Module: Gauss AGM Pi MAP

Gauss and the fastest iteration to Pi


Historic: In 1798 Gauss culminated his investigations of a curve known as the lemniscate by finding that its circumference is related to that of the circle by the result of applying a process called the Arithmetic Geometric Mean (AGM) to the number 1 and √2. This connections was so intriguing, that Gauss wrote,
We have gained some very elegant details about the leminscate, which have exceeded all expectations, and indeed using methods which open up an entirely new field. [...] A new field of analysis stands before us [...]

Because of the speed of convergence of the AGM process, as an application, he gives the fastest known algorithm for the computation of π. Today, over 200 years later, it is still the fastest.

World records useing AGM

Mathematical: Gauss's AGM Map was the confluence of two ideas. First, he very early noticed that taking the arithmetic and geometric means of two numbers gave two closer numbers, and this process converged quickly to the so called arithmetic-geometric mean (AGM). Later he notriced that the AGM of 1 and √2 was the ratio of the arc length of a circle to that of a lemniscate (a closed curve described in polar coordinates by r2 = cos 2Θ).




x2√(x1 x2)
x3x3 - x4 (x1-x2)^2/4
x42 x4
x5(x1+x2)^2/(2 x3)


x1: the arithmetic mean
x2 : the geometric mean
x3 : the arc length of a lemniscate
x4 : 2^k
x5 : the k-th convergent to π

Derivation of the Equations:


Figure 1. To animate this image in your local Phaser,
Load Gallery URL:


Related Modules:

Archimedes' Pi MAP; Bournoulli Lemniscate; Elliptic Integrals. World record for Pi using Gauss AGM.


ARNDT, J, and HAENEL, C [2000], π - Unleashed, Springer.

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


Please write to us with your comments on this module.