User:Harishamur/Pi-Sequence

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Pi-Sequence

Geometry of regular polygons provides a method to identify a sequence of real numbers starting with 2√2 and converging to π. For simplicity we call this sequence π-sequence.

Consider a regular polygon with even number of sides(RPES) inscribed in a circle. Each such polygon has a diagonal which divides the area of the polygon into two equal parts. This is analogous to diameter dividing the area of a circle into two equal parts.We set:

R_2n = Perimeter of RPES of sides 2n/ Length of the diagonal of the RPES We claim that the formula for R is given by

1) R_2n = 2n sin 90/n, n=2,3,4, . . .

As n tends to infinity , the number of sides of RPES increases, but the size of each side goes on decreasing and since RPES is inscribed in a circle it ultimately coincides with the circle.

We give illustrations which set the pattern for arriving at the general formula.

For n=2, RPES is a square . Consider half the part of a square with side a. We need to find the length of the diagonal. Let O be the centre of the diagonal, we denote its length as 2x. Draw OM perpendicular to side AB. Clearly AM=a/2 and AMO is a right angled triangle. Hence x= a/2sin45. Thus

R₄=4a/2x= 4 * sin45 = 2√2

Formula (1) gives the same result for n=2,

For n=3, RPES is a hexagon. The part of the hexagon under the semicircle is divided into 3 equilateral triangles each of side a . The angle subtended at the centre by each triangle is 60⁰. We need to find the length 2x of the diagonal AD. Draw OM perpendicular to the side AB.Then

x sin30=a/2 , R₆= 6a/2x = 6*sin30=3

Formula (1) gives this for n=3.

The pattern of the geometrical proof is the same for octagon(n=4), decagon(n=5) and so on.

Using the formula(1) we give a few more computations to show that the terms in the sequence gradually go on increasing and reach towards the limit π=3.142. . .

R₈= 8 * sin(22.5) = 3.061 R₁₀ = 10 * sin(18) = 3.09 R₁₂ = 12 * sin(15) = 3.105 R₁₄ = 14 * sin(12.8) = 3.110 R₁₆ = 16 * sin(11.25) = 3.121 . . . . . . . . . . . . R₉₀ = 90 * sin(2) = 3.1401 R₁₈₀ = 180 * sin(1) = 3.1410 So the π-sequence is: 2√2, 3, 3.061, 3.09, 3.105,. . . 3.1401,. . . 3.1410. . . π

References

External links

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