POEMs and Newtons Aerodynamic Frustum

There are numerous situations in the literature in which the golden mean has been naively interpreted as an indicator of optimal beauty; but only a few in which this mean arises as the solution of a true optimization problem. In this article we present one such problem. We demonstrate a close relationship between the golden mean and a special case of Newtons aerodynamical problem for the frustum of a cone. Then, we exhibit a parallel relationship between the general case of this problem and a family of means called POEMs (for pth Order Extreme Means). This shows that the POEMs not only share with the golden mean the algebraic and geometric properties established by Falbo, but also a remarkable minimizing physical property. Sadly, we have another situation in which an interesting property of the golden mean is matched by those of the POEMs.