Simple Closed Geodesics on the Tetrahedron
Every closed geodesic on the tetrahedron is simple, i.e. has no self intersections. This follows as on each face of the tetrahedron all the segments of a geodesic are parallel. In turn, this is implied as the regular tessellation of the plane by equilateral triangles consists only of translates of equilateral triangles and of 180-degree rotations of them. For example look at the following geodesic and its unfolding in the tesselation.This gives a pictorial proof why every closed geodesic is simple on the tetrahedron.
We next show six closed and simple closed geodesics. For a poster showing simple closed geodesics on regular polyhedra see here