thoughts on:

*Pi and Phi*

**Pi** is the ratio of the diameter
to the circumference of a circle. It is usually represented by the Greek
letter *pi*. There are many
ways to derive the value and the calculation of ** pi** has been
refined during history from a simple 22/7 ratio to computer calculations
running to many millions of digits.

*3.14159 26535 89793 23846 26433 83279 50288
41971 *etc..

There are many ways to calculate this unique number.

For example

* By continuing the series 1 - 1/3 + 1/5 - 1/7 + 1/9 and multiplying the
result by four gives pi.

* Adding the reciprocals of the squares (i.e. 1/1 + 1/4 + 1/9 + 1/16
+ 1/25 ....) gives (pi^2)/6.

Phi is also known as the *Golden
*or* Divine Ratio*, and has been used in many models for patterns
found in nature and artistic concepts. Mathematically, it is half of the
square root of five, plus one....

(5^(1/2)+1)/2 = 1.618034.....

Phi may be derived from the Fibonacci series, as it is the ratio of increase from the series:

*1,2,3,5,8,13,21,34,55,89,144............*

Phi is the only number which is one
greater that its reciprocal.

*phi = 1 + (1/phi)
which is 1.618...... = 1 + (1/1.618.....)*

*Pi *and *Phi* are connected mathematically,
for *pi* may be calculated from the Fibonacci sequence

The Fibonacci sequence is defined by the formula *u(0)
= 0, u(1) = 1
u(n+1) = u(n) + u(n-1)
*So

This connects to pi in the following manner:

Calculate all the Fibonacci numbers from *u(1)* up
to *u(m)*.

Multiply all the results together and call this *P(m).
*Find the Least Common Multiple

For *m=7 P(m) *is 1*1*2*3*5*8*13*21 = 32760

LCM(m)=10920 (i.e. 3*5*7*2*2*2*13)

*6*(log32760/log10920) = 6*4.515/4.038 = 6.709.* Square root is 2.590
= *Z(m)*

For *m=8 P(m) *is 1*1*2*3*5*8*13*21*34= 1113840

LCM(m)=185640

*6*(log1113840/log185640) = 6*6.0468/5.2687= 6.886. *Square root is
2.6241 = *Z(m)*

This serves to show the method for low numbers. As the
values of *m* increase eventually *Z(m)* reaches ** pi**.

*A sort-of-related afterthought:* A formula found by the Swiss mathematician Leonhard Euler in 1748, states that when one is added to the transcendental number "e" (roughly 2.71828) raised to the power of Pi (roughly 3.14159) multiplied by the square-root of minus one, the result is precisely zero.

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