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Each elliptic curve can be embedded uniquely in the projective plane, up toprojective equivalence. The hessian curve of the embedding is generically a newelliptic curve, whose isomorphism type depends only on that of the initialelliptic curve. One gets like this a rational map from the moduli space ofelliptic curves to itself. We call it the hessian dynamical system. We computeit in terms of the $j$-invariant of elliptic curves. We deduce that, seen as amap from a projective line to itself, it has 3 critical values, whichcorrespond to the point at infinity of the moduli space and to the two ellipticcurves with special symmetries. Moreover, it sends the set of critical valuesinto itself, which shows that all its iterates have the same set of criticalvalues. One gets like this a sequence of dessins d'enfants. We describe analgorithm allowing to construct this sequence.

Iterating the hessian: a dynamical system on the moduli space of elliptic curves and dessins d'enfants