  We present a theoretical study of director fields in toroidal geometries with
degenerate planar boundary conditions. We find spontaneous chirality: despite
the achiral nature of nematics the director configuration show a handedness if
the toroid is thick enough. In the chiral state the director field displays a
double twist, whereas in the achiral state there is only bend deformation. The
critical thickness increases as the difference between the twist and
saddle-splay moduli grows. A positive saddle-splay modulus prefers alignment
along the short circle of the bounding torus, and hence stimulates promotes a
chiral configuration. The chiral-achiral transition mimics the order-disorder
transition of the mean-field Ising model. The role of the magnetisation in the
Ising model is played by the degree of twist. The role of the temperature is
played by the aspect ratio of the torus. Remarkably, an external field does not
break the chiral symmetry explicitly, but shifts the transition. In the case of
toroidal cholesterics, we do find a preference for one chirality over the other
-- the molecular chirality acts as a field in the Ising analogy.
