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!!!!!
Pages to are hidden for
"Simple ideas"Please download to view full document