What is Phi?
Phi ( = 1.618033988749895... ), most often pronounced fi like "fly," is simply an irrational number like pi ( p = 3.14159265358979... ), but one with many unusual mathematical properties. Unlike pi, which is a transcendental number, phi is the solution to a quadratic equation.

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 "dividing a line in the extreme and mean ratio" and to Renaissance artists as the "Divine Proportion" It is also called the Golden Section, Golden Ratio and the Golden Mean.

Phi, like Pi, is a ratio defined by a geometric construction
Just as pi (p) 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 1.618 0339 887 ..., 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.
Compute any number in the Fibonacci Series easily!
Here are two ways you can use phi to compute the nth number in the Fibonacci series (fn).

If you consider 0 in the Fibonacci series to correspond to n = 0, use this formula:

fn = Phi n / 5½

Perhaps a better way is to consider 0 in the Fibonacci series to correspond to the 1st Fibonacci number where n = 1 for 0. Then you can use this formula, discovered and contributed by Jordan Malachi Dant in April 2005:

fn = Phi n / (Phi + 2)

Both approaches represent limits which always round to the correct Fibonacci number and approach the actual Fibonacci number as n increases.


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The ratio of successive Fibonacci numbers converges on phi
Sequence
in the
series Resulting
Fibonacci
number
(the sum
of the two
numbers
before it) Ratio of each
number to the
one before it
(this estimates
phi) Difference
from
Phi

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0
0

1 1
2 1 1.000000000000000 +0.618033988749895
3 2 2.000000000000000 -0.381966011250105
4 3 1.500000000000000 +0.118033988749895
5 5 1.666666666666667 -0.048632677916772
6 8 1.600000000000000 +0.018033988749895
7 13 1.625000000000000 -0.006966011250105
8 21 1.615384615384615 +0.002649373365279
9 34 1.619047619047619 -0.001013630297724
10 55 1.617647058823529 +0.000386929926365
11 89 1.618181818181818 -0.000147829431923
12 144 1.617977528089888 +0.000056460660007
13 233 1.618055555555556 -0.000021566805661
14 377 1.618025751072961 +0.000008237676933
15 610 1.618037135278515 -0.000003146528620
16 987 1.618032786885246 +0.000001201864649
17 1,597 1.618034447821682 -0.000000459071787
18 2,584 1.618033813400125 +0.000000175349770
19 4,181 1.618034055727554 -0.000000066977659
20 6,765 1.618033963166707 +0.000000025583188
21 10,946 1.618033998521803 -0.000000009771909
22 17,711 1.618033985017358 +0.000000003732537
23 28,657 1.618033990175597 -0.000000001425702
24 46,368 1.618033988205325 +0.000000000544570
25 75,025 1.618033988957902 -0.000000000208007
26 121,393 1.618033988670443 +0.000000000079452
27 196,418 1.618033988780243 -0.000000000030348
28 317,811 1.618033988738303 +0.000000000011592
29 514,229 1.618033988754323 -0.000000000004428
30 832,040 1.618033988748204 +0.000000000001691
31 1,346,269 1.618033988750541 -0.000000000000646
32 2,178,309 1.618033988749648 +0.000000000000247
33 3,524,578 1.618033988749989 -0.000000000000094
34 5,702,887 1.618033988749859 +0.000000000000036
35 9,227,465 1.618033988749909 -0.000000000000014
36 14,930,352 1.618033988749890 +0.000000000000005
37 24,157,817 1.618033988749897 -0.000000000000002
38 39,088,169 1.618033988749894 +0.000000000000001
39 63,245,986 1.618033988749895 -0.000000000000000
40 102,334,155 1.618033988749895 +0.000000000000000
Note: nx means n raised to the x power. Some browsers may not display exponents as superscripts or raised characters.

Deriving Phi mathematically
Phi can be derived by solving the equation:

n2 - n1 - n0 = 0

which is the same as

n2 - n - 1 = 0

This equation can be rewritten as:

n2 = n + 1 and 1 / n = n - 1

The solution to the equation is the square root of 5 plus 1 divided by 2:

( 5½ + 1 ) / 2 = 1.6180339... = Phi

This, of course, results in two properties unique to phi:

If you square phi, you get a number exactly 1 greater than phi: 2.61804...

Phi2 = Phi + 1

If you divide phi into 1, you get a number exactly 1 less than phi: 0.61804...:

1 / Phi = Phi - 1

Phi, curiously, can also be expressed all in fives as:

5 ^ .5 * .5 + .5 = Phi

This provides a great, simple way to compute phi on a calculator or spreadsheet!


