Convergence of sequences of polygons

Abstract

In 1878, Darboux studied the problem of midpoint iteration of polygons. Simply put, he constructed a sequence of polygons ${\Pi }^{\left(0\right)},{\Pi }^{\left(1\right)},{\Pi }^{\left(2\right)},\dots$ in which the vertices of a descendant polygon ${\Pi }^{\left(k\right)}$ are the midpoints of its parent polygon ${\Pi }^{\left(k-1\right)}$ and are connected by edges in the same order as those of ${\Pi }^{\left(k-1\right)}$. He showed that such a sequence of polygons converges to their common centroid. In proving this result, Darboux utilized the powerful mathematical tool we know today as the finite Fourier transform. For a long time period, however, neither Darboux’s result nor his method was widely known. The same problem was proposed in 1932 by Rosenman as Monthly Problem # 3547 and had been studied by several authors, including I. J. Schoenberg (1950), who also employed the finite Fourier transform technique. In this paper, we study generalizations of this problem. Our scheme for the construction of a polygon sequence not only gives freedom in selecting the vertices of a descendant polygon but also allows the polygon generating procedure itself to vary from one step to another. We show under some mild restrictions that a sequence of polygons thus constructed converges to a single point. Our main mathematical tools are ergodicity coefficients and the Perron–Frobenius theory on nonnegative matrices.