Complete classification of non-side-to-side spherical triangle tilings

Complete the classification of all tilings of the sphere by congruent triangles that are not side-to-side, including the unresolved case in which all triangle angles are rational in degree.

Background

The paper studies non-side-to-side tilings of the sphere by congruent spherical triangles and proves a complete classification when the triangle has at least one angle that is irrational in degree. The classified irrational-angle tilings consist of two-layer earth map families and their rotational modifications, a one-parameter family with eight tiles, and one sporadic sixteen-tile example.

The authors explicitly state that the general classification was not known at the time of writing. Their stated program divides the task between the present paper, which treats triangles with an irrational angle, and a sequel devoted to the remaining rational-angle case. Thus, the unresolved problem is to classify all non-side-to-side congruent-triangle tilings, particularly those whose three angles are rational in degree.