  A two dimensional system of discs upon which a triangle of spins are mounted
is shown to undergo a sequence of interesting phase transitions as the
temperature is lowered. We are mainly concerned with the `solid' phase in which
bond orientational order but not positional order is long ranged. As the
temperature is lowered in the `solid' phase, the first phase transition
involving the orientation or toroidal charge of the discs is into a `gauge
toroid' phase in which the product of a magnetic toroidal parameter and an
orientation variable (for the discs) orders but due to a local gauge symmetry
these variables themselves do not individually order. Finally, in the lowest
temperature phase the gauge symmetry is broken and toroidal order and
orientational order both develop. In the `gauge toroidal' phase time reversal
invariance is broken and in the lowest temperature phase inversion symmetry is
also broken. In none of these phases is there long range order in any Fourier
component of the average spin. A definition of the toroidal magnetic moment
$T_i$ of the $i$th plaquette is proposed such that the magnetostatic
interaction between plaquettes $i$ and $j$ is proportional to $T_iT_j$.
Symmetry considerations are used to construct the magnetoelectric free energy
and thereby to deduce which coefficients of the linear magnetoelectric tensor
are allowed to be nonzero. In none of the phases does symmetry permit a
spontaneous polarization.
