A rational positive-definite quadratic form is perfect if it can be
reconstructed from the knowledge of its minimal nonzero value m and the finite
set of integral vectors v such that f(v) = m. This concept was introduced by
Voronoi and later generalized by Koecher to arbitrary number fields. One knows
that up to a natural "change of variables'' equivalence, there are only
finitely many perfect forms, and given an initial perfect form one knows how to
explicitly compute all perfect forms up to equivalence. In this paper we
investigate perfect forms over totally real number fields.