World order from chaotic beginnings

Extract

This article explains the variety of patterns seen in sequences defined by for bilinear functions of the form . The initial confusion of very different patterns can be understood by generalising from the real numbers to the whole complex plane, and simplified further by projecting each number from the plane onto the surface of a sphere. The sphere is finally projected onto a Mercator map on which the iterative sequences take on the simplest possible pattern, creating a link with the problem of rhumb-line navigation. The examples given below are typical, and should first be tried as a reminder of what can happen.