  We present a model of classical Heisenberg spins on a Hollandite lattice,
which has been developed to describe the magnetic properties of
$\alpha$-MnO$_2$ and similar compounds. The model has nearest neighbor
interacting spins, however the strength and the sign of spin-spin interactions
is anisotropic and depends on the nature of the bonds. Our analysis shows that
the Hollandite lattice supports four different incommensurate and helical
magnetic ground states depending on the relative strengths and signs of
spin-spin interactions. We show that the incommensurate helical ground states
appear due to the geometrical frustration present in the model. We demonstrate
that each of the four helical incommensurate magnetic phases are continuously
connected to four different collinear antiferromagnetic ground states as the
strength of spin-spin interaction along some bonds is increased. The present
results give support to the presence of helical states that have been
previously suggested experimentally for Hollandite compounds. We provide an
in-depth analysis of the magnetic form factors for each helical phase and
describe how it could be used to identify each of these phases in neutron
diffraction experiments.
