Regular simple geodesic loops on a tetrahedron (Q1189641)

A simple geodesic loop on a tetrahedron which does not pass through any vertex of the tetrahedron is called regular. Such a loop must meet each face on the tetrahedron. It is called minimal if each face is met exactly once. Considering developments of the faces of the tetrahedron in the plane the authors obtain necessary and sufficient conditions for the existence of numerical loops on a tetrahedron. It is shown that for the existence of three distinct minimal loops through any point on the face of a tetrahedron it is necessary and sufficient that the tetrahedron be isosceles. Furthermore, on such a tetrahedron all other regular simple geodesic loops are classified using an associated tiling of the plane. Finally conditions are discussed under which similar loops exist on a general tetrahedron.