Tilings of the sphere with right triangles III: the asymptotically obtuse families

Abstract

Sommerville and Davies 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.