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.
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