  Despite recent observational and theoretical advances in mapping the magnetic
fields associated with molecular clouds, their three-dimensional (3D)
morphology remains unresolved. Multi-wavelength and multi-scale observations
will allow us to paint a comprehensive picture of the magnetic fields of these
star-forming regions. We reconstruct the 3D magnetic field morphology
associated with the Perseus molecular cloud and compare it with predictions of
cloud-formation models. These cloud-formation models predict a bending of
magnetic fields associated with filamentary molecular clouds. We compare the
orientation and direction of this field bending with our 3D magnetic field view
of the Perseus cloud. We use previous line-of-sight and plane-of-sky magnetic
field observations, as well as Galactic magnetic field models, to reconstruct
the complete 3D magnetic field vectors and morphology associated with the
Perseus cloud. We approximate the 3D magnetic field morphology of the cloud as
a concave arc that points in the decreasing longitude direction in the plane of
the sky (from our point of view). This field morphology preserves a memory of
the Galactic magnetic field. In order to compare this morphology to
cloud-formation model predictions, we assume that the cloud retains a memory of
its most recent interaction. Incorporating velocity observations, we find that
the line-of-sight magnetic field observations are consistent with predictions
of shock-cloud-interaction models. To our knowledge, this is the first time
that the 3D magnetic fields of a molecular cloud have been reconstructed. We
find the 3D magnetic field morphology of the Perseus cloud to be consistent
with the predictions of the shock-cloud-interaction model, which describes the
formation mechanism of filamentary molecular clouds.