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Determining the nth number of the Fibonacci series
You can use phi to compute the nth number in the Fibonacci series (fn):

fn = Phi n / 5½

As an example, the 40th number in the Fibonacci series is 102,334,155, which can be computed as:

f40 = Phi 40 / 5½ = 102,334,155

This method actually provides an estimate which always rounds to the correct Fibonacci number.

You can compute any number of the Fibonacci series (fn) exactly with a little more work:

fn = [ Phi n - (-Phi)-n ] / (2Phi-1)

Note: 2Phi-1 = 5½= The square root of 5


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Determining Phi with Trigonometry and Limits
Phi can be related to Pi through trigonometric functions:



Phi can be related to e, the base of natural logs,
through the inverse hyperbolic sine function:

Phi = e ^ asinh(.5)


It can be expressed as a limit:


or



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Other unusual phi relationships
There are many unusual relationships in the Fibonacci series. For example, for any three numbers in the series Phi(n-1), Phi(n) and Phi(n+1), the following relationship exists:

Phi(n-1) * Phi(n+1) = Phi(n)2 - (-1)n

( e.g., 3*8 = 52-1 or 5*13=82+1 )

Here's another:

Every nth Fibonacci number is a multiple of Phi(n),
where Phi(n) is the nth number of the Fibonacci sequence.

Given 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765

(Every 4th number, e.g., 3, 21, 144 and 987, are all multiples of Phi(4), which is 3)

(Every 5th number, e.g., 5, 55, 610, and 6765, are all multiples of Phi(5), which is 5)

And another:

The first perfect square in the Fibonacci series, 144,

is number 12 in the series and its square root is 12!

0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

or, if not starting with 0:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
Certain solar system orbital periods are related to phi
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:


Mercury Venus Earth Jupiter Saturn
Power of Phi -3 -1 0 5 7
Decimal Result 0.24 0.62 1.0 11.1 29.0
Actual Period 0.24 0.62 1.0 11.9 29.5



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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.





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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:

Planet Distance
from
the sun
in km (000) Distance
where
Mercury
equals 1 Period
where
Mercury
equals 1
Mercury 57,910 1.0000 1.0000
Venus 108,200 1.8684 2.5490
Earth 149,600 2.5833 4.1521

Curiously enough we find:

Ö Period of Venus * Phi = Distance of the Earth

Ö 2.5490 * 1.6180339 = 1.5966 * 1.6180339 = 2.5833



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!

See more relationships at the Solar Geometry site.


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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:

Planet
Mean
distance
in million
kilometers
per NASA
Relative
mean
distance
where
Mercury=1

Mercury
57.91
1.00000

Venus
108.21
1.86859

Earth
149.60
1.38250

Mars
227.92
1.52353

Ceres
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

Pluto
5,869.66
1.30580

Total

16.18736

Average

1.61874

Phi

1.61803

Degree of variance
(0.00043)


Note: We sometimes forget about the asteroids when thinking of the planets in our solar system. Ceres, the largest asteroid, is nearly spherical, comprises over one-third the total mass of all the asteroids and is thus the best of these minor planets to represent the asteroid belt.

2005 unveiled the discovery of a 10th planet called 2003UB313. It was found at a distance of 97 times that of the Earth from the Sun. Its ratio to Pluto would thus be 2.47224, much higher than any previous planet to planet orbital distance ratio. Could it be that this is actually the 11th planet and the 10th planet will be found at an orbit whose ratio is 1.52793 times that of Pluto, preserving the phi average? Time will only tell, but if it happens remember that you heard it here first.


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The shape of the Universe itself is a dodecahedron based on Phi
New findings in 2003 based on the study of data from NASA's Wilkinson Microwave Anisotropy Probe (WMAP) on cosmic background radiation reveal that the universe is finite and shaped like a dodecahedron, a geometric shape based on pentagons, which are based on phi. The the Universe page for more.
The Golden Section is a ratio based on a phi
The Golden Section is also known as the Golden Mean, Golden Ratio and Divine Proportion. It is a ratio or proportion defined by the number Phi ( = 1.618033988749895... )

It can be derived with a number of geometric constructions, each of which divides a line segment at the unique point where:

the ratio of the whole line (A) to the large segment (B)

is the same as

the ratio of the large segment (B) to the small segment (C).



In other words, A is to B as B is to C.

This occurs only where A is 1.618 ... times B and B is 1.618 ... times C.


