the perfect value of 1.618 was obtained for the
first ratio average and 1.619 was obtained for the second. This value
held for both girls and boys of similar ages ("Golden Mean of the
Human Body" by T. Antony Davis and Rudolf Altevogt).

But there is more. From the top of the head to the bottom of the feet
is the body's total height A. From the top of the head to the bottom
of the fingertips is a golden section B of the total height A. From
the top of the head to the navel and elbows is a golden section C of
the golden section B. From the top of the head to the pectorals and
armpits, the width of the shoulders, the length of the forearm and
shinbone is a golden section D of the golden section C. From the top
of the head to the base of the skull and the width of the abdomen is a
golden section E of the golden section D. The width of the head and
half the width of the chest and the hips is the golden section F of
the golden section E.

The human body is also based on the number 5: The torso has 5
appendages, in the arms, legs and head., In turn, each of these has
five appendages, in the fingers and toes and 5 openings on the face.
and 5 senses in sight, sound, touch, taste and smell. The golden
section is also based on 5, as the number phi, or 1.6180339 is
computed from the square root of 5 multiplied by .5 plus .5 = Phi.

Relative Planetary Distances

The 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: The mean distance is in million
kilometers per NASA. If Mercury is 1, these are the relative mean
distances Mercury 57.91 (1.00000) Venus 108.21 (1.86859) Earth 149.60
(1.38250) Mars 227.92 (1.52353) Ceres (the largest asteroid) 413.79
(1.81552) Jupiter 778.57 (1.88154) Saturn 1,433.53 (1.84123) Uranus
2,872.46 (2.00377) Neptune 4,495.06 (1.56488) and Pluto 5,869.66
(1.30580) The total is 16.18736 and so if we divide by 10, we get the
average which is1.61874. Phi is1.61803.

The Fibonacci Series also predicts accurately the distances of the
moons of Jupiter, Saturn and Uranus from each one's respective parent
planet. Individual offsets can be attributed to planetary densities.
Jupiter has 12 moons. Saturn has 9 moons and Uranus has 5 moons. For
actual plotted graphs of these amazing results refer to the Fibonacci
Quarterly October 1970, vol. 8, Number 4.

The Greatest Human Achievements Include the Golden Ratio

The dimensions of the King's Chamber of the Great Pyramid in Giza,
Egypt are based upon the Golden Ratio. The Great Pyramid's vertical
height and the width of any of its sides are in Golden Sections. The
architect, Le Corbusier designed his Modulor system around the use of
the ratio; the painter Mondrian based most of his work on the Golden
Ratio; Leonardo Da Vinci included it in many of his paintings and
Claude Debussy used its properties in his music. So did Bela Bartok.
The Golden Ratio also is found in widescreen televisions, postcards,
credit cards, and photographs which all commonly conform to its
proportions. Like the brilliant Pythagoras before him, Leonardo had
made an in-depth study of the human figure, showing how all of its
major parts were related to the Golden Ratio. The Mona Lisa's face
fits inside a perfect Golden Rectangle. The Golden Rectangle is found
in his painting called The Last Supper also. Many other Renaissance
artists did the same. After Leonardo, artists such as Raphael and
Michelangelo made great use of the Golden Ratio to construct their
works. Michelangelo's beautiful sculpture of David conforms to the
Golden Ratio, from the location of the navel with respect to the
height and placement of the joints in the fingers. The builders of the
medieval and Gothic churches and cathedrals of Europe also made these
structures conform to the Golden Ratio. The Golden Rectangle is one
whose sides are in the proportion of the Golden Ratio. This means the
longer side is 1.618 times longer than the shorter side. The front of
the Parthenon in Athens, Greece can be framed within a Golden
Rectangle. Da Vinci's drawing of the Vitruvian Man has the outlines of
a rectangle based on the head, one on the torso, and another over the
legs.

The Golden Ratio In The Bible

The Ark of the Covenant is a Golden Rectangle. In Exodus 25:10, God
commands Moses to build the Ark of the Covenant, in which to hold His
Covenant with the Israelites, the Ten Commandments, saying,"Have them
make a chest of acacia wood -- two and a half cubits long, a cubit and
a half wide, and a cubit and a half high." The ratio of 2.5 to 1.5 is
1.666..., which is as close to phi (1.618 ...) as you can come with
such simple numbers and is certainly not visibly different to the
eye. The Ark of the Covenant is thus constructed using the Golden
Section, or Divine Proportion. This ratio is also the same as 5 to 3,
numbers from the Fibonacci series.

