  The low temperature entropy of the the spin ice compounds, such as
Ho$_2$Ti$_2$O$_7$ and Dy$_2$Ti$_2$O$_7$, is well described by the
nearest-neighbor antiferromagnetic Ising model on the pyrochlore lattice, i.e.\
by the ``ice rules''. This is surprising since the dominant coupling between
the spins is their long ranged dipole interaction. We show that this phenomenon
can be understood rather elegantly: one can construct a model dipole
interaction, by adding terms of shorter range, which yields {\it precisely} the
same ground states, and hence T=0 entropy, as the nearest neighbor interaction.
A treatment of the small difference between the model and true dipole
interactions reproduces the numerical work by Gingras et al in detail. We are
also led to a more general concept of projective equivalence between
interactions.
