  We obtain $D_k$ ALF gravitational instantons by a gluing construction which
captures, in a precise and explicit fashion, their interpretation as non-linear
superpositions of the moduli space of centred $SU(2)$ monopoles, equipped with
the Atiyah-Hitchin metric, and $k$ copies of the Taub-NUT manifold. The
construction proceeds from a finite set of points in euclidean space,
reflection symmetric about the origin, and depends on an adiabatic parameter
which is incorporated into the geometry as a fifth dimension. Using a
formulation in terms of hyperK\"ahler triples on manifolds with boundaries, we
show that the constituent Atiyah-Hitchin and Taub-NUT geometries arise as
boundary components of the 5-dimensional geometry as the adiabatic parameter is
taken to zero.