Noah's Ark uses a Golden Rectangle. In Genesis 6:15, God commands Noah
to build an ark saying,"And this is the fashion which thou shalt make
it of: The length of the ark shall be three hundred cubits, the
breadth of it fifty cubits, and the height of it thirty cubits."Thus
the end of the ark, at 50 by 30 cubits, is also in the ratio of 5 to
3, or 1.666..., again a close approximation of phi not visibly
different to the naked eye. Noah's ark was built in the same
proportion as ten arks of the covenant placed side by side.

Exodus 27:1-2 mentions the dimensions of the altar -- constructed
according to phi: "Build an altar of acacia wood, three cubits high;
it is to be square, five cubits long and five cubits wide."

Furthermore, the location of Jerusalem is 31 degrees 45 minutes north
of the equator. God said Jerusalem is "the city which I have chosen to
put my name there" (1 Kings 11:36). Why did God select this location?
First build a rectangular building in Jerusalem with sides that
exhibit the golden rectangle ratio. The longer two sides (1.618) must
run from east to west. The shorter two sides (1) must run from north
to south. Then at each of the four corners place a flag pole. Make the
roof flat. Remember that the sun rises in the east and sets in the
west. Now if you draw one diagonal line from the northwest to the
southeast corner, and another diagonal line from the northeast to
southwest corner, you will create angles of 31 degrees 45 minutes with
respect to the straight line joining the two southern corners of the
building. At the summer solstice, the sun is over the Tropic of
Cancer, 23.5 degrees north of the equator. At this point the shadows
will point most easterly and westerly. At the winter solstice, the sun
is over the Tropic of Capricorn, 23.5 degrees south of the equator. At
this point the shadows will point most northerly. On the spring (March
21) and fall (Sept. 23) equinoxes, when the sun is directly over the
equator, the shadows created by sunrise and sunset will fall exactly
on these diagonal lines. This method can be used at any latitude with
rectangles of different proportions. But a rectangle with the
proportions of "the golden section " will only give this result at the
exact latitude of Jerusalem. It is important to know when the
equinoxes occur each year in order to celebrate God's festivals. We
are to take the first new moon on or after the spring equinox as New
Year's Day. No complex postponement rules or calculated calendar is
necessary. Babylon is about one degree further north than Jerusalem.
The pyramids of Gizeh are further south and Mecca still further south.

Fibonacci Numbers In The Musical Scale

The musical scales are based on Fibonacci numbers. The piano keyboard
scale of C to C has 8 white keys in an "octave" and 5 black keys,
making 13 keys total. The 5 black keys are divided into groups: of 2
keys and 3 keys. A scale is comprised of 8 notes, 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 the1st note of
the scale. Although there are only 12 notes in the scale, if you don't
have a root and octave, a start and an end, you have no means of
calculating the gradations in between, so this 13th note as the octave
is essential to computing the frequencies of the other notes. The
word "octave" comes from the Latin word for 8, referring to the eight
whole tones of the complete musical scale, which in the key of C are C-
D-E-F-G-A-B-C.

In a scale, the dominant note is the 5th note of the major scale,
which is also the 8th note of all 13 notes that comprise the octave.
This provides another instance of Fibonacci numbers in key musical
relationships. Interestingly, 8/13 is .61538, which approximates
phi. What's more, the typical three chord song in the key of A is
made up of A, its Fibonacci & phi partner E, and D, to which A bears
the same relationship as E does to A. This is analogous to the "A is
to B as B is to C" basis for the golden section, or in this case "D is
to A as A is to E."

Musical frequencies are based on Fibonacci ratios Notes in the scale
of western music have a foundation in the Fibonacci series, as the
frequencies of musical notes have relationships based on Fibonacci
numbers: A440 Hertz is an arbitrary standard. The American Federation
of Musicians accepted the A440 as standard pitch in 1917. It was then
accepted by the U.S. government its standard in 1920 and it was not
until 1939 that this pitch was accepted internationally. Before
recent times a variety of tunings were used. It has been suggested by
James Furia and others that A432 be the standard. A432 was often used
by classical composers and results in a tuning of the whole number
frequencies that are connected to numbers used in the construction of
a variety of ancient works and sacred sites, such as the Great Pyramid
of Egypt. The controversy over tuning still rages, with proponents of
A432 or C256 as being more natural tunings than the current standard.

Musical compositions often reflect Fibonacci numbers and phi Fibonacci
and phi relationships are often found in the timing of musical
compositions. As an example, the climax of songs is often found at
roughly the phi point (61.8%) of the song, as opposed to the middle or
end of the song. In a 32 bar song, this would occur in the 20th bar.