Golden Ratio


Golden Ratio

Phi, like Pi, is a ratio defined by a geometric construction

 In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,

where the Greek letter phi ( or ) represents the golden ratio. Its value is:
\varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .

 We find the golden ratio when we divide a line into two parts so that:

 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 larger line segment (B)
to the length of the smaller line segment (C).

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.



Interesting fact:the Golden Ratio is also equal to 2 × sin(54°), get your calculator and check!

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