Connecting the midpoints of a polygon in plane or space, and iterating this process, yields a sequence of polygons which converge to the center of gravity of the initial object. This Mathematica notebook, which has been converted to html, contains a number of programs for studying this phenomenon. In particular, here are programs for graphing sequences of midpoint polygons both in the plane and space, and computing the minimum number of iterations needed to reach a convex polygon. See [1] for the mathematical theory behind this phenomenon, which involves stochastic matrices.
[1] Davis, Philip J., Circulant matrices. A Wiley-Interscience Publication. Pure and Applied Mathematics. John Wiley & Sons, New York-Chichester-Brisbane, 1979.