Abstract:
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.
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