Cows Grazing in Elliptagons

Abstract

Summary If a string is looped around three or more pins placed at the vertices of a regular polygon, then the resulting curve traced by a pencil pulling the string taut is an amalgam of elliptic arcs, which we call an elliptagon. An elliptagon may be viewed as the boundary of the grazing area of a cow tethered to a loop of rope around a barn whose base is a regular polygon. We determine the grazing area for a cow, inside the elliptagon but outside the barn, as a function of the number of edges n of the polygon and the length L of the rope, and we show that elliptagons are smooth curves. We investigate the length of rope for which the grazing area is equal to the area of the base of the barn, and compute the limit of the ratio of this length to the perimeter of the barn as n approaches infinity.