Normalized Iterated Averaging Polygons


Yonah Biers-Ariel, Matthew P. Charnley, Brooke Logan, Anthony Zaleski,and Doron Zeilberger

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Written: June 13, 2018

Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

Inspired by a 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

Comment (Feb. 8, 2019): This beautiful article (with the last author's name deleted) was submitted to the expository journal Mathematics Magazine, and was stupidly and narrow-mindedly rejected by its editor, Michael Jones, because it "did not contain enough new material". The goal of Mathematics Magazine is to communicate beatiful math, not to report on new research.

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