  We derive the leading asymptotic limit of the Wigner $3j$-symbol from a
stationary phase approximation of a twelve dimensional integral, obtained from
an inner product between two exact Bargmann wavefunctions. We show that, by the
construction of the Bargmann inner product, the stationary phase conditions
have a geometric description in terms of the Hopf fibration of ${\mathbb C}^6$
into ${\mathbb R}^3 \times {\mathbb R}^3 \times {\mathbb R}^3$. In addition, we
find that, except for the usual modification of the quantum numbers by 1/2, the
imaginary part of the logarithm of a Bargmann wavefunction, evaluated at the
stationary points, is equal to the asymptotic phase of the $3j$-symbol.
