Dihedral f-tilings of the sphere by rhombi and triangles

An isometric folding is a non-expansive locally isometry that sends piecewise geodesic segments into piecewise geodesic segments of the same length. An isometric folding is a continuous map that need not to be differentiable. The points where it is not differentiable are called singular points. The foundations of isometric foldings of Riemannian manifolds are introduced by Robertson (1977). For surfaces, the set of singular points gives rise to two colored tilings (called f-tilings) with the property that each vertex has even valency and obey the angle folding relation, i.e., the sums of alternating angles around each vertex is π. The classification of f-tilings started out by Breda (1992), where a complete classification of all monohedral f-tilings of the sphere by triangles was done. The complete classification of monohedral tilings of the sphere by triangles (which, obviously, includes the monohedral triangular f-tilings) was made clear by Ueno and Agaoka (2002). This classification was partially done by Sommerville (1922), and an outline of the proof was provided by Davies (1967). For additional information on tilings, see Grünbaum and Shephard (1986). Our interest focuses in dihedral spherical f-tilings.