Dihedral f-Tilings of the Sphere Induced by the M\"obius Triangle (2,3,4)

We classify the special families of dihedral folding tilings of the sphere derived from the M\"obius triangle $(2,3,4)$. Our study emerges from the study of isometric foldings in the Riemann sphere and meets at the juncture of the triangle group $\Delta(2,3,4)$. The juxtaposition enables us to apply the classification theorem of edge-to-edge tilings of the sphere by congruent triangles and introduces a group theoretical method. The two prototiles of each family consist of the M\"obius triangle and another polygon induced by a reflection of the triangle group acting on the M\"obius triangle. To enumerate the tilings, we give two solutions to solve the associated constraint satisfaction problem. The methods are not exclusive to this problem and therefore applicable to similar problems of more general settings.