Geodesic lines on a bounded closed convex polyhedron

Any equilateral tetrahedron has infinite non-self-intersecting geodesic lines. We shall see below how this follows from a consideration of the network of such a tetrahedron. The main result to be proved in this paper (cf. theorem 2. 1) is that the converse is also true. In other words, among all bounded closed convex polyhedra the equilateral tetrahedron is characterized by the existence of an infinite non-self-intersecting geodesic line. In the final Section 3 a necessary condition for the existence of a closed geodesic line without double-points on a closed convex polyhedron is given, and some related problems are suggested. The local structure of a polyhedron may be isometrically described by (a) a circular disk in the Euclidean plane, if we consider a sufficiently small neighbourhood of a point that is not a vertex, or by (b) a cone, if we consider a sufficiently small neighbourhood of a vertex T. In this case we can assign a positive quantity y, called curvature, to the vertex T, taking the missing part of the full angle when we make a network of the cone. Euler’s polyhedral relation may be stated in the form