  The number of triangulations of a planar n point set is known to be $c^n$,
where the base $c$ lies between $2.43$ and $30.$ The fastest known algorithm
for counting triangulations of a planar n point set runs in $O^*(2^n)$ time.
The fastest known arbitrarily close approximation algorithm for the base of the
number of triangulations of a planar n point set runs in time subexponential in
$n.$ We present the first quasi-polynomial approximation scheme for the base of
the number of triangulations of a planar point set.
