Dihedral F-tilings of the sphere by equilateral and scalene triangles. I.
A polygonal tiling of the Euclidean sphere \(S^2\), where the edges of the polygon are geodesics, is called an \(f\)-tiling if all vertices are of even valency and the sums of alternate angles around each vertex is \(\pi\). The tiling is called dihedral if each prototile is congruent to one of two fixed figures. The authors show that a dihedral \(f\)-tiling of the sphere with an equilateral spherical triangle and a scalene spherical triangle such that the long side of the scalene triangles are adjacent to the equilateral triangles belongs to one of two families, which they explicitly describe. The proof follows a case by case analysis, considering the placement of the first few tiles.
scientific article; zbMATH DE number 5849492