http://search.deviantart.com/?q=by:WhiteTigerWolfDragon&section=today deviantART RSS for by:WhiteTigerWolfDragon en-us Copyright 2009, deviantART.com Wed, 23 Dec 2009 10:35:10 PST deviantART.com http://blogs.law.harvard.edu/tech/rss http://s.deviantart.com/minish/widgets/apple-touch-icon-precomposed.png http://WhiteTigerWolfDragon.deviantart.com/journal/9841955/ http://WhiteTigerWolfDragon.deviantart.com/journal/9841955/ Thu, 24 Aug 2006 18:25:32 PDT 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.


--------------------------------------------------------------------------------

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

--------------------------------------------------------------------------------

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
... ]]> ~WhiteTigerWolfDragon http://WhiteTigerWolfDragon.deviantart.com/journal/9680498/ http://WhiteTigerWolfDragon.deviantart.com/journal/9680498/ Thu, 10 Aug 2006 16:23:12 PDT My name is White Tiger, and I am a multmillionare.
I live in a 40 million dollar mansion over-looking the English Channel.
I have over 100 million dollars!
I do hope that I have the up most respect on this site, or Deviantart Will be seeing and hearing from my lawyer.
Thank you.
~White Tiger~ ]]> ~WhiteTigerWolfDragon