Approximation of ∏ – Archimedes Algorithm

I wish to start this blog by describing the Archimedes method of calculating the perimeter of circle with a diameter of one unit, which happens to be equal to ∏.  You might ask why to study an old algorithm? or how does a perimeter of circle help us in today’s applications?. You will surely appreciate the below reasons to study this algorithm.

• The connection between the perimeter of a circle to the origin of signal processing can be revealed by looking  at Archimedes method.
• Sine waves have a fundamental period of 2*∏.  Therefore ∏ (Perimeter of circle with a diameter of one unit) becomes vital in signal processing
• Learning history will motivate us to learn advance algorithms.

In this article i will help you to understand the first bulleted point.

Archimedes of Syracuse contributed quite substantially to the theory of mechanics and mathematics. One of his greatest achievement is his method of calculating the perimeter of a circle with a diameter of one unit, which happens to be equal to ∏. He inscribed a hexagon in a circle of diameter of one unit and divided the hexagon into six equilateral triangles as shown in figure 1. Since the diameter of the circle is one unit , the sides of each triangle is half a unit long.

Therefore, perimeter of the inscribed Hexagon = 6 * 1/2 = 3 units

Archimedes concluded that the perimeter of the circle ∏, is larger than 3, the perimeter of hexagon.

a6 = 3 < ∏

Where ‘a6′ is the perimeter of the inscribed hexagon

He then circumscribed the circle by another hexagon as illustrated in figure 2.

To find out the perimeter of the larger hexagon, Pythagoras theorem helped him.

The Length of the sides of the equilateral triangle ABC is calculated as follows:

In right angled triangle ABD,

L^2 = (L/2)^2 + (1/2)^2 [ Pythogoras theorem]

Upon simplification,

L^2 = 1/3 , which implies L = 1/√3

Therefore the perimeter of the larger hexagon is 6*( 1/√3)= 2*√3 = 3.4641

From this Archimedes concluded that the perimeter of circle would be less than 3.4641, i.e.,

a6=3< ∏ <3.4641=b6, where ‘b6′ is the perimeter of the larger hexagon.

Most of us would have stopped at this point. However Archimedes thought about the next step. He doubled the sides of the inside hexagon to obtain a 12 sided polygon and calculated the perimeter of the inside polygon(a12). Similarly he doubled the sides of of the outside hexagon and calculated the perimeter of outside polygon (b12). He found out that the perimeter of outside polygon (b12) is given by the harmonic mean of a6 and b6.

Harmonic mean is defined as the reciprocal of the arithmetic mean of the two numbers, i.e,

b12 = 1/1/2*(1/a6+1/b6) = 2*a6*b6/a6+b6                  ——–  (1)

and a12 is given by the geometric mean of a6 and b12, i.e;

a12 = √a6*b12                                                                ——– (2)

Using equations (1) and (2) we get,

b12=3.2154 and a12=3.1058. Therefore ∏ value should lie between,

3<3.1058<∏<3.2154<3.4641

Obviously we know that a 12-sided polygon will provide a closer bound on the perimeter of circle compared to the 6 sided polygon.

The genius also noted that equations (1) and (2) readily extend to 24, 48, 96 sided polygons. Infact the relation holds good for 2n sided polygons which is given as,

b2n = 2*an*bn / an+bn

and

a2n = √an*b2n

Using these relations the lower and upper bounds of ∏, given in table 1 is obtained

The average value of ∏ for the 96 sided polygon is referred as the archimedes algorithm. It has an erorr of about 0.008%.

Relation to signal processing:

The bounds of ∏ is due to the discrete functions of ‘n’ and there averages are successive approximations of the perimeter of the circle. The successive approximations using geometric and harmonic mean is a method of interpolation and the discrete functions of ‘n’ can be depicted as the discretization process in modern terminology.

Approximation of circle interms of polygon was done independently in china where Tsu Chung-Chi(430-501) found the more precise bound

3.1415926<∏<3.1415927

and said that 22/7 is an inaccurate value of ∏ and termed 355/113 as the accurate value of ∏

Software:

Access the below link to download the ‘C’ program. The program works upto 6144 sided polygons.

Download Archimedes method

References:

1. A.Antoniou, On the Roots of Digital signal processing – Part 1
2. Indexes of Biographies. The Mactutor History of Mathematics Archive