The Non-Edge-to-Edge Tilings of the Sphere by Regular Polygons

In 1966, Zalgaller completed the task of determining all edge-to-edge tilings of the sphere by regular spherical polygons, proving that the only possibilities are the Platonic tilings, the Archimedean tilings, the Johnson tilings, the prism and anti-prism tilings and two tiles sharing their boundaries. We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides, showing there are five continuous families of kaleidoscope tilings, fifteen continuous families of 2-hemisphere tilings, four lunar tilings, five sporadic tilings, five composed tilings and one magic triangle tiling.