Written: April-May 2018

This paper presents a collection of experimental results regarding extensions of the random polgon phenomenon conducted by the students 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, 151-170.

ToDo List

• Solve the mystery of the 45 degrees claimed in Van Loan's article. See if his process is similar to ours.

• Turn the file C25a.txt, renamed IteratedPolygon.txt into a semi-professional Maple package, in the format that Yukun Yao is familiar with.

• Study more general averaging processes (not necessarily simple average), and weighted average of more than two consecutive points in the polygon

• Create graphing procedures (if possible animated) that show the convergence of the random polygon to an oval. Display the outputs in this webpage.

• What happens in 3D, even higher?

• Can you find "explicit" or closed-form expressions for the characteristic polynomials of the relevant matrices?, and hence (at least implicitly) for the eigenvalues (and who knows? for the eigenvectors?)

• Generalize Edna Jones' proof and Ahshan Khan's proof to these more general scenarios.

• Anything else you can think of (of course, related to this project)

Maple package

• IteratedPolygon.txt (Under Construction, very preliminary)

Sample Input and Output Files

• COMING UP

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