  In 1968, Ringel and Youngs confirmed the last open case of the Heawood
Conjecture by determining the genus of every complete graph $K_n$. In this
paper, we investigate the minimum genus embeddings of the complete $3$-uniform
hypergraphs $K_n^3$. Embeddings of a hypergraph $H$ are defined as the
embeddings of its associated Levi graph $L_H$ with vertex set $V(H)\sqcup
E(H)$, in which $v\in V(H)$ and $e\in E(H)$ are adjacent if and only if $v$ and
$e$ are incident in $H$. We determine both the orientable and the
non-orientable genus of $K_n^3$ when $n$ is even. Moreover, it is shown that
the number of non-isomorphic minimum genus embeddings of $K_n^3$ is at least
$2^{\frac{1}{4}n^2\log n(1-o(1))}$. The construction in the proof may be of
independent interest as a design-type problem.

Combinatorics