  In this letter, we have constructed and experimentally investigated
frustrated arrays of dipoles forming two-dimensional artificial spin ices with
different lattice parameters (rectangular arrays with horizontal and vertical
lattice spacings denoted by $a$ and $b$ respectively). Arrays with three
different ratios $\gamma =a/b = \sqrt{2}$, $\sqrt{3}$ and $\sqrt{4}$ are
studied. Theoretical calculations of low-energy demagnetized configurations for
these same parameters are also presented. Experimental data for demagnetized
samples confirm most of the theoretical results. However, the highest energy
topology (doubly-charged monopoles) does not emerge in our theoretical model,
while they are seen in experiments for large enough $\gamma$. Our results also
insinuate that magnetic monopoles may be almost free in rectangular lattices
with a critical ratio $\gamma = \gamma_{c} = \sqrt{3}$, supporting previous
theoretical predictions.
