Iterating a function like 1+1/x gives a sequence which converges to the Golden Mean but does so at a much slower rate than those sequences derived from Newton's method or the secant method. There is, however, a surprising relation between all these sequences. This relation, easily explained by the use of good notation, is generalized by means of Pascal's "Mysterium Hexagrammicum". Throughout, we make contact with many areas of mathematics and physics including abstract groups, calculus, continued fractions, differential equations, elliptic curves, Fibonacci numbers, functional equations, fundamental groups, Lie groups, matrices, Moebius transformations, pi, polynomial approximation, relativity, and resistors.
This article has been published in the March 2002 issue of American Mathematical Monthly.