Tilings of the sphere with right triangles. III: The asymptotically obtuse families (Q1010602)

Summary: \textit{D.M.Y. Sommerville} [Edinb. R. S. Proc. 43, 85--116 (1923; JFM 49.0411.03)] and \textit{H.\,L. Davies} [Proc. Colloq. Convexity, Copenhagen 1965, 42--51 (1967; Zbl 0149.17804)] classified the spherical triangles that can tile the sphere in an edge-to-edge fashion. However, if the edge-to-edge restriction is relaxed, there are other such triangles; here, we continue the classification of right triangles with this property begun in our earlier papers. We consider six families of triangles classified as ``asymptotically obtuse'', and show that they contain two non-edge-to-edge tiles, one (with angles of \(90^{\circ}, 105^{\circ}\) and \(45^{\circ})\) believed to be previously unknown. [For parts I and II see ibid. 13, No.\,1, R48 (2006; Zbl 1096.05015), R49 (2006; Zbl 1096.05016).]