  We study a new connection between multidimensional continued fractions, such
as Jacobi--Perron algorithm, and additively indecomposable integers in totally
real cubic number fields. First, we find the indecomposables of all signatures
in Ennola's family of cubic fields, and use them to determine the Pythagoras
numbers. Second, we compute a number of periodic JPA expansions, also in
Shanks' family of simplest cubic fields. Finally, we compare these expansions
with indecomposables to formulate our conclusions.
