Simple Ideas That Generates

  • Uploaded by: mayur
  • 0
  • 0
  • May 2020
  • PDF

This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA


Overview

Download & View Simple Ideas That Generates as PDF for free.

More details

  • Words: 442
  • Pages: 3
- the most fundamental constant in the universe.

SIMPLE IDEA THAT GENERATES



  





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 m= length of the base of the triangle C= circumference of the circle

(180o/n) L m (360o/n)

m/2 = sin (180o/n)

Using trigonometry, we get: Approximately we have, L ≈ m

i.e. m = 2sin (180o/n) ∴ 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) This is a quadratic equation in X2n X2n

2

1 − ( X 2 n /(2n) 2

, solving which we get,

= 2n2 ± 2n n 2 − X n …………... (D) 2

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, X24

2

= 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!!!!!

Related Documents


More Documents from ""