Summary: The spectral analysis of Aharonov-Bohm Hamiltonians with flux \(\frac 12\) leads surprisingly to a new insight on some questions of isospectrality and of minimal partitions. We illustrate this point of view by discussing the question of spectral minimal 3-partitions for the rectangle \(\big]-\frac a2,\frac a2\big[\times \big]-\frac b2,\frac b2\big[\), with \(0 < a \leq b\). It has been observed in B. Helffer, T. Hoffmann-Ostenhof and S. Terracini [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 1, 101–138 (2009; Zbl 1171.35083)] that when \(0<\frac ab < \sqrt{\frac 38}\) the minimal 3-partition is obtained by the three nodal domains of the third eigenfunction corresponding to the three rectangles \(\big]{-}\frac a2,\frac a2\big[\times \big] {-}\frac b2,-\frac b6\big[\), \(\big]{-}\frac a2,\frac a2\big[\times \big]{-}\frac b6,\frac b6\big[\) and \(\big]{-}\frac a2,\frac a2\big[\times \big] \frac b6, \frac b2\big[\). We describe a possible mechanism of transition for increasing \(\frac ab\) between these nodal minimal 3-partitions and non-nodal minimal 3-partitions at the value \(\sqrt{\frac 38}\) and discuss the existence of symmetric candidates for giving minimal 3-partitions when \(\sqrt{\frac 38} <\frac ab \leq 1\). Numerical analysis leads very naturally to nice questions of isospectrality which are solved by the introduction of Aharonov-Bohm Hamiltonians or by going on the double covering of the punctured rectangle.