  We show that the two dimensional Ising model is complete, in the sense that
the partition function of any lattice model on any graph is equal to the
partition function of the 2D Ising model with complex coupling. The latter
model has all its spin-spin coupling equal to i\pi/4 and all the parameters of
the original model are contained in the local magnetic fields of the Ising
model. This result has already been derived by using techniques from quantum
information theory and by exploiting the universality of cluster states. Here
we do not use the quantum formalism and hence make the completeness result
accessible to a wide audience. Furthermore our method has the advantage of
being algorithmic in nature so that by following a set of simple graphical
transformations, one is able to transform any discrete lattice model to an
Ising model defined on a (polynomially) larger 2D lattice.