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This ratio has been used by mankind for centuries
Its use may have started as early as with the Egyptians in the design of the pyramids,





The Greeks recognized it as
"dividing a line in the extreme and mean ratio" The Renaissance artists
knew it as the
Divine Proportion

and used it for beauty
and balance in the
design of architecture and used it for beauty
and balance in the
design of art

It was used in the design of Notre Dame in Paris



and continues today in many examples of art, architecture and design.

It also appears in the physical proportions of the human body, movements in the stock market and many other aspects of life and the universe.
Musical scales are based on Fibonacci numbers
The Fibonacci series appears in the foundation of aspects of art, beauty and life. Even music has a foundation in the series, as:
There are 13 notes in the span of any note through its octave.
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.

Note too how the piano keyboard scale of C to C above of 13 keys has 8 white keys and 5 black keys, split into groups of 3 and 2.
While some might "note" that 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 an added 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."


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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:

Fibonacci
Ratio Calculated
Frequency Tempered
Frequency Note in
Scale Musical
Relationship When
A=432 * Octave
below Octave
above
1/1 440 440.00 A Root 432 216 864
2/1 880 880.00 A Octave 864 432 1728
2/3 293.33 293.66 D Fourth 288 144 576
2/5 176 174.62 F Aug Fifth 172.8 86.4 345.6
3/2 660 659.26 E Fifth 648 324 1296
3/5 264 261.63 C Minor Third 259.2 129.6 518.4
3/8 165 164.82 E Fifth 162 (Phi) 81 324
5/2 1,100.00 1,108.72 C# Third 1080 540 2160
5/3 733.33 740.00 F# Sixth 720 360 1440
5/8 275 277.18 C# Third 270 135 540
8/3 1,173.33 1,174.64 D Fourth 1152 576 2304
8/5 704 698.46 F Aug. Fifth 691.2 345.6 1382.4

The calculated frequency above starts with A440 and applies the Fibonacci relationships. In practice, pianos are tuned to a "tempered" frequency, a man-made adaptation devised to provide improved tonality when playing in various keys. Pluck a string on a guitar, however, and search for the harmonics by lightly touching the string without making it touch the frets and you will find pure Fibonacci relationships.

* A440 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.


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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.




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Musical instruments are often based on phi


Fibonacci and phi are used in the design of violins and even in the design of high quality speaker wire.


The Fibonacci series was discovered by studying population growth
Population growth is also related to the Fibonacci series. It was the question of how fast rabbits could breed under ideal circumstances that Leonardo Fibonacci originally investigated in the year 1202. Here was the question he posed:

Suppose a newborn pair of rabbits, one male and one female, is put in the wild. The rabbits mate at the age of one month and at the end of its second month a female can produce another pair of rabbits. Suppose that the rabbits never die and that each female always produces one new pair, with one male and one female, every month from the second month on. How many pairs will there be in one year?

The answer is found in series of numbers now known as the Fibonacci series. Picture that pair A of rabbits gives birth to pairs B, C, D and E. Each of these in turn begins to give birth to other pairs B1, B2, B3, C1, and C2, who in turn give birth to B11, etc. At the end of each month, the total population of rabbits will be a number in the Fibonacci series:

Month Rabbits from A: from B: from C: D: B1: Total
0 A 1
1 A 1
2 A B 2
3 A B C 3
4 A B C D B1 5
5 A B C D E B1 B2 C1 8
6 A B C D E F B1 B2 B3 C1 C2 D1 B11 13
etc.
1 2 3 4 5 6 7 8 9 10 11 12 13 etc.


The Fibonacci series can be used to predict urban populations
It appears that the Fibonacci series can even be used to predict populations of major cities, as shown by the relationships of various U.S. urban areas in 1970:

Area
Census
Rank
Actual
Population
Predicted Population

Method 1
Method 2

New York, NE NJ
1
16,206,841

LA Long Beach CA
2
8,351,266
10,016,379
10,016,379

Chicago NW IN
3
6,714,578
6,190,462
5,161,366

Detroit, MI
5
3,970,584
3,825,916
4,149,837

Washington DC
8
2,481,459
2,364,546
2,453,956

Houston, TX
13
1,677,863
1,461,370
1,533,626

Cincinnati, OH
21
1,110,514
903,176
1,036,976

Dayton, OH
34
685,942
558,194
686,335

Richmond, VA
55
416,563
344,983
423,935

Las Vegas, NV
89
236,681
213,211
257,450

New London, CT
144
139,121
131,772
146,277

Great Falls, MT
233
70,905
81,439
85,982


Method 1 takes the population of the largest city and divides it again and again by phi. Method 2 takes the population of each successive city and divides it by phi.

