Tilings of Nonoriented Surfaces by Steiner Triple Systems

A Steiner triple system of order $n$ (for short, $STS(n)$) is a system of three-element blocks (triples) of elements of an $n$-set such that each unordered pair of elements occurs in precisely one triple. Assign to each triple $(i,j,k)\in STS(n)$ a topological triangle with vertices $i$, $j$, and $k$. Gluing together like sides of the triangles that correspond to a pair of disjoint $STS(n)$ of a special form yields a black-and-white tiling of some closed surface. For each $n\equiv3\pmod6$ we prove that there exist nonisomorphic tilings of nonorientable surfaces by pairs of Steiner triple systems of order $n$. We also show that for half of the values $n\equiv1\pmod6$ there are nonisomorphic tilings of nonorientable closed surfaces.