  We show that there is a constant $c$ such that any 3-uniform hypergraph
$\mathcal H$ with $n$ vertices and at least $cn^{5/2}$ edges contains a
triangulation of the real projective plane as a subgraph. This resolves a
conjecture of Kupavskii, Polyanskii, Tomon, and Zakharov. Furthermore, our
work, combined with prior results, asymptotically determines the Tur\'an number
of all surfaces.
