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					    SIMPLE IDEA THAT GENERATES            - the most               ∏

           fundamental constant in the universe.
                                                                                    


             Consider a circle of radius 1 unit being divided into “n” number of congruent
triangles i.e. the circle circumscribes a “n” sided regular polygon as shown in the figure.




Consider now one such triangle in this circle. See the angles notified inside it.




                                             L= length of the arc subtended
            (180o/n)                         m= length of the base of the triangle
                                 L           C= circumference of the circle
  m
                (360o/n)


Using trigonometry, we get:              m/2 = sin (180o/n)

                                     i.e. m = 2sin (180o/n)
Approximately we have, L ≈ m

                                  ∴ L ≈ m = 2sin (180o/n)

   there are “n” such arcs, we get:     n × L=C

                                 i.e. n × (2sin (180o/n)) = C

                       C/(diameter) = (n × (2sin (180o/n)))/ (diameter)

But diameter = 2 units and hence,
                         C/ (diameter) = (n × (2sin (180o/n)))/2

                  ∴ C/ (diameter) = n × sin (180o/n)……………… (A)
     R.H.S of equation A will reveal that it is only dependent
on the variable “n”. So if “n” is kept constant for every such
circle then,                          C/ diameter = constant for
every circle in the universe.
       Let us call this quantity as something unknown Xn as it
solely dependent on “n”.
                 ∴ C/( diameter) = Xn = n × sin (180o/n)…………… (A)

      Now what if we double the number of triangles….. then equation A becomes,

                 ∴ C/ (diameter) = 2n × sin (180o/2n)……………… (B)
We know that,
                                 Sin ( θ ) = 2 sin ( θ /2) cos ( θ /2)
                              Sin ( θ ) = 2 sin ( θ /2) 1 − sin 2 (θ / 2)

If θ = (180o/n), then

                   Sin (180o/n) = 2 sin (180o/n) 1 − sin 2 (180o /n) … (C)

From equation A, B and C,
                              Xn/n = X2n/(2n)      1 − ( X 2 n /(2n) 2

This is a quadratic equation in X2n     , solving which we get,
                            = 2n2 ± 2n n 2 − X n …………... (D)
                       2                                   2
                 X2n

Initially if we select n = 6, i.e. a hexagon of side 6,

                                    X6 = 6 × sin (180o/ 6) = 3

Putting this in equation D, we obtain for a circle circumscribing 12 sided regular polygon.
                    2
                 X12 = 9.646170928                  X12 = 3.105828541

put n = 12,
                        2
                  X24       = 9.813362029               X24 =   3.132628613
Similarly putting n = 24, 48, 96, ……., 6144, we get,

              X122882 = 9.8696044                      X12288 = 3.141592653
The unknown quantity X12288 is the value of Π correct upto 8
decimal places….

  EUREKA!!!! YOU JUST SAW An UNKNOWN
   QUANTITY DEVELOPED INTO THE MOST
FUNDAMENTAL CONSTANT WITHOUT ACTUALLY
               KNOWING IT.
             SIMPLE ENOUGH!!!!!

				
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posted:3/23/2010
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