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!!!!!