Source: [link]

Multicellular organisms
In biology, once an egg is fertilized, it divides and multiplies in count until it reaches a point at which the ratio of the succeeding number of cells to the previous number of cells is phi (1.618 ...).
Human expectations occur in a ratio that approaches Phi
Changes in stock prices largely reflect human opinions, valuations and expectations. A study by mathematical psychologist Vladimir Lefebvre demonstrated that humans exhibit positive and negative evaluations of the opinions they hold in a ratio that approaches phi, with 61.8% positive and 38.2% negative.

Phi and Fibonacci numbers are used to predict stocks
Phi, the Golden Mean and Fibonacci numbers have been used with great success to analyze and predict stock market moves. Forbes ASAP recently featured a story on the work of scientist Stephen Wolfram in cellular automata (underlying rules that determine seemingly random phenomenon) stating "This seashell may hold the secret of stock market behavior, computers that think and the future of science."

Markets may be as geometrically perfect as a spider's web
Ermanometry Research shows the markets to be perfectly patterned, explaining that humans, being part of nature, create perfect geometric relationships in their behaviors, not unlike a spider spinning a geometrically perfect web with no conscious awareness of its amazing feat. Ermanometry applies the logarithmic spirals found in sea shells with dynamic ratios in 3D to relate one market move to others.

Phi, or Golden Ratio, patterns often define the timing of highs and lows and price resistance points
The golden ratio, or phi, appears frequently enough in the timing of highs and lows and price resistance points that adding this tool to technical analysis of the markets may help to identify key turning points. The photos below illustrate how the Golden Mean Gauge and Phi-based analysis software can be used to identify these turns in the market. The middle arm of the gauge keeps the phi point of the outer arms as the gauge is opened and closed. The lines of the phi-based software are all in phi relationship to one another. The ratios of Fibonacci numbers, commonly used in technical market analysis, converge on phi as explained on the Fibonacci Series page.


DJIA Daily Chart
from 1/2004
through 11/04
using PhiMatrix DJIA Monthly Chart
from 1/2000
through 6/2003
using a GM Gauge

Phi and Fibonacci numbers define the price 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, adding to the potential to identify the turns described above. 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 to 1/ and 0.382 is 1/2.

For additional information on Elliott Wave Theory, its application and related concepts, please consult the resources below.
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.
Note: A cubit is the measure of the forearm below the elbow.


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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.



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The Number 666 is related to Phi

Revelation 13:18 says the following:
"This calls for wisdom. If anyone has insight, let him calculate the number of the beast, for it is a man's number. His number is 666."

This beast, regarded by some as the Anti-Christ described by John, is thus related to the number 666, one of the greatest mysteries of the Bible.
Curiously enough, if you take the sine of 666º, you get -0.80901699, which is one-half of negative phi, or perhaps what one might call the "anti-phi."

The trigonometric relationship of sin 666º to phi is based on an isosceles triangle with a base of phi and sides of 1. When this triangle is enclosed in a circle with a radius of 1, we see that the lower line, which has an angle of 306º on the first rotation and 666º on the second rotation, has a sine equal to one-half negative phi.

In this we see the unity of phi divided into positive and negative, analogous perhaps to light and darkness or good and evil. Could this "sine" be a "sign" as well?

In addition, 666 degrees is 54 degrees short of the complete second circle and when dividing the 360 degrees of a circle by 54 degrees you get 6.66... The other side of a 54 degree angle in a right angle is 36 degrees and 36 divided by 54 is .666.

Phi appears throughout creation, and in every physical proportion of the human body. In that sense it is the number of mankind, as the mysterious passage of Revelation perhaps reveals.

Also see the Theology page.


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The colors of the Tabernacle are based on a phi relationship
The PhiBar program produces the colors that the Bible says God gave to Moses for the construction of the Tabernacle.

As it says in Exodus 26:1, "Make the tabernacle with ten curtains of finely twisted linen and blue, purple and scarlet yarn, with cherubim worked into them by a skilled craftsman."

Set the primary color of the PhiBar program to blue, the secondary color of the PhiBar to purple and it reveals the Phi color to be scarlet.

This reference to the combination blue, purple and scarlet in the construction of the tabernacle appears 24 times in Exodus 25 through 39, describing the colors to be used in the curtains, waistbands, breastpieces, sashes and garments.

