Given a polygon with an even number of sides, the derived polygon is obtained by joining the points which are a fractional distance r along each side. If r = 1/2, then the derived polygons are called midpoint polygons and tend to a shape with opposite sides parallel and equal in length. Furthermore, alternate polygons have approximately the same length, and the original and all derived polygons have the same centroid. Amazingly, if r!=1, the derived polygons still approach a shape with opposite sides parallel and equal in length, and all have the same centroid. The above illustrations show 20 derived polygons for ratios r = 0.3, 0.5, 0.7, and 0.9. More amazingly still, if the original polygon is skew, a plane polygonal is approached which has these same properties.

midpoint polygon | nested polygon | whirl