  In this note we announce some results that will appear in [6] (joint work
with also Matteo Novaga) on the minimization of the functional
$F(\Gamma)=\int_\Gamma k^2+1\,\mathrm{d}s$, where $\Gamma$ is a network of
three curves with fixed equal angles at the two junctions. The informal
description of the results is accompanied by a partial review of the theory of
elasticae and a diffuse discussion about the onset of interesting variants of
the original problem passing from curves to networks. The considered energy
functional $F$ is given by the elastic energy and a term that penalize the
total length of the network. We will show that penalizing the length is
tantamount to fix it. The paper is concluded with the explicit computation of
the penalized elastic energy of the 'Figure Eight', namely the unique closed
elastica with self--intersections.
