  We consider the problem of constructing an abstract $(n+1)$-polytope $Q$ with
$k$ facets isomorphic to a given $n$-polytope $P$, where $k \geq 3$. In
particular, we consider the case where we want $Q$ to be $(n-2,n)$-flat,
meaning that every $(n-2)$-face is incident to every $n$-face (facet). We show
that if $P$ admits such a flat extension for a given $k$, then the facet graph
of $P$ is $(k-1)$-colorable. Conversely, we show that if the facet graph is
$(k-1)$-colorable and $k-1$ is prime, then $P$ admits a flat extension for that
$k$. We also show that if $P$ is facet-bipartite, then for every even $k$,
there is a flat extension $P|k$ such that every automorphism of $P$ extends to
an automorphism of $P|k$. Finally, if $P$ is a facet-bipartite $n$-polytope and
$Q$ is a vertex-bipartite $m$-polytope, we describe a flat amalgamation of $P$
and $Q$, an $(m+n-1)$-polytope that is $(n-2,n)$-flat, with $n$-faces
isomorphic to $P$ and co-$(n-2)$-faces isomorphic to $Q$.
