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Derivation of Phi
Phi , in the words of Euclid:
"A straight line is said to have been cut in extreme and mean ratio (ratio of Phi) when, as the whole line is to the greater segment, so is the greater to the less"
Hopefully a picture will help us visualize what Euclid is saying.
Basically Euclid is saying that if you were to cut the line segment A above at the point, line segments B and C are created. Then for the special case where the ratio of line segments are related such that A/B = B/C, the ratio is Φ (Phi). So lets solve the problem algebraically to find the value of Φ.
A/B = B/C
noting that A=B+C (see diagram above) and substituting we get
(B+C)/B = B/C
which can be rewritten as
B/B + C/B = B/C
1 + C/B = B/C
for simplicity, lets call the ratio B/C = Φ, which gives us
1 + 1/Φ =Φ
solving,
1/Φ = Φ - 1
1 = Φ2 - Φ
0 = Φ2 - Φ - 1
which gives roots
Φ= (1 ± √5)/2 (I used the quadratic equation to solve this with a=1, b=-1, c=-1)
The positive root is traditionally taken to be the value of Phi
Φ=1.61803398874989.....
though the negative root is just as valid a solution
Φ= - 0.618033988749894....
Here is a link that gives phi to 50,000 places
http://www.cs.arizona.edu/icon/oddsends/phi.htm
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Re: Phi - The Golden Ratio Part I
