Archimedes the Numerical Analyst

Highlights:

Historic: Proposition 3 of Archimedes' Measurement of a Circle give an estimation of π, 223/71 < π < 22/7 Archimede's approximation 22/7 to π is accurate to an astonishing one part in 2484.

Mathematical: Hexagons are circumscribed and inscribed around a unit circle. The polygons are iteratively subdivided to give 12, 24, 38 and finally a pair of 96-gons. The semi-perimeters of these polygons give the upper and lower bounds for π, respectively. The semi-perimeters are iteratively approximated using trigonometric identities which Archimedes derives.

Prerequisites:

Foundational: None.

Equations:

x1	→	x1+√(1+x12)
x2	→	2 x2
x3	→	x2/x1
x4	→	x2/√(1+x12)

Variables:

x₁: cotangent of π/n.
x₂ : n, the number of sides of the polygons.
x₃ : semi-perimeter of an n-gon circumscribed about a unit circle.
x₄ : semi-perimeter of an n-gon inscribed in a unit circle.

Derivation of the Equations:

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. By geometric argument, P_n = n /(cot π/n)
p_n = n /(csc π/n) Archimedes did not use the notation of trigonometric functions. Greek mathematics used the ratios which only much later were given the names familiar today. Archimedes starts from an excellent approximations for the angles associated with the hexagon, cot π/6 = √3 > 265/153 = 1.7320261438
csc π/6 = 2 and iteratively applies a half-angle formula to get semi-perimeters for 12, 24, 48 and 96-gons. The rational approximation to √3 is from Archimedes, and we do not know how he arrived at this raio.

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. In his work, Archimedes shows explicitly only the first of these identites. The second is inferred from the his calculations. The two trignometric identities can be combined into a single formula, cot x/2 = cot x + √(1 + cot x²)

Error Analysis:

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)) When n ≥ 6 we have the bounds, 3 < P_n < 2 √3
3 < p_n < π This gives a simpler inequality for the error, (P_n - p_n) < 2 (π/n)². As a numerical illustration, for n=96 the error bound is 0.00214184, and the actual error is, (P_n - p_n) = 0.00168265.

Dynamics:

Figure 1. To animate this image in your local Phaser,
Load Gallery URL: http://www.phaser.com/modules/historic/archimedes/archimedes.ppf

Look at the xi values first. Note the converge of x₃, the semi-perimeter of the circumscribed n-gon,and x₄, the semi-perimeter of the inscribed n-gon. Now look at the values using xi vs. time.

Note: the equations have been modified slightly to make the Xi values easier to read. The initial n is one half the true initial n, and the equations giving the semi-perimeters compensate with a factor of two. In this way, each line reads with the current value of n, P_n and p_n. The cotangent (x₁) is, however, the next value of the cotangent.

Experiments:

Change the initial condition for x₁ to a more accurate value of √3. Note how the final value for π is more accurate.

Archimedes began with a hexagon. Change the initial conditions to begin with squares.

Add an equation for the error. Verify the quadratic convergence.

Related Modules:

AGM

References:

ALFELD, P, Archimedes and the Computation of Pi, http://www.math.utah.edu/~alfeld/Archimedes/Archimedes.html

BERGGREN, L., BORWEIN, J., and BORWEIN, P. [1997]. Pi: A source book, Springer-Verlag.

DIJKSTERHUIS, E. J. [1987]. Archimedes, Princeton University Press.

GOURDON,X. and SEBAH, P, Numbers, Constants and Computation, http://numbers.computation.free.fr/Constants/constants.html

NOVA, Infinite Secrets, http://www.pbs.org/wgbh/nova/archimedes/pi.html

O'CONNOR, J. and ROBERTSON, E. F., Archimedes of Syracuse http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.html

PHILLIPS, G. M., [1981], Archimedes the nuermical analyst, in BBB [1997]. pp. 15-19.

ROSENBERG, B. [2003]. Archimedes and Pi PDF

THE WALTERS ART GALLERY, Archimedes Palimpsest, http://www.thewalters.org/archimedes