  In this paper, the Hermite problem has been approached finding a periodic
representation (by means of periodic rational or integer sequences) for any
cubic irrationality. In other words, the problem of writing cubic irrationals
as a periodic sequence of rational or integer numbers has been solved. In
particular, a periodic multidimensional continued fraction (with pre--period of
length 2 and period of length 3) is proved convergent to a given cubic
irrationality, by using the algebraic properties of cubic irrationalities and
linear recurrent sequences. This multidimensional continued fraction is derived
from a modification of the Jacobi algorithm, which is proved periodic if and
only if the inputs are cubic irrationals. Moreover, this representation
provides simultaneous rational approximations for cubic irrationals.