Is there meaning hidden in Phi, the symbol for the Golden Number?
The use of the Greek letter Phi 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 Phi, 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 Phi? (Oddly enough, to type Phi on your computer, you hold the Alt key and enter 1000 on the number pad, an interesting "alt"ernate look at 1 with a trinity of 0's!)

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

Note: This original insight by the site author was added on 3/15/2003.
Adding Unity to nothingness produces the Fibonacci series, which converges on Phi
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 Phi as you move towards infinity, ∞!

Number in the series O l l 2 3 5 8 13 ... ∞
Ratio of
each number in the series
to the
previous number in the series ∞ l 2 l.5 l.66... l.600 l.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 Book of John begins with these words that capture the essence of this:

In the beginning was the Word,
and the Word was with God,
and the Word was God.

Jesus, in John 14:9, expressed a similar thought:

Anyone who has seen me has seen the Father.

Here the human Jesus (the Son of Man) is to the divine Jesus (the Son of God) as the divine Jesus (the Son of God) is to God (the Father or whole).

Insight on the relationship of Christ to God as analogous to the golden section contributed by Steve McIntosh.

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.
Phi creates a sense of beauty
Phi appears throughout life and the universe. Some believe that it is the most efficient outcome, the result of natural forces. Some believe it is a universal constant of design, the signature of God.

Whatever you believe, the pervasive appearance of phi in all we see and experience creates a sense of balance, harmony and beauty in the design of all we find in nature.

It should be no surprise then that mankind would use this same proportion found in nature to achieve balance, harmony and beauty in its own creations of art, architecture, colors, design, composition, space and even music.
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
It appears that the Egytians may have used both pi and phi in the design of the Great Pyramids. The Greeks based the 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. Phi is also the 21st letter of the Greek alphabet, and 21 is one of numbers in the Fibonacci series. The character for phi 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.
For those of you who just can't get enough of Phi, here it is to 20,000 places:

1.6180339887498948482045868343656381177203 0917980576286213544862270526046281890
2449707207204189391137484754088075386891 75212663386222353693179318006076672635
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The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of our cosmos. Unlike those abstract numbers, however, phi appears clearly and regularly in the realm of things that grow and unfold in steps, and that includes living things.
The decimal representation of phi is 1.6180339887499... .
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You can find it in a number of places:


Number Series
If you start with the numbers 0 and 1, and make a list in which each new number is the sum of the previous two, you get a list like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... to infinity-->


This is called a 'Fibonacci series'.
If you then take the ratio of any two sequential numbers in this series, you'll find that it falls into an increasingly narrow range:

1/0 = Whoa! That one doesn't count.
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538...
34/21 = 1.61904...
and so on, with each addition coming ever closer to multiplying by some as-yet-undetermined number.


The number that this ratio is oscillating around is phi (1.6180339887499...). It's interesting to note that the ratio 21/13 differs from phi by less than .003, and 34/21 by only about .001 (less than 1/10 of one percent!), thus providing our less technically-advanced ancestors an easy way to derive phi on a large scale in the real world with a high degree of precision.


Geometry
If you have a rectangle whose sides are related by phi (say, for instance, 13 x 8), that rectangle is said to be a Golden Rectangle. It has the interesting property that, if you create a new rectangle by 'swinging' the long side around one of its ends to create a new long side, then that new rectangle is also Golden. In the case of our 13 x 8 rectangle, the new rectangle will be (13 + 8 = ) 21 x 13. You can see this is the same thing that's going on in our number list, but when you discover it geometrically it looks downright magical. If you start with a square (1 x 1) and start swinging sides to make rectangles, you wind up with Golden rectangles without even trying. Here's the list, in case it isn't obvious:
1 x 1
2 x 1
3 x 2
5 x 3
8 x 5
13 x 8
21 x 13
34 x 21
and so on, with, again, each addition coming ever closer to multiplying by phi.
Ancient architecture is filled with Golden rectangles.
Unfold the Golden Rectangle
Text-only version (character-based graphics)

When you swing the long side of a Golden Rectangle around to create a new rectangle, the line it forms with the short side is made up of two sections having lengths of phi and one, respectively. This division of a straight line into a phi proportion is what is actually meant by the term 'Golden Section'.


Pure Math
Proportion is the relationship of the size of two things.
Arithmetic proportion exists when a quantity is changed by adding some amount.
Geometric proportion exists when a quantity is changed by multiplying by some amount.
Phi possesses both qualities.
If you study the Fibonacci series and the Golden Rectangle carefully, you will eventually realize that
phi + 1 = phi * phi.
Consider: suppose that you start with a Golden rectangl