On the bi-embeddability of certain Steiner triple systems of order 15 (Q1864565)

From among the 80 nonisomorphic Steiner triple systems of order 15, systems \(\#1\) and \(\#2\) in standard notation are the designs of points and lines of \(\text{PG}(3,2)\), and the system obtained from it by switching any of its Pasch configurations, respectively. It is shown that there exists no bi-embedding of STS \(\#1\) and \(\#2\) into an orientable surface, and that there are precisely five nonisomorphic bi-embeddings into a nonorientable surface.