S. Allen Broughton
A tiling is a covering by polygons, without gaps or overlapping, of a compact, orientable surface. We are particularly interested in tilings by triangles that generate a large symmetry group of the surface. An oval of the tiling is a simple, closed curve that is a union of edges of the tiling. We investigate the number of points of intersection of two ovals. We have found that the number of intersections is bounded when the subgroup of orientation preserving symmetries is abelian. However, there is no upper bound on the number of intersections in the non-abelian case.
Schmidt, Dennis A., "Oval Intersections in Tilings on Surfaces" (1997). Mathematical Sciences Technical Reports (MSTR). 69.