April 1, 2014 - In the Fall of 2013, ICERM had a semester program on experiental geometry, topology, and dynamics. One of the many projects undertaken during this semester was the study of polygon iterations - geometrically defined maps on the space of polygons.

Perhaps the simplest polygon iteration is the midpoint map. Starting with an n-gon P1, we let P2 be the n-gon whose vertices lie at the centers of the edges of P1. The construction can be done over and over again, producing a sequence {Pn} of polygons which shrink to a point. If these polygons are rescaled (and suitably rotated) so as to have unit diameter, then, for almost every choice, they converge to an affinely regular n-gon. That is, the limit has the form T(Rn), where T is an affine transformation and Rn is a regular polygon.