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

TLDR

The problem of determining the possible morphological types of convex polyhedra in three‑dimensional Euclidean space is known to be quite hopeless, and even more specialized questions about faces of various polygons remain elusive. We lack any general method to determine whether a convex polyhedron exists with given numbers of triangular, quadrilateral, and higher‑gon faces. No other information.

Abstract

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.