Triangular spherical dihedral f-tilings: the $(\pi/2, \pi/3, \pi/4)$ and $(2\pi/3, \pi/4, \pi/4)$ family

We classify all the dihedral f-tilings with spherical triangles π 2 , π 3 , π 4 and 2π 3 , π 4 , π 4 , and give the combinatorial structure, including the symmetry group of each tiling.