A class of tilings of \(S^ 2\) (Q1205442)

A polygonal subdivision \({\mathbf S}\) of the Euclidean sphere \(S^ 2\) is called monohedral iff all polygonal cells in \({\mathbf S}\) are isometric to one another. If, in addition, every vertex of \({\mathbf S}\) is of even valency and at each vertex the sum of the alternating angles is \(\pi\), then \({\mathbf S}\) is called a monohedral \(f\)-tiling. According to \textit{S. A. Robertson} [Polytopes and Symmetry (1984; Zbl 0548.52002)] the valency assumption implies that only triangles can serve as prototiles. In the paper under consideration all monohedral \(f\)-tilings of \(S^ 2\) are enumerated.