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.




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)





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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