Education

Abstract

Take a random polygon, find the the midpoints of its sides, and construct a new polygon from them. Repeat ad infinitum. What happens? This simple geometric iteration is at the heart of this issue's Education section offering, “From Random Polygon to Ellipse: An Eigenanalysis” by Adam N. Elmachtoub and Charles F. Van Loan. A cursory numerical experiment (although the traditionalist may wish to try this with a compass and straightedge) shows that the polygon shrinks to a point which the astute observer will realize is the centroid of the original vertices. If we now zoom in on the sequence of shrinking polygons, most easily accomplished by rescaling at each iteration, we see even more order. The rescaled vertices approach a binary oscillation between two polygons, both of which can be inscribed in the same ellipse. While students posed with these questions may think they are pursuing a geometry problem, the not-too-secret agenda is to introduce them to some fundamental ideas of numerical linear algebra, namely iteration, rescaling, and convergence. The authors guide us to write the problem as an iterative algorithm whose properties can be analyzed via eigenvalues and eigenvectors and whose convergence is akin to the power method for finding the eigenvalue of largest magnitude of a matrix. One might consider threading this example through a course in numerical analysis or numerical linear algebra, having the students explore the behavior at the beginning of the course (a great programming exercise), and eventually leading them through the analysis after introducing iterative methods for computing eigenvalues and eigenvectors. The problem opens the door to many generalizations, including exploring what happens for different geometric constructions and asking for geometric interpretations for other iterative linear algebra algorithms. The midpoint and rescaling construction is also simple enough to show to freshmen or even high school students; who knows, perhaps a few of them may be drawn to the study of mathematics by the allure of binary blinking ellipses?