  The nature of compact groups (CGs) of galaxies, apparently so dense that the
galaxies often overlap, is still a subject of debate: Are CGs roughly as dense
in 3D as they appear in projection? Or are they caused by chance alignments of
galaxies along the line-of-sight, within larger virialized groups or even
longer filamentary structures? The nature of CGs is re-appraised using the z=0
outputs of three galaxy formation models, applied to the dissipationless
Millennium Simulation. The same selection criteria are applied to mock galaxy
catalogs from these models as have been applied by Hickson and co-workers in
redshift space. We find 20 times as many mock CGs as the `HCGs' found by
Hickson within a distance corresponding to 9000 km/s. This very low (5%) HCG
completeness is caused by Hickson missing groups that were either faint, near
the surface brightness threshold, of small angular size, or with a dominant
brightest galaxy. We find that most velocity-filtered CGs are physically dense,
regardless of the precise threshold used in 3D group size and line-of-sight
elongation, and of the galaxy formation model used. This result also holds for
mock CGs with the same selection biases as was found for the HCGs.
