THE PLATONIC METAPHYSICS AND THE CUBE DUPLICATION

Doubling the cube was, along with the angle trisection and circle squaring, one of the three classical problems of Greek geometry whose solutions were sought using only compass and straightedge. A significant development in finding a solution to the problem was the discovery by Hippocrates of Chios that it is equivalent to finding two mean proportionals between a line segment and another with twice the length. However, the solution of Archytas is the most remarkable of all because it is not a construction in a plane but a bold construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution.