God's Handiwork -- Fibonacci Numbers hat 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 34 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.
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.
1/2
elm, linden, lime, grasses The
numerator is # of turns and denominator is # of leaves til we arrive at
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).
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. For instance, the pineapple rind has eight gently-sloping spirals (left), thirteen steeper-sloping spirals (center) and twenty-one almost vertical spirals (right). A Fibonacci series. Many plants produce new branches in quantities
that are based on Fibonacci numbers.
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. 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
ventricles of the human heart reset themselves at the Golden Ratio point
of the heart's rhythmic cycle.
The Golden ratio (1.618) is related to the sidereal and synodic years of all planets until Uranus to a statistical significance of over 99%. Mercury's sidereal year is 88 days. The Mercury-Venus synodic is 144.5255474 days. 88x1.61803 = 142.387, which is within 2 percent of the Mercury-Venus synodic.
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. 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.
The musical scales are based
on Fibonacci numbers. 1,1,2,3,5,8,13. 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 the 1st 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. 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 as 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. Next Lesson: Why Should We Celebrate HANUKKAH? | Back to Home | Email Us |