  This paper investigates the quadratic irrationals that arise as periodic
points of the Gauss type shift associated to the odd continued fraction
expansion. It is shown that these numbers, which we call O-reduced, when
ordered by the length of the associated closed primitive geodesic on some
modular surface $\Gamma \backslash \mathbb{H}$, are equidistributed with
respect to the Lebesgue absolutely continuous invariant probability measure of
the Odd Gauss shift.
