Counting regions of three families of circles on the sphere

On the unit sphere, consider the circles obtained by taking the same equally spaced polar angles about each of three mutually perpendicular axes. We derive exact counts of the vertices, edges and regions for every positive number of levels per axis. Using Gordan’s classification of rational-angle solutions, we count the distinct triple points on this grid and account for tangencies. The arrangement has 344 regions at eleven levels per axis, 626 at fourteen and 10,376 at fifty-nine.