  Dirichlet's version of Gauss's reduction theory for indefinite binary
quadratic forms includes a map from Gauss-reduced forms to strings of natural
numbers. It attaches to a form the minimal period of the continued fraction of
a quadratic irrationality associated with the form. When Zagier developed his
own reduction theory, parallel to Dirichlet's, he omitted an analogue of this
map. We define a new map on Zagier-reduced forms that serves as this analogue.
We also define a map from the set of Gauss-reduced forms into the set of
Zagier-reduced forms that gives a near-embedding of the structure of Gauss's
reduction theory into that of Zagier's. From this perspective, Zagier-reduction
becomes a refinement of Gauss-reduction.
