The magic number Phi-1.68103399?

Question

the number Phi(2cos36)
what is this number really.how did people find this?

Answer 1

I think you mean Φ = 1.61803399 and it is commonly called the 'golden ratio'.

If you think of a rectangular piece of paper with sides of 1 and 1.61803399 (in this ratio) and you cut out a square, you end up with a piece of paper that is 0.61803399 and 1. And this is the same ratio as the original rectangle. See the first link, for a picture.

phi can also be written as: ( 1 + √5) / 2 = 1.61803399...

Using the example above with the rectangular paper:
(Φ - 1) / 1 = 1 / Φ
Φ ( Φ - 1) = 1
Φ² - Φ - 1 = 0

When you use the quadratric equation to solve for Φ, you get:
Φ = (1 + √5) / 2 = 1.61803399...

And this ratio shows up in lots of places in nature such as the ratio between a trunk and a branch of a tree. Or the branch and its twigs, etc.

If you take the ratio between terms of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.) the number gets closer and closer to Φ.

Various mathematicians noticed this ratio, so you can't really determine who "found" it. There's more info in the Wikipedia article, or at the MathWorld site. Links below:

Answer 2

It is human fascination with the infinite that also leads us to perhaps the most "underrated" irrational number in mathematics: Phi. More commonly called the "Golden Ratio" or "Golden Section," Phi has a numeric value of approximately 1.618. Although not as well known as pi or e, this number can be found in architecture, art, and nature. Examples are plentiful, but we should first examine Phi's definition and history in order to get a better understanding of its appeal and application. Phi can be defined as the only known number that is one more than its own multiplicative reciprocal:

The unique result that this creates is that Phi is 1.618033? and its inverse is 0.618033?. Having no proper decimal representation, the digits of these two numbers are the same infinitely. Beyond this unique algebraic representation of Phi, the Golden Ratio can be written simply in terms of fundamental trigonometric functions: Phi is equal to 2 times the sine of 54 degrees, as well as 2 times the cosine of 36 degrees. The fact that the expression of Phi is so mathematically simple is surprising when we realize the extent to which it exists in everyday life.

Answer 3

I agree with the above. See the Wikipedia entry for more information.
Except for the part about Phi being the only known number that is one plus it's multiplicative reciprocal. What about -(1/Phi)?

Answer 4

1st, it is 1.618034

It can be found from the fibbibacci series:
1,1,2,3,5,8,13,21,34,55,89.... where each term is the sum of the 2 previous terms. The ration of adjacent numbers is aprox the golden ratio. the farther out you take the, the closer it gets, ex:
3/2=1.5, 5/3=1.67,8/5=1.625, 89/55=1.61818

It can also be found from the property:
f-1=1/f
f^2-f=1
f^2-f-1=0
f=(1+sqrt(5))/2=1.618034
or f=(1-sqrt(5))/2=-.618034

The derivation from 2cos(36) is new to me, I'll have to work on this 1 for a while.