  We present a simple two-dimensional model in which the lattice degrees of
freedom mediate the interactions between magnetic moments and electric dipoles.
This model reproduces basic features, such as a sudden electric polarization
switch-off when a magnetic field is applied and the ubiquitous dimerized
distortion patterns and magnetic $\uparrow \uparrow \downarrow \downarrow$
ordering, observed in several multiferroic materials of different composition.
The list includes E-type manganites, RMnO$_3$, nickelates such as in YNiO$_3$
and other materials under strain, such as TbMnO$_3$. In spite of its
simplicity, the model presented here captures the essence of the origin of
multiferroicity in a large class of type II multiferroics.
