Written: April-May 2018
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger. The article in this journal can be found here.
This paper presents a collection of experimental results regarding extensions of the random polygon phenomenon conducted by the students in Dr. Z.'s Spring 2018 Experimental Mathematics Class.
Inspired by the fascinating paper Untangling Random Polygons and Other Things by Charles F. Van Loan (SIAM News, April 2018, p. 4), and the earlier paper referenced there (joint with A.N. Elmachtoub), SIAM Rev. 52, 151-
Abstract: Inspired by the beautiful article by Adam Elmachtoub and Charles Van Loan, we take an initial polygon consisting of n points in the Cartesian plane with centroid at the origin and define an averaging procedure, generating a new polygon out of the midpoints of each segment that defines the initial polygon. We perform an analysis of this procedure, corroborating previous results on this procedure when the polygon is normalized at each step. We also introduce a new averaging procedure suggested by the power method which also converges to an ellipse but can be in any orientation. We conclude by considering other averaging procedures and analyze the results that come from these generalizations as well.