The Number of Hexagons and the Simplicity of Geodesics on Certain Polyhedra

The problem of determining the possible morphological types of convex polyhedra in three-dimensional Euclidean space E 3 is well known to be quite hopeless. We lack not only any general way of determining whether there exists a convex polyhedron having as faces ƒ 3 triangles, ƒ 4 quadrangles, . . . , and ƒ n n -gons, but even much more special questions of this kind seem to be rather elusive.