Archimedes and Pi

Introduction

Proposition 3 of Archimedes' Measurement of a Circle give an estimation of π,

    223/71 < π < 22/7

The approximation π_a = 22/7 is referred to as Archimedes Approximation and is very good. It has been reported that a 2000 B.C. Babylonian approximation is π_b = 25/8. We will compare these two approximations. The author, in the spirit of idiot's advocate, will venture his own approximation of π_c = 19/6. The Babylonian approximation is good to one part in 189, the author's, one part in 125, and Archimedes an astonishing one part in 2484.

Archimedes' approach is to circumscribe and inscribe regular n-gons around a unit circle. Let P_n be the semi-perimeter of a regular n-gon circumscribed around a unit circle. Let p_n be the semi-perimeter of a regular n-gon inscribed in a unit circle. Then,

    P_n = n /(cot π/n)
    p_n = n /(csc π/n)

and with increasing n the approximations become very close,

    p_n < π < P_n

Archimedes derives and uses a cotangent half-angle formula and a cosecant identity,

    cot x/2 = cot x + csc x,
    csc² x = 1 + cot² x.

to calculate P_n and p_n for n = k 2ⁱ, starting from a k-gon. From the geometry of hexagons,

   cot π/6 = sqrt(3).

Archimedes then calculates to n = 96.

Discussion

The two trignometric identities can be combined into a single formula,

   cot x/2 = cot x + sqrt(1 + cot x²)

This formula is used for the semi-perimeter calculation of both the inscribed and circumscribed polygons. The square root will be approximated, with a bias consistent with the direction of approach to π. If P_n^* is the estimate of P_n, then P_n^* > P_n Likewise, if p_n^* is the estimate of p_n, then p_n^* < p_n must be maintained. The half-angle formula respects the direction of error,

  If c >[<] cot x, then (c + sqrt(1+c²)) >[<] cot x/2.

Assuming that the rounding errors are not too large, the cosecant formula can be used to give a simply analysis of the converge around π of the pair of semi-perimeters. Write the identity as,

  (1/p_n² - 1/P_n²) = 1/n².

Then,

  (P_n - p_n) = (P_n p_n)² / ( n² (P_n+p_n))

The simple bounds,

  3 < P_n < 2 sqrt(3)
  3 < p_n < π

true when n ≥ 6 simplifies the error bound,

  (P_n - p_n) < 2 π² / n².

As a numerical illustration, for n=96 the error bound is 0.00214184, and the actual error is,

  (P_n - p_n) = 0.00168265.

Using Phaser

In the equation editor, write the following system of equations,

   x1 -> x1 + sqrt(1+x1*x1)
   x2 -> (6*2^t)/x1
   x3 -> (6*2^t)/sqrt(1+x1*x1)
   x4 -> 1/sqrt(x2-x3)

The cotangent for successive half-angles in given by x1. It is the only iteration formula of the four. Variables x2 and x3 are the semi-perimeters of cicumscribed and inscribed n-gons, where n = 6 × 2^t. Variable x4 measures the convergence of the two semi-perimeters, and the discussion of the previous section indicates that it should be proportional to n.

Set the initial value to sqrt(3) = 1.73205. Run the iteration in the Xi values view. Note that P_n converges decreasing and p_n converges increasing. Note how the inverse error measure x4 doubles with each time step, as the number of faces doubles.

Set two additional initial values, one below and one above sqrt(3). Run the iteration in the Xi vs. Time view.

Last Update: 6 August 2003
Burton Rosenberg