All maps of type are covered by a universal map which lies on one of the three simply connected Riemann surfaces; in fact covers all maps of type where and . In this paper we construct a tessellation which is universal for all maps on all surfaces. We also consider the tessellation which covers all triangular maps. This coincides with the well-known Farey tessellation and we find many connections between and .
1980 Mathematics Subject Classification (1985 revision): 05C10, 20H05, 57M20, 10D07.