  We construct a countable family of multi-dimensional continued fraction
algorithms, built out of five specific multidimensional continued fractions,
and find a wide class of cubic irrational real numbers a so that either (a,
a^2) or (a, a-a^2) is purely periodic with respect to an element in the family.
These cubic irrationals seem to be quite natural, as we show that, for every
cubic number field, there exists a pair (u,u') with u a unit in the cubic
number field (or possibly the quadratic extension of the cubic number field by
the square root of the discriminant) such that (u,u') has a periodic
multidimensional continued fraction expansion under one of the maps in the
family generated by the initial five maps. Thus these results are built on a
careful technical analysis of certain units in cubic number fields and our
family of multi-dimensional continued fractions. We then recast the linking of
cubic irrationals with periodicity to the linking of cubic irrationals with the
construction of a matrix with nonnegative integer entries for which at least
one row is eventually periodic.
