The theory of f-tilings is related to the theory of isometric foldings, initiated by S. Robertson [8] in 1977.
The study of dihedral f-tilings of the Euclidean sphere
S2 by triangles and r-sided regular polygons was initiated in
2004, where the case r=4wasconsidered[4].Inasubsequentpaper[1],thestudyofallsphericalf−tilingsbytrianglesandr−sidedregularpolygons,foranyrge 5,wasdescribed.Recently,in[2]and[3]aclassificationofalltriangulardihedralsphericalf−tilingsforwhichoneoftheprototilesisanequilateraltriangleisgiven.Inthispaper,weextendtheseresultsconsideringthedihedralcaseoftwononcongruentisoscelestrianglesinaparticularwayofadjacencyendinguptoaclassoff−tilingscomposedbythreeparametrisedfamilies,denotedbymathcal{F}_{k, alpha},mathcal{E}_{alpha}andmathcal{L}_{k}, ; kgeq 3, ; alpha>frac{pi}{2},respectively,andoneisolatedtiling,denotedbymathcal{G}$.
The combinatorial structure including the symmetry group of each
tiling is also given.
Dawson and Doyle in [6], [7] have also been working on spherical tilings, relaxing the edge to edge condition